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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova Science

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova Science

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PARTICLE SWARM OPTIMIZATION: THEORY, TECHNIQUES AND APPLICATIONS

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Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

ENGINEERING TOOLS, TECHNIQUES AND TABLES

PARTICLE SWARM OPTIMIZATION: THEORY, TECHNIQUES AND APPLICATIONS

ANDREA E. OLSSON Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Particle swarm optimization : theory, techniques, and applications / editor, Andrea E. Olsson. p. cm. Includes bibliographical references and index. ISBN H%RRN 1. Swarm intelligence. 2. Mathematical optimization. I. Olsson, Andrea E. Q337.3.P37 2010 006.3--dc22 2010012247

Published by Nova Science Publishers, Inc. † New York

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

CONTENTS

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Preface

vii

Chapter 1

Using Mono-objective and Multi-objective Particle Swarm Optimization for the Tuning of Process Control Laws Guillaume Sandou and Saïd Ighobriouen

Chapter 2

Study on Vehicle Routing Problem with Time Windows Based on Enhanced Particle Swarm Optimization Approach Chun-Ta Lin

31

Chapter 3

Reliability Optimization Problems Using Adaptive Genetic Algorithm and Improved Particle Swarm Optimization YoungSu Yun

49

Chapter 4

Convergence Issues in Particle Swarm Optimization David Bradford Jr. and Chih-Cheng Hung

67

Chapter 5

Globally Convergent Modifications of Particle Swarm Optimization for Unconstrained Optimization Emilio F. Campana, Giovanni Fasano and Daniele Peri

97

Chapter 6

Nonlinear 0-1 Programming through Particle Swarm Optimization Using Decoding Algorithms Takeshi Matsui and Masatoshi Sakawa

119

Chapter 7

Comparative Study of Different Approaches to Particle Swarm Optimization in Theory and Practice Stefanie Thiem and Jörg Lässig

127

Chapter 8

Particle Swarm Optimization for Computer Graphics and Geometric Modeling: Recent Trends Akemi Gálvez and Andrés Iglesias

169

Chapter 9

Particle Swarm Optimization Used for Mechanism Design and Guidance of Swarm Mobile Robots Peter Eberhard and Qirong Tang

193

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

1

vi

Contents

Chapter 10

A New Neighborhood Topology for the Particle Swarm Optimization Algorithm Angel Eduardo Muñoz-Zavala, Arturo Hernández-Aguirre and Enrique Raùl Villa-Diharce

227

Chapter 11

PSO Assisted Multiuser Detection for DS-CDMA Communication Systems Taufik Abrão, Leonardo Dagui de Oliveira, Bruno Augusto Angélico and Paul Jean Etienne Jeszensky

249

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Index

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

299

PREFACE Particle swarm optimization (PSO) is an algorithm modeled on swarm intelligence that finds a solution to an optimization problem in a search space or model and predicts social behavior in the presence of objectives. The PSO is a stochastic, population-based computer algorithm modeled on swarm intelligence. Swarm intelligence is based on socialpsychological principles and provides insights into social behavior, as well as contributing to engineering applications. This book presents information on particle swarm optimization such as using mono-objective and multi-objective particle swarm optimization for the tuning of process control laws; convergence issues in particle swarm optimization and vehicle routing problems using enhanced particle swarm optimization. The design of efficient control laws is a crucial point in the industry community, as a fine tuning of systems can strongly increase their global productivity. The computation of automatic control laws is often based on the solution of optimization problems. Costs and constraints of these problems are often non convex, non smooth or non analytic ones. The classical approach is to simplify the problem so as to get a tractable and exactly solvable optimization problem. In Chapter 1, a Particle Swarm Optimization (PSO) method is used to solve complex initial problems. Examples of Proportional Integral Derivative (PID) tuning, reduced order robust control synthesis and non linear model predictive control are given. The design of the control laws is firstly depicted in a mono-objective framework, showing much than satisfactory results. However, the tuning of control law is intrinsically a multi-objective problem where the designer has to find the best compromise between several challenges (mainly low energy consumption against best rapidity and performance). An extension of PSO for multi-objective optimization is thus presented and used. Numerical results are shown for real life systems. Very satisfactory results are obtained and show that metaheuristic optimization algorithms, and especially Particle Swarm Optimization, could be of great interest for the design of Automatic Control laws. Further, the developed methods have no parameters to be tuned, that is considered as a quite interesting advantage in the industry community. Particle Swarm Optimization (PSO) technique proved its ability to deal with very complicated optimization and search problems. Several variants of the PSO algorithm have been proposed. Chapter 2 proposes a variant of the basic PSO technique named Predicting Particle Swarm Optimization (Predicting PSO) algorithm dealing with combinatorial optimization to solve the well known Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is the extension of the capacitated vehicle routing problem which is

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.

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Andrea E. Olsson

aimed to the definition of the minimum cost paths where the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. The original PSO is basically developed for continuous optimization problem. However, VRPTW problems are formulated as combinational optimization problem. The authors first propose the Predicting PSO algorithm based on original PSO which modified by three new solution strategies. The main contributions of this proposed algorithm include: (1) to provide a totally different concept to transform the search space from multi-dimensions to one-dimension and (2) successfully transfer the combinatorial problem to continuous problem through boundary constraints handling. (3) Fully using the memory from population. Predicting PSO algorithm not only can modify the variance between current status and memorial best status, but also can forward predict what the best status should be at next time status based on memory. The Predicting PSO algorithm is designed to fit this combinational optimization problem type to find near-optimal solutions in the VRPTW problems. The Predicting PSO algorithm has also been tested on Solomon’s benchmark problems. The findings indicate that the proposed algorithm outperforms other heuristic algorithms. In chapter 3, several reliability optimization problems are considered to be optimized. For the optimization, a hybrid approach using adaptive genetic algorithm (aGA) and improved particle swarm optimization (iPSO) is proposed. For the aGA, an adaptive scheme is incorporated into genetic algorithm (GA) loop and it adaptively regulates crossover and mutation rates during genetic search process. For the iPSO, a conventional PSO is improved and it is applied to the hybrid approach. Therefore, the proposed hybrid approach takes an advantage of compensatory property of the aGA and the iPSO. For proving the performance of the proposed hybrid approach, conventional hybrid algorithms using GA and PSO are presented and their performances are compared with that of the proposed hybrid algorithm using several reliability optimization problems which have been often used in many conventional studies. Finally, efficiency of the proposed hybrid approach is proved Convergence. Optimal. Exploration. These three words, or variations thereof, weave their way into many an algorithms definition of functional success; without ‘exploration’ of the problem space there can be no ‘convergence’ upon some measured ‘optimal’ solution to the problem. Particle Swarm Optimization is not exempt. Chapter 4 will present convergence issues that PSO must face due to its very structure as a population‐based swarm whose greatest desire is to move quickly and in tandem toward a favorable solution point. It will also present methods many PSO designers have used so far to address these issues and, as well, present a few solutions that other algorithms in the optimization family have tried and are trying today. Such topics as ‘how much exploration is too much’ (controlling population, velocity, and exploration) and ‘how does one determine when convergence is necessary’ (redefining a swarms search space) will be looked at, as will ‘what is a sub‐optimal solution’ (the local optima trap and the multiple‐optimal landscape) and ‘can local optima be overcome efficiently’ (methods for resuming exploration after an optimal success). These questions and their various answers, as implemented in the various PSO designs and hybridizations mentioned, will help illuminate the nature of not only PSO algorithms but, through inference, many evolution/ population based algorithms that explore through a problem space for one or more optimal solutions. The authors’ conclusion will present an analysis of those included PSO solutions (as applied to the convergence issues experienced) and attempt to classify their level of success in the problem domain being explored.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Preface

ix

The authors focus on the solution of a class of unconstrained optimization problems, where the evaluation of the objective function is possibly costly and the use of exact algorithms may require a too large computational burden. Several real applications, included in the latter class, claim for optimization methods where the derivatives of the objective function are unavailable and/or the objective function must be treated as a ‘black-box’. Many design optimization and shape optimization problems belong to the latter class; moreover, the derivatives computed with finite differences may be much inaccurate. Here, expensive simulations provide information to the optimizer, so that each function evaluation could require up to several CPU-hours. On the other hand, for continuously differentiable functions the use of heuristics may yield inadequate and/or unsatisfactory results. The authors consider in Chapter 5 the evolutionary Particle Swarm Optimization (PSO) algorithm. The authors introduce some globally convergent modifications of PSO by drawing their inspiration from, so that sequences of points are generated which admit stationary limit points for the objective function. The latter result is carried out for a generalized PSO scheme, where suitable ranges of the parameters are identified in order to possibly avoid diverging trajectories for the particles. In general, actual various decision making situations are formulated as large scale mathematical programming problems with many decision variables and constraints. In mathematical programming problems, for the programming problems that decision variables are 0 or 1, the authors can theoretically get strict solution by application of dynamic programming. In particular, for nonlinear 0-1 programming problems, there are not general strict solution method or approximate solution method, such as branch and bound method in case of linear 0-1 programming problems. In such a case, a solution method depended on property in problems is proposed. Thus, Sakawa proposed genetic algorithms with double string representation based on updating basepoint solution using continuous relaxed as general approximate solution method for nonlinear 0-1 programming problems. In recent years, a particle swarm optimization (PSO) method was proposed by Kennedy et al. and has attracted considerable attention as one of promising optimization methods with higher speed and higher accuracy tha those of existing solution methods. Hence, Kato et al. proposed revised particle swarm optimization (rPSO) method solving drawbacks that it is not directly applicable to constrained problems and its liable to stopping around local solutions and showed its effectivity. Moreover, the authors expanded revised particle swarm optimization method for application to nonlinear integer programming problems (rPSONLIP) and showed more efficiency than genetic algorithm. And the authors expanded rPSO for application to nonlinear 0-1 programming problems (rPSONLZOP [5], rPSOmNLZOP [6]) and showed more efficiency than genetic algorithm and QUADZOP. However, in those researches, rPSONLZOP and rPSOmNLZOP method needs many computational time as same as genetic algorithm. In Chapter 6, the authors focus on nonlinear 0-1 programming problems and propose higher approximation solution method based on particle swarm optimization using the decoding algorithm than those methods. A research area, which in particular has gained increasing attention in recent years, investigates the application of biological concepts to various optimization tasks in science and technology. Technically, global extrema of objective functions in a ddimensional discrete or continuous space have to be determined or approximated, which is a standard problem with an ample number of applications. In Chapter 7 the particle swarm optimization paradigm is described in different variations of the underlying equations of motion and studied

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Andrea E. Olsson

comparatively in theory and empirically on the basis of selected optimization problems. To facilitate this task, the different variants are described in a general scheme for optimization algorithms. Then different side constraints as initial conditions, boundary conditions of the search space and velocity restrictions are investigated in detail. This is followed by an efficiency comparison of swarm optimization to a selection of other global optimization heuristics, such as various single state iterative search methods (simulated annealing, threshold accepting, great deluge algorithm, basin hopping, etc.), ensemble based algorithms (ensemble-based simulated annealing and threshold accepting), and evolutionary approaches. In specific, the application of these algorithms to combinatorial problems and standard benchmark functions in high-dimensional search spaces is examined. Further, the authors show how particle swarm optimization can be effectively used for the optimization of so-called singlewarehouse multi-retailer systems to optimize the ordering strategy and the transportation resources. Optimization tasks of this class are very common in economy as e.g. in the manufacturing industry and for package delivery services. Particle Swarm Optimization (PSO) is an evolutionary computation scheme originally described in 1995 by James Kennedy and Russell Eberhart. This technique has received increasing attention from the scientific community during the last few years because of its ability to find a solution to an optimization problem in a search space, and model and predict social behavior in the presence of objectives. Amazingly enough, PSO has barely been applied so far in the fields of computer graphics and geometric modeling, with only a few references found in the literature regarding these topics. Chapter 8 is aimed at filling this gap in two different ways: on one hand, by providing the interested reader with a gentle and updated overview on some applications of the PSO methodology to relevant problems in the fields of computer graphics and geometric modeling; on the other hand, by identifying new lines of research in which PSO has the potential to become a fundamental tool in solving relevant problems. From the authors’ discussion it becomes clear that the original PSO technique along with its modifications have a great potential as a powerful computational tool for many problems in those fields. Chapter 9 presents particle swarm optimization (PSO) based algorithms. After an overview of PSO’s development and application history also two application examples are given in the following. PSO’s robustness and its simple applicability without the need for cumbersome derivative calculations make it an attractive optimization method. Such features also allow this algorithm to be adjusted for engineering optimization tasks which often contain problem immanent equality and inequality constraints. Constrained engineering problems are usually treated by sometimes inadequate penalty functions when using stochastic algorithms. In this work, an algorithm is presented which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrangian particle swarm optimization (ALPSO). It is used for the stiffness optimization of an industrial machine tool with parallel kinematics. Based on ALPSO, we can go a further step. Utilizing the ALPSO algorithm together with strategies of special velocity limits, virtual detectors and others, the algorithm is improved to augmented Lagrangian particle swarm optimization with special velocity limits (VLALPSO). Then the work uses this algorithm to solve problems of motion planning for swarm mobile robots. All the strategies together with basic PSO are corresponding to real situations of swarm mobile robots in coordinated movements. The authors build a swarm motion model based on Euler forward time integration that involves some mechanical properties such as

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Preface

xi

masses, inertias or external forces to the swarm robotic system. The results show that the stiffness of the machine can be optimized by ALPSO and simulation results show that the swarm robots moving in the environment mimic the real robots quite well. In the simulation, each robot has the ability to master target searching, obstacle avoidance, random wonder, acceleration/deceleration and escape entrapment. So, from this two application examples one can claim, that after some engineering adaptation, PSO based algorithms can suit well for engineering problems. Chapter 10 introduces a new neighborhood structure for the Particle Swarm Optimization (PSO) algorithm, called Singly-Linked Ring. In the PSO algorithm, a neighborhood enables different communication paths among its members, and therefore, the way the swarm searches the landscape. Since the neighborhood topology changes the flying pattern of the swarm, convergence and diversity differ from topology to topology. The approach proposes a neighborhood whose members share the information at a different rate. Its main property is a single link between two particles. The objective is to avoid the premature convergence of the flock and stagnation into local optimal. The approach is compared against two neighborhoods which are the state-of-theart: ring structure and von Neumann structure. These structures possess a mutual attraction between two particles. A set of controlled experiments is developed to observe the transmission behavior (convergency) of every structure. Besides, a well-known set of global optimization problems is used to compare the 3 structures with the same PSO parameters. A statistical test was performed for every experiment to compare the mean values of the 3 structures. The authors’ approach is easy to implement, and its results and its convergence performance are better than the other 2 structures. In Chapter 11, a heuristic perspective for the multiuser detection problem in the uplink of direct sequence code division multiple access (DS-CDMA) systems is discussed. In particular, the particle swarm optimization multiuser detector (PSO-MUD) is analyzed regarding several figures of merit, such as symbol error rate, near-far and channel error estimation robustness, and computational complexity aspects. The PSO-MUD is extensively evaluated and characterized under different channel scenarios: additive white Gaussian noise (AWGN), single input single/multiple output (SISO/SIMO) flat Rayleigh, and frequency selective (multipath) Rayleigh channels. Although literature presents single-objective (SOO) and multi-objective optimization (MOO) approaches to deal with multiuser detection problem, in this chapter the single-objective optimization criterion is extensively used, since its application requirement is simpler than the MOO, and its performance results for the proposed optimization problem are quite satisfactory. Nevertheless, the MOO is shortly addressed as an alternative approach. Furthermore, the complexity × performance trade-off of the PSO-MUD is carefully analyzed via Monte-Carlo simulation (MCS), and the complexity reduction concerning the optimum multiuser detector (MUD) is quantified. Simulation results show that, after convergence, the performance reached by the PSO-MUD is much better than the conventional detector (CD), and somewhat close to the single user bound (SuB), having computational complexity substantially lower than OMUD.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 1-30

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Chapter 1

USING MONO-OBJECTIVE AND MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION FOR THE TUNING OF PROCESS CONTROL LAWS Guillaume Sandoua and Saïd Ighobriouenb SUPELEC Systems Sciences (E3S), Automatic Control Department, Gif-sur-Yvette, France

Abstract

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The design of efficient control laws is a crucial point in the industry community, as a fine tuning of systems can strongly increase their global productivity. The computation of automatic control laws is often based on the solution of optimization problems. Costs and constraints of these problems are often non convex, non smooth or non analytic ones. The classical approach is to simplify the problem so as to get a tractable and exactly solvable optimization problem. In this chapter, a Particle Swarm Optimization (PSO) method is used to solve complex initial problems. Examples of Proportional Integral Derivative (PID) tuning, reduced order robust control synthesis and non linear model predictive control are given. The design of the control laws is firstly depicted in a mono-objective framework, showing much than satisfactory results. However, the tuning of control law is intrinsically a multi-objective problem where the designer has to find the best compromise between several challenges (mainly low energy consumption against best rapidity and performance). An extension of PSO for multi-objective optimization is thus presented and used. Numerical results are shown for real life systems. Very satisfactory results are obtained and show that metaheuristic optimization algorithms, and especially Particle Swarm Optimization, could be of great interest for the design of Automatic Control laws. Further, the developed methods have no parameters to be tuned, that is considered as a quite interesting advantage in the industry community.

a b

E-mail address: [email protected]. E-mail address: [email protected].

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

2

Guillaume Sandou and Saïd Ighobriouen

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I. Introduction The design of efficient control laws is a crucial point in the industry community, as a fine tuning of systems can strongly increase their productivity. Optimization is traditionally the core of efficient methods to compute controllers and more generally control laws. Usually, a simplified model of the plant to control is established (using for instance linearized plants or neglecting high frequency dynamics), and an optimization problem formulating the desired behavior and performance is defined. A special attention has to be paid to the structure of the model and the corresponding optimization problem so as to be able to solve it with exact and deterministic solvers. In the 70s, the Linear Quadratic method, belonging to the class of optimal control methods [12] has appeared. The method uses a linear model of the system, and the problem is formulated as a quadratic cost optimization. For that particular problem, the optimal solution can be analytically found with the help of Riccati equations. More recently, efficient H2/H∞ synthesis methods have appeared [23]. The approach is also based on a linear model of the plant, and the problem is expressed as the minimization of the H∞ norm of the closed loop system. Reformulations are used so as to express the problem in an LMI (Linear Matrix Inequalities) framework for which dedicated solvers can be used. Another trend is the use of the Youla-Kucera parameterization [6]. An “optimal” controller can be found from this parameterization by solving a convex optimization problem. Finally, recent developments on predictive control have allowed the synthesis of control laws in the temporal domain [14]. Using once again linear prediction models and quadratic costs, and knowing in advance the desired output, one can compute an optimal discrete controller, weighting the reference tracking and the energy consumption. The method is based on the solution of Diophantine equations. Finally, for all these classical approaches, a linear model of the plant is used, and linear and quadratic optimization problems are addressed to express the specification to be captured. However, due to the necessity of a particular structure of the optimization model, some of the specifications cannot be directly taken into account. Specifications are thus firstly relaxed in the synthesis procedure and have to be a posteriori checked in an analysis phase. This approach may lead to some iteration between the synthesis and the analysis phases which can be highly time consuming. Nowadays, three main trends have to be considered: • •

•

Systems to be controlled are more and more complex, and it is not always possible to define linear (or at least simplified) models to capture all the system. Specifications are more and more various and precise, and it appears to be crucial to take them into account in the synthesis phase to decrease the global time spent in the controller design. Industries do not only want to find a controller which satisfy the desired specifications, but also a controller which optimize them.

Finally, the corresponding optimization problems are non convex, non differentiable, with numerous local optima. Some attempts have been performed with sub-gradients methods

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

3

[13], but the proposed method remains a local search method and so is strongly dependent on the initial point. In such a context, metaheuristic optimization methods appear to be interesting candidate methods to solve these kinds of problems. In this chapter, the main focus is on the use of Particle Swarm Optimization method. The interest of the method is the fact that it can be used as a zero parameter method, thus allowing a wide spread in the industry community. Some attempts have already been presented in [22]. A more generic framework is presented in this chapter, capturing various well known problems of the Automatic Control field. Particle Swarm Optimization is briefly presented in section II, together with the choices of implementation. Several examples of the possible use of the method are then presented in a mono-objective framework in section III: tuning of PID controller (section III.2), reduced order H∞ synthesis (section III.3), and non linear model predictive control in section (section III.4). The method is extended in the multi-objective framework in section IV as the synthesis of control law is intrinsically a multi-objective optimization problem where several contradictory objectives have to be simultaneously dealt with. Finally, conclusion remarks are drawn in section V.

II. Choice for the Use of Particle Swarm Optimization Particle swarm Optimization (PSO) was firstly introduced by Eberhart and Kennedy [4]. The method is inspired by the social behavior of bird flocking or fish schooling. Consider the following optimization problem: min

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x∈χ

f ( x)

(1)

P particles are moving in the search space. Each of them has its own velocity, and is able to remember where it has found its best performance. Each particle has some “friends”. The

following notations are used: •

x kp (resp v kp ): position (resp. velocity) of particle p at iteration k;

•

b kp = arg min( f (b kp−1), f ( x kp )) : best position found by particle p until iteration k (“local guide”)

•

V ( x kp ) ⊂ {1,2, …, P} set of “friend particles” of particle p at iteration k;

•

g kp =

arg min

x∈{bik ,i∈V ( x kp )}

f ( x) best position found by the friend particles of particle p

until iteration k (“global guide) The particles move in the search space according to the following transition rule:

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Guillaume Sandou and Saïd Ighobriouen

v kp+1 = wv kp + c1 ⊗ (b kp − x kp ) + c2 ⊗ ( g kp − x kp ) x kp+1 = x kp + v kp+1

(2)

In this equation: • • •

⊗ is the element wise multiplication of vectors. w is the inertia factor ; c1 (resp. c2 ) is a random number in the range [0, c1 ] (resp. [0, c2 ] ).

Finally, the geometric representation of the transition rule (2) is depicted in figure 1. x kp+1 b kp

v kp+1

g kp

v kp

x kp

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Set of friend particles

Figure 1. Geometric representation of the Particle Swarm Optimizer.

The choice of parameters is very important to ensure the satisfying convergence of the algorithm. Lots of work have been done on the topic; see[20], [21]. However, it is not in the scope of this study to look for fine strategies of tuning. Indeed, the method has to be used by Automatic Control people and people from the industry community who are not aware of metaheuristics details. Thus, it is crucial to have “zero parameter” methods and thus standard values, which are given in [11] will be used: •

Swarm size P : 10 +

•

w = 1 /(2 ln(2)) ;

•

c1 = c 2 = 0,5 + ln(2) ;

•

dim(V ( x kp )) ≤ 3 .

n , where n is the number of optimization variables;

V (x k )

p . For a comprehensive study of this Several topologies exist for the design of sets topic, see [10] for instance. In particular, if these sets do not depend on k, neighborhoods are said to be “social”. This choice is the simplest for the implementation of the algorithm and so

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

5

will be used in the sequel for the mono-objective optimization. Note that the use of the algorithm does not require any tuning of parameter, which is of great interest in the industry community.

III. Mono-objective PSO for the Design of Control Laws III.1. Definition of Optimization Problems Consider the generic closed loop framework of figure 2. s is the Laplace variable.

r + -

Controller

ε

K (θ , s)

u+ +

d v

System

y

G(s )

Figure 2. Classical closed loop framework.

The aim is to compute a controller K , depending of some parameters θ , for the system G . r is the reference input, ε the tracking error, u the control input computed by the controller, d a disturbance, v the control actually applied to the system, and y the output. In this closed loop, any transfer function from an input x to an output z is a function of the controller parameters θ and thus can be written:

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Tx→ z ( s ) = H ( s,θ )

(3)

Similarly, any time response x(t ) with respect of a particular reference r (t ) and disturbance d (t ) is a function of the controller parameter:

x(t ) = f (θ , r (τ ), d (τ ),τ ∈ [0,+∞[ )

(4)

Specifications can be expressed in the frequency domain. Classical criterions in the case of Single Input Single Output (SISO) are as follows. •

Cut-off frequency:

ω 0 (θ ) = arg min ω1 ω1

s.t Tε → y ( jω , θ ) < 1, ∀ω > ω1

•

(5)

Phase margin: Δφ (θ ) = arg(Tε → y ( jω , θ )) − ( −180°) avec Tε → y ( jω , θ ) = 1

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(6)

6

Guillaume Sandou and Saïd Ighobriouen •

Gain margin:

ΔG (θ ) = −20 log Tε → y ( jω , θ )

with arg(Tε → y ( jω , θ )) = −180° •

(7)

Module margin:

Δm(θ ) = min Tε → y ( jω , θ ) − (−1) ω

(8)

• H ∞ norm of the system:

Tr → y

∞

(θ ) = sup σ ( jω , θ ) ω

with

σ ( jω , θ ) = max λi (Tr → y ( jω , θ ) ∗ Tr → y ( jω ,θ ))

(9)

i

and λi ( M ) i

th

eigen value of M .

More generally, specifications can be given as temporal templates for transfer function of the system of figure 2:

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Gmin (ω ) ≤ Tx → z ( jω ) = H ( jω , θ ) ≤ Gmax (ω )

φ min (ω ) ≤ arg(Tx → z ( jω ) = H ( jω , θ )) ≤ φ max (ω )

(10)

For instance, the figure 3 represents the classical open loop template (transfer Tε → y (s ) ) of a Simple Input Simple Output system, with a cut-off frequency higher that ω for the rapidity of the system, a phase margin higher than Δϕ for its stability, and high low frequency gains (for precision) and low high frequency gains (for noise filtering and rejection). Specifications can also be given in the time domain. Some classical specifications are given for the Heaviside step response. Once again, all criterions are functions of θ . •

Maximum overshoot D% : y (t , θ ) − lim y (t1 , θ ) D%(θ ) = max t

t1 → +∞

lim y (t1 , θ ) t1 → +∞

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(11)

Using Mono-objective and Multi-objective Particle Swarm Optimization…

7

G dB

ω

ω

φ

Δφ

-180°

ω

Figure 3. Classical open loop frequency template of a SISO system.

•

α % time response: Te (θ ) = inf {T \ ∀t > T : ε (t , θ ) ≤ α / 100 ⋅ r (t ) } T >0

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

•

Maximum of the control input: u max (θ ) = max u (t , θ ) t

•

(13 )

Control input energy: Eu (θ ) =

•

(12)

tf

∫t = t u

T

(t , θ )u (t , θ )dt

T

(t , θ )ε (t , θ )dt

(14)

0

Quadratic error for the regulation: Eε (θ ) =

tf

∫t = t ε

(15)

0

Specifications can also be given for disturbance rejection. For instance for an Heaviside step of d (t ) : • •

Steady state error; Maximum of the control input;

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

8

Guillaume Sandou and Saïd Ighobriouen •

Time of rejection.

More generally, specifications can be given as temporal templates for the behaviour of the system with respect to a particular reference r or disturbance d :

x(t ) ≤ x(t ) = f (θ , r (τ ), d (τ ),τ ∈ [0,+∞[ ) ≤ x (t )

(16)

For example, the figure 4 depicts a classical template for the step response, with an overshoot lower than D , a time response lower than t r , and a steady state error lower than

ε .

D

±ε

1

t

tr

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 4. Classical open loop temporal template of a SISO system.

Finally, in this section, some classical optimization criterions of the Automatic Control field have been presented. Note that most of them are non linear or even non analytic with numerous local minima. The aim is now to solve some particular optimization problems with the PSO algorithm.

III.2. Tuning of PID Controllers The PSO algorithm will be tested to compute some “optimal” controllers for a magnetic levitation. The system is represented in figure 5, together with the notations for the modeling. The modeling of the systems is as follows. The electromagnetic force is expressed by:

Fm = c

i2 x2

, x = ( z E − l ) − z = x0 − z

(17)

The Principle of Dynamics leads to:

mz = c

i2 ( x0 − z ) 2

− mg

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(18)

Using Mono-objective and Multi-objective Particle Swarm Optimization…

9

E l e c t r ic m a g ne t

i

Z

Vz

Electric magnet P e n d u lu m

zE

P o s i ti o n

x l

+10 z

vz

0 Position sensor

-10V

Figure 5. Magnetic levitation system.

The equilibrium point about z = 0 is then: i 2 c 0 − mg = 0 x0 2

(19)

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

The derivation of a linearized plant about this equilibrium point, considering z = 0 + z1 ; i = i0 + i1 and z1 0

if max( u (t ) ) > 10 : J1 (θ ) = exp(λ max( u (t ) ) − 10) t

t

else J1 (θ ) = 0 if min G ( jω ) − (−1) < 0.5 :

(24)

ω

J 2 (θ ) = α ⋅ (min G ( jω ) − (−1) − 0.5) 2 ω

else J 2 (θ ) = 0.

The constraint of the module margin is not given by a physical limitation, so it is possible to use smoother penalization functions than the penalization function on the control value. Statistical optimization results are given in table I. Computation times are about 10 seconds with Matlab 2007b on a Pentium IV, 2.0GHz.

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11

Levitation V zc +

ε -

Controller

U ( s) = C (s) ε( s)

u

Actuator

I ( s) K I = U ( s ) 1 + τs

i

Z1 ( s ) G =− I1 ( s) s2 1− ω02

z1

V z Sensor β Figure 6. Magnetic levitation system.

Table 1. Results for the time response minimization with penalization on the control input and module margin. Best 30.8 10-3s

Worst 39.7 10-3s

Mean 31.4 10-3s

Standard deviation 1.1 10-3s

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Note that the PSO has to be statistically validated as it is a stochastic one. Thus, the results given in table I have been obtained by solving the same optimization problem (24) one hundred times. For comparison and validation of the quality of computed solutions, a Nelder Meald simplex optimization has been performed for all initial positions of particles. Due to the complexity of the problem, no better solution than those computed by PSO has been found by the deterministic algorithm. See [18] for a more detailed presentation of the results.

III.3. Reduced Order H∞ Control III.3.1. H∞ Control H∞ synthesis is an efficient tool in Automatic control to compute controllers in a closed loop framework, achieving high and various performances (reference tracking, low energy controls, disturbance rejection…). Two principal solution methods have been developed for this purpose, one based on Linear Matrix Inequalities problems [7], [9], and one based on Riccati equation solutions [8], [23]. The main drawback of such approaches is the controller order: H∞ synthesis provides a controller whose order is the same as the synthesis model. This drawback is worsened by the fact that fictive filters are added to the system model to express control specifications, leading to a higher order for the synthesis model and so for the controller. A classical way to get low order controllers is to perform a full-order synthesis and then to reduce the obtained controller, for example with a Hankel decomposition method. However, this approach may lead to a high H∞ norm of the closed loop system and a high sensitivity to high frequency noises.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

12

Guillaume Sandou and Saïd Ighobriouen

To avoid high order controllers, the H∞ optimization problem can be solved, adding some order constraints on the solution. However, this kind of constraints is expressed with rank constraints. Thus, the reduced-order synthesis problem appears to be a non-convex; nonsmooth optimization problem, and classical algorithms may fail in the solution. Several approaches exist to solve the problem: • • •

Convex approximations of non-convex stability regions. These approximations often obtained from conservative sufficient conditions; Global nonlinear optimization techniques such as an exhaustive search, which very expensive; Local optimization techniques such as BMI solvers. The solution depends on initial controller and there is no guarantee on the actual global optimality of solution.

are are the the

In this chapter, a new approach is proposed, using particle swarm optimization to solve the reduced-order H∞ synthesis problem.

III.3.2. Full Order H∞ Synthesis Consider once again the classical closed loop structure of a MIMO system called back in figure 7. G (s ) is the system to control, K (s ) is the controller.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

r + -

d

u

ε

+ v

K (s )

G (s )

y

+

Figure 7. Classical closed loop system.

The closed loop transfer matrix of the system is: ⎡ε ( s )⎤ ⎡ S ⎢u ( s )⎥ = ⎢ KS ⎦ ⎣ ⎣ with

− SG ⎤ ⎡ r ( s ) ⎤ ⎡ r (s) ⎤ = T (s)⎢ ⎥ ⎥ ⎢ ⎥ − KSG ⎦ ⎣d ( s )⎦ ⎣d ( s ) ⎦

S ( s ) = ( I + G ( s ) K ( s ))

(25)

−1

The H∞ synthesis problem is defined as follows. Find a stabilizing controller K (s ) such that:

γ = min T ( s) ∞ K (s)

(26)

where the H∞ norm has been defined in equation (9). This problem can be reformulated into a convex problem and solved with Riccati equations or LMI formulations. This solution is Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

13

called “full order” synthesis, as the solution of problem (26) is a controller whose order is equal to the order of G (s ) . To get some tuning parameters, some design (fictive) filters Wi (s ) are added to the synthesis model, as in figure 8.

e1

e2

W1 ( s ) r + -

ε

d

W2 ( s ) K (s )

u

+ v

W3 ( s ) G (s )

e3 y

+

Figure 8. Synthesis model.

The new system is now given by: ⎡ e1 ( s) ⎤ ⎡ W1S ⎢e ( s)⎥ = ⎢W KS ⎣ 2 ⎦ ⎣ 2

− W1SGW3 ⎤ ⎡ r ( s) ⎤ − W2 KSGW3 ⎥⎦ ⎢⎣d ( s )⎥⎦

(27)

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Indeed, solving the H∞ problem for this system induces frequency dependent constraints for each transfer of matrix (25). For example one obtains for a SISO system:

⎛ W1S ⎜⎜ ⎝W2 KS

− W1SGW3 ⎞ ⎟ − W2 KSGW3 ⎟⎠

γ ⎧ ⎪ S ( jω ) ≤ W ( jω ) 1 ⎪ ⎪ γ ⎪ KS ( jω ) ≤ W 2 ( jω ) ⎪ ≤ γ ⇒ ∀ω ⎨ γ ⎪ SG ( jω ) ≤ ∞ ⎪ W1 ( jω ) W3 ( jω ) ⎪ γ ⎪ KSG ( jω ) ≤ ⎪ W2 ( jω ) W3 ( jω ) ⎩

(28)

Defining suitable filters Wi leads therefore to the possibility of shaping all transfers of the loop. Note however that these filters increase the order of the synthesis model and thus of the solution controller.

III.3.3. Reduced Order H∞ Synthesis The reduced-order H∞ problem refers to the solution of the following optimization problem:

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

14

Guillaume Sandou and Saïd Ighobriouen

γ = min T ( s) ∞ K (s)

(29)

s.t. ∂°K ( s ) = nr

where ∂°K denotes the order of K (s ) , and n r is strictly less than the order of the synthesis model. This problem can be reformulated into LMI equations by adding rank constraints on matrices. Thus, the property of convexity is lost, and the optimization problem can hardly be solved. Some heuristics have been developed to solve the problem. One possibility is to add some penalty functions to deal with the rank constraints [3]. This kind of methods allows using classical convex algorithms to solve non convex problems, but are often time consuming. Another possibility is based on the cone complementary algorithm [5], where the problem is reformulated into a solvable problem. In this chapter, a new approach is proposed: the problem will be solved with a stochastic optimization method, Particle Swarm Optimization

III.3.4. Case Study The algorithm has been tested for a pendulum in the cart which is defined in figure 9. Motor Moteur

i

Réducteur Reduction

ω

Chariot Cart Génératrice tachymétrique

xc

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

ϕ

Potentiomètre

Potentiomètre

Charge Pendulum

Figure 9. Pendulum in the cart.

The system can be modeled by:

L

d i (t ) + R i (t ) + K e ω (t ) = u (t ) dt

(30)

J

d ω (t ) + f ω (t ) + d (t ) = K e i (t ) dt

(31)

d xc (t ) r = ω (t ) dt N cos (φ (t ))

d 2 xc (t ) d t2

+l

d 2φ (t ) d t2

+α

d φ (t ) + g sin (φ (t )) = 0 dt

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(32)

(33)

Using Mono-objective and Multi-objective Particle Swarm Optimization… Where 

i (t ) , u (t ) : current, voltage in the motor;



 (t ) : rotation speed of the motor;



xc (t ) : position of the cart;



 (t ) : angle of the pendulum;



d (t ) : disturbance moment;



L  0.0002H , R  2.3 : inductor (resistance) of the motor;



K e  0.0162 Nm/A : electromagnetic constant;



J  5.10 6 kg.m 2 : inertia of the motor;



f  6.10 5 N.m.s : friction coefficient;

r  0.022 m : radius of the pulley;

  

N  17 : gear reduction; l  0.275 m : length of the pendulum;



  0.3 m.s 1 : friction coefficient on the pendulum;



g  9.81 m.s 2 : weight acceleration.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

xc (t ) and  (t ) are measured. Specifications are as follows:

  

tracking of the position reference of figure 10; no steady state error; rejection of disturbance d (t ) ;



 (t )  0.05rad ;



time response less than 6s.

Position reference f

0,4 m

time

0 0

4s

Figure 10. Position reference.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

15

16

Guillaume Sandou and Saïd Ighobriouen

III.3.5. One Output H∞ Synthesis In a first time, a one output case is considered: the feedback law is computed only from xc (t ) . The model for H∞ problem is represented in figure 11.

e1 W1 ( s )

r + ε -

e2 W2 ( s ) K (s )

W3 ( s )

e3

G (s )

φ xc

d

u

Figure 11. Synthesis model for the one output case.

Filters can be defined so as to get a satisfying behaviour of the system with respect to the input r and the disturbance d . Classical methods are used to choose these filters from the specifications, with W1 ( jω ) and W2 ( jω ) being large at low and high frequencies respectively to penalize the tracking error and the control effort, and W3 ( jω ) being constant to handle the disturbance. Finally, define the filters as:

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

W1 ( s ) =

1 s + 1.7 s+2 , W2 ( s ) = 100,⋅ ⋅ 2 s + 0.0009 s + 2000

(34)

The full-order synthesis is computed from these values, and leads to a minimal H∞ norm γ = 0.97 . K (s ) has order 4, as G (s ) has order 2 from control input to x c , W1 ( s) and

W2 ( s) are order 1 and W3 ( s ) is order 0. A Hankel reduction of the controller can be performed, leading to order 2, and a H∞ norm γ = 2.30 . Reduced order H∞ synthesis. The reduced optimization problem can be stated as: ⎛ W1S ⎜⎜ K 0 ,τ 1 ,τ 2 ,τ 3 ,τ 4 ⎝W2 KS min

− W1SGW3 ⎞ ⎟ − W2 KSGW3 ⎟⎠

∞

⎧S ( s ) = (1 + G ( s ) K ( s ))−1 ⎪ s.t ⎨ (1 + τ1s )(1 + τ 2 s ) ⎪K (s) = K0 ( 1 + τ 3s )(1 + τ 4 s ) ⎩

(35)

As already explained in section III.2, PSO is a stochastic algorithm, and the validation can only be made by statistical results. Results are given in table II for 100 tests and 100 iterations of the algorithm. Computation times are 30s on a Pentium IV, 2GHz, and Matlab

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

17

6.5. The initial population is always chosen at random so as to explore the whole search space. Table II. Optimization Results for the One Output Case. Worst ∞

Best

= 2.01

∞

Mean

= 0.97

∞

= 1.42

The optimal value of the H∞ problem is obtained for the full order synthesis. Thus, the optimal value of the reduced order problem is higher or equal than 0.97 as it is a more constrained problem. As can be seen from the results, the algorithm finds solutions very close to the (a priori unknown) optimal solution of the reduced order problem.

III.3.6. Three Output H∞ Synthesis Of course, as the measure of the pendulum angle is not taken into account, oscillations on φ (t ) are large when using the previous controller. To enhance the behavior of the closed loop, a three output synthesis is now computed. The new synthesis model is depicted in figure12. e1 W1 ( s ) r +

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

-

e2 W2 ( s ) u

ε

W3 ( s )

e3

d

xc

G (s ) φ

K (s ) +

+

W4 ( s )

e4

W5 ( s )

e5

W6 ( s )

e6

Figure 12. Synthesis model for the “3 output” case.

e4 is a new output, e5 , e6 two disturbances. Filters W1 ( s ), W2 ( s ), W3 ( s ) have the expression as in section III.3.5. Filters W4 ( s ), W5 ( s ), W6 ( s ) are added to satisfy the classical hypotheses of [3] while adding tuning parameters for the control law; roughly speaking, small magnitude for W4 ( jω ) and large magnitudes for W5 ( jω ) , W6 ( jω ) lead to the previous solution. Increasing the first one allows controlling the oscillations on φ (t ) while decreasing the second and third ones gives more efficiency to the control law. The following values are chosen:

W4 ( s) = 2 , W5 ( s) = 1 , W6 ( s ) = 0.1 Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(36)

18

Guillaume Sandou and Saïd Ighobriouen The solution of the full order synthesis leads to a H∞ norm γ = 1.06 . The controller order

is 6, as G (s ) has order 4 for the full model, W1 ( s) and W2 ( s) are order 1 and W3 ( s ), W4 ( s ), W5 ( s ), W6 ( s ), are order 0. The Hankel reduction leads to a very large H∞ norm

γ = 56 .7 for the 2 order controller (4 states of the controller are removed). Reduced order H∞ synthesis. In this case, K (s ) has 3 inputs and 1 output. Thus, a two order controller can be written:

⎡ (1 + τ 11s )(1 + τ 12 s ) ⎤ ⎢ K1 ⎥ (1 + τ 1s )(1 + τ 2 s ) ⎥ ⎢ ⎢ (1 + τ 21s )(1 + τ 22 s ) ⎥ K (s) = ⎢ K 2 ⎥ (1 + τ 1s )(1 + τ 2 s ) ⎥ ⎢ ⎢ (1 + τ 31s )(1 + τ 32 s ) ⎥ ⎢K 3 ⎥ (1 + τ 1s )(1 + τ 2 s ) ⎥⎦ ⎢⎣

T

(37)

A controller is computed by the PSO algorithm, with the filters of the full order synthesis. Statistical results are given in table III for 100 tests. Table III. Optimization results for the three output case. Worst

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

∞

Best

= 4.53

∞

Mean

= 2.60

∞

= 3.50

Results appear to be much than satisfactory, with a low H∞ norm. Note however that it is impossible to say where is the optimal between 1.06 (the best value for the full order synthesis) and 2.60 (the best computed value).

10 10

vr --> eps and gamma/W 1

2

10

0

10 10

10

-4

5

-5

10

0

10

5

10

d --> eps and gamma/W1/W3 10

-5

10

0

10

5

d --> u and gamma/W 1/W 2

5

0

0

10

10

-5

10

10 10

0

-2

10 10

vr --> u and gamma/W2

5

-5

10

-5

10

0

10

5

10

-5

Full order bound Hankel reduction

-10

10

-5

10

0

10

5

Figure 13. Bode transfer of full order and classical Hankel reduction. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

19

Cart Position 0.4 0.2 0 Reference Full order Hankel reduction

-0.2 -0.4

0

5

10

15

20

25

20

25

Pendulum angle 0.04 0.02 0 -0.02 -0.04

0

5

10

15

Figure 14. Time response of full order and classical Hankel reduction.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 13 gives Bode transfers for the closed loop system, for the full order and the reduced controller (the later obtained by Hankel decomposition and reduction) in the case of the “3 output” controller. Simultaneously, figure 14 represents time responses of the system for the input of figure 10. Results of the Hankel reduction are quite similar as those of the full order controller. Figure 15 and 16 give the same results obtained with the mean controller of the PSO method.

10

10

10

10

-5 -5

10

0

10

5

10

d --> eps

5

10

-5

10

0

10

5

d --> u

5

0

0

10

10

-5

10

10 10

vr --> u

0

0

10 10

vr --> eps

5

-5

10

-5

10

0

10

5

10

-5

-10

10

-5

10

0

10

5

Figure 15. Bode transfer for PSO controller.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

20

Guillaume Sandou and Saïd Ighobriouen Cart Position 0.4 Reference PSO reduction

0.2 0 -0.2 -0.4 0

5

10

15

20

25

20

25

Pendulum angle 0.04 0.02 0 -0.02 -0.04

0

5

10

15

Figure 16. Time response for PSO controller.

Note first that the response of ϕ (t ) is quite similar as previous ones. A slight overshoot is observed on the reference tracking: controllers computed by PSO seem less satisfying than classical H∞ controllers. However, consider figure 17, where a measurement noise d m is added. d r + Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

-

u

ε

K (s )

xc

G (s ) φ

+

+ dm

Figure 17. System model with a measurement noise.

There is almost no influence on the system outputs x c and φ , as G (s ) is a low-pass filter. However, the control input u computed by the controller K (s ) is represented in figure 18 both for Hankel reduction and PSO controllers. As can be seen from figure 13, Hankel reduction leads to a modification of the closed loop transfers for high frequencies, especially for the transfer function from the reference to the control input. As a result, high gains for high frequencies lead to an amplification of measurement noises and thus to chattering control inputs. Of course, there is no difference between full order and Hankel reduction controllers for smooth reference inputs as in figure 14. The increase of the H∞ norm for Hankel reduction is mainly due to a non robust behavior

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Using Mono-objective and Multi-objective Particle Swarm Optimization…

21

of the system against measurement noises. The behavior of PSO controllers is less satisfying for nominal inputs (slight overshoot on figure 16), but the system is more robust against measurement disturbances. u for Hankel reduction controller 4 2 0 -2 -4 0

5

10

15

20

25

20

25

u for PSO controller 4 2 0 -2 -4 0

5

10

15

Figure 18. Control inputs for measurement noises.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

III.4. Non Linear Predictive Control The system that has to be controlled in this section is a district heating network which is represented in figure 19. The goal is to operate producers, pumps and valves so as to satisfy consumers’ demands at lower costs. The problem is discretized in N periods and can be stated as: K N

min

∑∑ c kprod (Qnk )

⎫⎪ ⎧⎪ Qnk ,d nv ⎬ k =1 n =1 ⎨ ⎪⎩n∈[1:N ],k∈[1:K ],v∈[1:V ]⎪⎭

(38)

where K is the number of producers, V is the number of valves, Qnk is the power produced by producer k and d nv is the opening degree of valve v during period n .

c kprod (Qnk ) = a 2k (Qnk ) 2 + a1k Qnk + a0k

(39)

is the production cost of production site k . Constraints to be satisfied are the fulfillment of consumers’ demands and bounds on mass flows, pressures and temperatures over the whole distribution network.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

22

Guillaume Sandou and Saïd Ighobriouen

Cons 1 _1 Pump 1

Pro d 1

1

3

C ons 1 _3

Pump 2

C ons 1 _3 6

7 8

ac

Con s 2_2

Pu mp 3

Pro d 2 C ons 2 _1 9

Con s 2_3

Figure 19. District heating network.

For the computation of these constraints, one has to simulate the district heating network. This simulation is concerned with the solution of non linear algebraic and implicit systems of equations for the computation of mass flows and pressures, and with the solution of non linear systems of partial derivative equations for the computation of the energy propagation in the network. For instance, the energy propagation in pipes can be modeled by:

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

m p (t ) ∂T 2μ p ∂T (T ( x, t ) − T0 ) = 0 ( x, t ) + ( x, t ) + ∂t c p ρR p πρ R 2p ∂x

(40)

Where T ( x , t ) is the temperature in the pipe, m p is the mass flow through the pipe, ρ is the relative density of water, R p is the radius of the pipe, μ p its thermal loss coefficient, and T0 the external temperature. The full description of the model is given in [17].

Figure 20. Receding horizon strategy.

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The solution of problem (38) is an open loop control of the district heating network. However, this solution is computed from predictions of consumers’ demands. Thus it is necessary to use a closed loop control for the system. Following the principles of predictive control [10], a receding horizon strategy has been used to extend the results of optimization in a closed loop framework. This idea is represented in figure 20. In order to get a tractable algorithm for the on-line solution of the scheduling algorithm (21), a PSO algorithm is chosen. To prove the interest of the approach, the following case study is considered: •

•

Producer 1 is a cogeneration site. Such a site can produce both electric and thermal power. The produced electricity can be sold. In France, the price of the sold electricity is approximately 40 €/MWh from the 1st of November to the 31st of March and null from the 1st of April to the 31st of October. For the cogeneration site, the higher the price, the lower the production cost of thermal power. Producer 2 is a heat-only unit.

Simulation of the receding horizon control law is tested and lead to the results of table IV. Computation times are about 120 seconds (solution of the on line optimization problem) with Matlab 2007b on a Pentium IV, 2.0GHz Table IV. Results of the receding horizon strategy with a Particle Swarm Optimization method

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Price of Electricity 0 €/MWh 40 €/MWh

Total production prod. 1 Over 24 hours Over 48 hours 535 MWh 947 MWh 541 MWh 963 MWh

Total production prod. 2 Over 24 hours Over 48 hours 537 MWh 1016 MWh 492 MWh 950 MWh

Results can be easily explained: when the price of sold electricity is high, the interest is to produce as much as possible with the cogeneration site, and to use the interconnection of the valves. Although obvious, these results can only be achieved by considering the whole network and the whole technological string “production – distribution – consumption”. The corresponding optimization problem being complex and hard to solve in a real time framework, it has to be solved with a stochastic algorithm such as PSO. The use of stochastic algorithms in a close loop framework arises some theoretical questions. As the global optimality of the solution is not guaranteed, the proof of the stability of the closed loop fails in the general case. However, this is not the case for district heating networks, as local regulations of supply temperatures lead to the guarantee of bounded temperatures in the network, whatever the sequences of control inputs. The considered system can be said to be “unconditionally stable”.

IV. Multi-objective PSO for the Design of Controllers Multi-objective optimization is concerned with the simultaneous optimization of several objective functions:

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Guillaume Sandou and Saïd Ighobriouen

min{ f1 ( x),…, f k ( x)}

(41)

x

This optimization is performed using the concept of Pareto optimality. A solution x dominates a solution y if:

x ≺ y ⇔ ∀i, f i ( x) ≤ f i ( y) et ∃ j t.q. f j ( x) < f j ( y)

(42)

A solution is said to be Pareto optimal if it is not dominated by any other solution. The PSO algorithm used in previous sections can be modified so as to solve this kind of problem. The main difficulty is to define some local and global guides b kp , g kp in the sense of multi-objective optimization. Some implementations can be found in literature, see for instance [16], [19], [2]. In this chapter, the method developed in [1] will be used. This method has the advantage of using only the concept of Pareto domination and does not require any computation of distance in the objective functions space. Using this work, the local guide can be computed by: If ( x kp+1 ≺ b kp ) or ( x kp+1 ≺ b kp et x kp+1 ≺ b kp ) then b kp+1 = x kp+1

(43)

else b kp+1 = b kp where the symbol ≺

means that it is not possible to compare the two solutions using the

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concept of Pareto domination. The definition of global guides g kp can be done with the help of an archive matrix denoted A . In this matrix all non dominated solution, known at iteration k of the algorithm, are stored. At iteration k + 1 , all particles x kp+1 which are not dominated by any stored solutions enter in the archive matrix. Conversely, all solutions stored in the matrix but that are dominated by a particle x kp+1 go out from this archive matrix. From the definition of this evoluting archive matrix, the following sets can be defined:

Axk = {a ∈ A so that a ≺ x}

{

X ak = x kp so that a ≺ x kp

}

(44)

Thus, Axk represents for a given potential solution x the set of stored solution which dominates it. Conversely, X ak is, for a stored element of the archive matrix a , the set of particles which dominates it. From these definitions, the global guide for the particle x kp can be computed by:

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⎧

⎪ g kp = ⎨

a ∈ A with probability α / X ak if x kp ∈ A

k k k k ⎪a ∈ A x , x = x p , with probability α / X a if x p ∉ A ⎩

25

(45)

Where X ak denotes the cardinal of X ak and α is chosen so that the sum of probabilities is equal to 1. Equation (44) can be explains as follows. It is clear that any element of the archive matrix which dominates the particle is a potentially interesting guide for the particle. Further, choosing a probability proportional to X ak

−1

gives more chance to the elements

which dominate few particles. The goal is to get a Pareto domain as large as possible. In this section we consider the levitation system depicted in section III.2 (see figure 5). The goal is still to compute a PID controller given by equation (22). As already explained, the synthesis of controllers is intrinsically a multi-objective problem (namely achieve best performance with low control inputs). In this study we choose two objectives, energy of control input and quadratic error for a step response. Then, the objectives are given by: min

{ f1 (θ ), f 2 (θ )}

θ = ( K ,Ti ,Td ,T f )

(46)

0,5 2 0,5 2 avec : f1 (θ ) = ε (t )dt , f 2 (θ ) = u (t )dt t =0 t =0

∫

∫

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a)

Control energy c)

b)

Quadratic error Figure 21. Set of Pareto optimal solutions found by the PSO algorithm.

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Guillaume Sandou and Saïd Ighobriouen

Both criterions have to be computed using a simulator and have numerous local minima. Finally, the use of a stochastic algorithm and in particular of multi-objective PSO seems to be well suited. The dedicated simulator takes into account a saturation of the control input to ±5V . The set of Pareto optimal solutions found by the multi-objective PSO algorithm is given in figure 21. Computation times are about 1 minute with Matlab 2007b and a Pentium 2.5 GHz. The figure 22 represents the step response (output y and control input u ) for the point a) of the set of Pareto optimal solutions (see figure 21). This point is the solution for which the quadratic error is the lowest. As a result, the overshoot remains very low, and the step response exhibits a small time response. Conversely, the control input takes some high values (it goes to saturation ± 5V ) , especially at the beginning of the time response.

Step response

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Time (s)

Control input

Time (s) Figure 22. Step response for the point a) of the Pareto set.

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The same kind of result is presented for the point b) of the Pareto set in figure 23. For this point, the criterion 2 (control energy) is the lowest. Finally, it leads to low control inputs and also to a very oscillating step response. Finally, figure 24, corresponding to the solution of the point c) of the Pareto set, represents a compromise between the two objectives.

Step response

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Time (s)

Control input

Time (s) Figure 23. Step response for the point b) of the Pareto set.

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Step response

Time (s)

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Control input

Time (s) Figure 24. Step response for the point c) of the Pareto set.

V. Conclusions In this chapter the use of Particle Swarm Optimization for the design of controllers has been presented. Indeed, costs and constraints which have to be optimized for the synthesis of controllers are very often non linear, non analytic and with numerous local minima. Thus, the use of stochastic algorithms is of great interest for the Automatic Control field. In this chapter, the classical mono-objective algorithm has been used for the design of traditional Proportional-Integral-Derivative controller, for the design of reduced order H∞ controllers and for the definition of predictive control laws. For all these examples, the algorithm has proven its viability, exhibiting much than satisfactory results. In a second stage, we have presented the extension of the algorithm to the multi-objective case, still for the design of PID controllers. Indeed, the computation of controller is

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intrinsically a multi-objective problem as one has to capture different objectives in the regulation (namely high quality of regulation and low control energy). Once again, the algorithm has proven its viability. To conclude, we can add two important remarks concerning the use of PSO for Automatic Control. The first one deals with the lack of guarantee of the actual optimality of the solution found by the PSO. In the case of the design of controllers, it is not so important, as it is mainly a problem of feasibility. The user finds with this method a controller which satisfies all the constraints and which leads to better performance than those of a controller tuned “by hand”. The second remark deals with the use of the method in the industry community. The developed algorithms have no parameter to be tuned as we only use some standard values for the PSO parameters. Finally, this is of great interest for the industry community as the method has to be used by non specialist people. Some collaboration has already been initiated in that sense; see [10] for instance.

References

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[1]

Alvarez-Benitez, J. E., Everson, R. M. and Fieldsend, J. E. “A MOPSO Algorithm Based Exclusively on Pareto Dominance Concepts”, in: Lecture Notes in Computer Science, vol 3410, pp 459-473, 2005. [2] Cagnina, L., Esquivel, S. and Coello Coello, C. A. “A Particle Swarm Optimizer for Multi-Objective optimization”, in: Journal of Computer Science & Technology, vol. 5, no. 4, 2005. [3] David, J. and De Moor, B. “Designing reduced order output feedback controllers using a potential reduction method”, in: American Control Conference, Baltimore, Maryland, pp. 845-849, 1994. [4] Eberhart, R. C. and Kennedy, J. “A new optimizer using particle swarm theory”, in Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, Japan. pp. 39-43, 1995. [5] El Ghaoui, L., Oustry, F. and AitvRami, M. “A cone complementary linearization algorithm for static output feedback and related problems”, in: IEEE Transactions on Automatic Control, vol. 42(8), pp. 1171-1176, 1997. [6] Francis, B. and Doyle, J. C. “Linear control theory with an H∞ optimality criterion”, in: SIAM Journal on Control and Optimization, vol. 25, pp. 815-844, 1987. [7] Gahinet, P., and Apkarian, P. “A linear matrix inequality approach to H∞ control”, in: International Journal of Robust and Nonlinear Control, vol. 4, pp. 421-448, 1994. [8] Glover, K., and Doyle, J.C. “State-state formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensitivity”, in: Systems and Control Letters, vol.11, pp. 167-172, 1988. [9] Iwasaki., T., and Skelton, R.E. “All controllers for the general H∞ Control problem: LMI existence conditions and state-space formulas, in: Automatica, vol. 30(8), pp. 1303-1317, 1994. [10] Kennedy, J., “Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance”, in Proceedings of IEEE Congress on Evolutionary Computation (CEC 1999), Piscataway, NJ, USA, pp 1931-1938, 1999.

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[11] Kennedy, J. and Clerc, M., “Standard PSO”. http://www.particleswarm.info/ Standard_PSO_2006.c, 2006 [12] Kwakernaak, H. and Sivan, R. “Linear optimal control”, New York, Willeyinterscience, 1972. [13] Lassami, L. and Font, S. “Backstepping controller retuning using ε subdifferential optimization”, in: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Sevilla, Spain, pp. 5119-5125, 2005. [14] Maciejowski, M. “Predictive Control with Constraints”. Prentice Hall, England, 2002. [15] Pita-Gil, G., Talon, V., Sandou, G., Godoy, E., Dumur, D. “Robust Non-linear Control Applied to Internal Combustion Engine Air Path Using Particle Swarm Optimization”, in Proceedings of the 3rd IEEE Multi-conference on Systems and Control, SaintPetersburg, Russia, July 8-10th 2009. [16] Reyes-Sierra, M. and Coello Coello C. “A. Multi-Objective Particle Swarm Optimizers: A Survey of the State-of-the-Art”, in: International Journal of Computational Intelligence Research, vol.2, no. 3, pp. 287–308, 2006. [17] Sandou, G., Font, S., Tebbani, S., Hiret, A. and Mondon, C. “Global modelling and simulation of a district heating network”, in: Proceedings of the 9th International Symposium on District Heating and Cooling, Espoo, Finland, August 30th-31st 2004. [18] Sandou, G., Lassami, B. “Optimisation par essaim particulaire pour la synthèse ou la retouche de correcteurs”, in : Proceedings of the 7ième Conférence Internationale de Modélisation et Simulation MOSIM, Paris, France, 2008. [19] Santana-Quintero, L. V., Ramırez-Santiago, N., Coello Coello, C. A., Molina Luque, J., and Garcıa Hernandez-Dıaz, A. “A New Proposal for Multiobjective Optimization Using Particle Swarm Optimization and Rough Sets Theory”, in: Lecture Notes in Computer Science, vol. 4193, pp. 483–492, 2006. [20] Shi, Y. and Eberhart, R. C. “Parameter selection in particle swarm optimization”, in Evolutionary Programming VII: Proceedings of the Seventh Annual Conference on Evolutionary Programming, New York, USA. pp. 591-600, 1998. [21] Trelea, I.C. “The particle swarm optimization algorithm: convergence analysis and parameter selection”, in Information Processing Letters, vol. 85, n°6, pp. 317-325, 2003. [22] Zhao, J., Li, T. and Qian, J. “Application of particle swarm optimization algorithm on robust PID controller tuning”, in: Lecture Notes in Computer Science, Vol. 3612, L. Wang, K. Cheng, Y. S. Ong (Eds.) pp. 948-957, Springer-Verlag Berlin Heidelberg, 2005. [23] Zhou, K., Doyle, J. C. and Glover, K. “Robust and optimal control”, New Jersey, Prentice-Hall, 1996.

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Chapter 2

STUDY ON VEHICLE ROUTING PROBLEM WITH TIME WINDOWS BASED ON ENHANCED PARTICLE SWARM OPTIMIZATION APPROACH Chun-Ta Lin Department of Information Management, Yu-Da University, Miao-li, Taiwan, R.O.C

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Abstract Particle Swarm Optimization (PSO) technique proved its ability to deal with very complicated optimization and search problems. Several variants of the PSO algorithm have been proposed. This chapter proposes a variant of the basic PSO technique named Predicting Particle Swarm Optimization (Predicting PSO) algorithm dealing with combinatorial optimization to solve the well known Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is the extension of the capacitated vehicle routing problem which is aimed to the definition of the minimum cost paths where the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. The original PSO is basically developed for continuous optimization problem. However, VRPTW problems are formulated as combinational optimization problem. In this chapter, we first propose the Predicting PSO algorithm based on original PSO which modified by three new solution strategies. The main contributions of this proposed algorithm include: (1) to provide a totally different concept to transform the search space from multi-dimensions to one-dimension and (2) successfully transfer the combinatorial problem to continuous problem through boundary constraints handling. (3) Fully using the memory from population. Predicting PSO algorithm not only can modify the variance between current status and memorial best status, but also can forward predict what the best status should be at next time status based on memory. The Predicting PSO algorithm is designed to fit this combinational optimization problem type to find near-optimal solutions in the VRPTW problems. The Predicting PSO algorithm has also been tested on Solomon’s benchmark problems. The findings indicate that the proposed algorithm outperforms other heuristic algorithms.

Keywords: Predicting Particle Swarm Optimization, Particle Swarm Optimization, Vehicle Routing Problem with Time Windows.

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Chun-Ta Lin

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1. Introduction Vehicle Routing Problem (VRP) is a well-known combinatorial problem with considerable economic in logistics system. A typical Vehicle Routing Problem can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points (cities, stores, warehouses, schools, customers etc). The routes must be designed in such a way that each point is visited only once by exactly one vehicle, all routes start and end at the depot, and the total demands of all points on one particular route must not exceed vehicle capacity. Many different types of VRP can be generated in terms of the demand and restrictions of practical applications. The Vehicle Routing Problem with Time Windows (VRPTW) is a generalization of the VRP involving the additional constraint that every customer must be served within a given time window. The time window constraint is denoted by a defined time interval, given the ready time and the due time for service. The VRPTW has been applied extensively in practice, such as bank deliveries, postal deliveries, school bus routing, security patrol service and industrial refuse collection. Moreover, the VRPTW has been proved to be NP-hard and exact algorithms cannot find the combinational optimization solution for large VRPTW within reasonable computational times. Because of NPcompleteness, the majority of research has focused on heuristics. Many heuristic approaches like Blanton and Wainwright [3] were the first to apply a genetic algorithm to VRPTW. They hybridized a genetic algorithm with a greedy heuristic. Under this scheme, the genetic algorithm searches for a good ordering of customers, while the construction of the feasible solution is handled by the greedy heuristic. Berger et al. [2] present a hybrid genetic algorithm based on removing certain customers from their routes and then rescheduling them with well-known route-construction heuristics. Homberger and Gehring [16] propose two evolutionary metaheuristics based on the class of evolutionary algorithms called Evolutionary Strategies and three well-known route improvement procedures Or-opt [22], λ -interchanges [23] and 2-opt [24]. Gehring and Homberger use a similar approach with parallel tabu search implementation. Chiang and Russell [5] describe a reactive tabu search that dynamically adjusts its parameter values based on the current search status to avoid both cycles as well as an overly constrained search path, and Cordeau et al. [7] introduce a fast and very simple tabu search that applies reinsertions of single customers and allows unfeasible solutions during the search process. Most approaches to the VRPTW attempt to minimize both the number of vehicles and the total traveling cost such that each customer is served within its time window and the total load on any vehicle associated with a given route does not exceed vehicle capacity [17]. The review papers of Gendreau et al [14], Laporte [19] and Desrosier et al. [9] provide excellent pointers to the research efforts in the area of solution methods developed for the VRPTW in the 1990s, and generally classified into two approaches: (a) improvement and construction heuristics, which usually result in relatively mediocre-to-good solutions in short computational times, and (b) meta-heuristics, which result in high-quality solutions in relatively large computational times. Montemanni et al. [20] propose a solving technique for the dynamic VRP based on Ant Colony System which is constructed by three main elements: an event manger, which collects new orders and keeps trace of the already served orders and of the current position of each vehicle. The second element is Ant Colony System Algorithm

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and pheromone conservation procedure as the third element is strictly connects with the Ant Colony System Algorithm The Particle Swarm Optimization (PSO) was first introduced by Eberhart and Kennedy [10, 11, 18]. It has been successfully used in different fields of applications. The PSO is basically developed for continuous optimization problem. By, recently, there are increasingly interests on the research of applying PSO to solve the discrete problems, for example, Onwubolu [21], Clerc [6], Shen et al., [25] and Tasgetiren et al., [29]. Moreover, Zhu et al., [31] proposed a computation-efficient VRPTW algorithm based on PSO which can effectively and quickly get optimal resolution of VRPTW. A new hybrid approximation algorithm is developed by Chen et al., [4] to solve the capacitated VRP. In this hybrid algorithm, discrete particle swarm optimization (DPSO) combines global search and local search to search for the optimal results and simulated annealing (SA) uses certain probability to avoid being trapped in a local optimum. In this chapter, we introduce three new solution strategies to modify the continuous properties of basic PSO to be more suitable for applying in combinatorial applications. These strategies include: (1) search space transformation, (2) boundary constraints handling, and (3) forward predicting update on velocity update equation. Finally, the Predicting PSO algorithm based on the new solution strategies of PSO has been proposed to solve the VRPTW and tested on the Solomon’s 56 VRPTW 100-customer benchmark problems. The chapter is organized as follow: In Section 2, the difficulty for using basic PSO algorithm to solve the VRPTW has been discussed and the new solution strategies of Predicting PSO have been introduced. Section 3 proposes the Predicting PSO algorithm. Computational results on Solomon’s 56 VRPTW benchmark problems with Predicting PSO algorithm and comparison against other meta-heuristics are reported in section 4. Finally, conclusions are made.

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2. Problem Solving Methodology 2.1. The Basic Particle Swarm Optimization The PSO algorithm was first introduced in 1995. Instead of using evolutionary operators to manipulate the individuals, like in other evolutionary computational algorithms, each individual in PSO flies in the search space with a velocity which is dynamically adjusted according to its own flying experience and its companions’ flying experience. Each individual is treated as a volume-less particle, that is a point, in the D-dimensional search space. The ith particle is represented as xi = ( xi1 , xi 2 , " , xiD ) . The best previous position (the position giving the best fitness value) of the ith particle is recorded and represented as pi = ( pi1 , pi 2 , " , piD ) . The index of the best particle among all the particles in the population is represented by the symbol g. The rate of the position change (velocity) for the particle i is represented as vi = (vi1 , vi 2 , " , viD ) . The particles are manipulated according to the following equation:

vi (t + 1) = vi (t ) + c1 × rand () × ( pi (t ) − xi (t )) + c 2 × rand () × ( p g (t ) − xi (t )) (1)

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Chun-Ta Lin

xi (t + 1) = xi (t ) + vi (t + 1)

(2)

where t denotes discrete time statuses, c1 and c 2 are cognitive and social parameters, respectively, and rand() are random function in the range [0,1]. It is the velocity vector that drives the optimization process, and reflects both the experiential knowledge of the particle and socially exchanged information from the particle's neighborhood. The experiential knowledge of a particle is generally referred to as the cognitive component, which is proportional to the distance of the particle from its own best position (referred to as the particle's personal best position) found since the first time step. The socially exchanged information is referred to as the social component of the velocity equation. Therefore, the equation (1) can be translated as: Velocity at time status t + 1 = Velocity at the time status t + Cognitive Component Update at time status t + Social Component Update at time status t. where cognitive component update at time status t = c1 × rand () × ( pi (t ) − xi (t )) and social component update at time status t = c2 × rand () × ( p g (t ) − xi (t ) ) Furthermore, in evolutionary programming, the balance between the global and local search is adjusted through adapting the variance (strategy parameter) of the Gaussian random function or step size, which can even be encoded into the chromosomes to undergo evolution itself. In PSO, a parameter called inertia weight is brought in for balancing the global and local search and equation (1) is changed to:

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vi (t + 1) = w × vi (t ) + c1 × rand () × ( pi (t ) − xi (t )) + c 2 × rand () × ( p g (t ) − xi (t ))

(3)

where w is the inertia weight[26,27]. The inertia weight has characteristics that are reminiscent of the temperature parameter in the Simulated Annealing [12].A large inertia weight facilitates a global search while a small inertia weight facilitates a local search. By linearly decreasing the inertia weight from a relatively large value to a small value through the course of the PSO run, the PSO tends to have more global search ability at the beginning of the run while having more local search ability near the end of the run. The accumulative effect of all the position updates of a particle is that each particle converges to a point on the line that connects the global best position and the personal best position of the particle. But, through the velocity update equation (3) and the position update equation (2), the updated position xi (t + 1) may be infeasible in a combinatorial search space. This property makes the basic PSO algorithm become less efficient than the other meta-heuristics to deal with combinatorial optimization problems.

2.2. The Vehicle Routing Problem with Time Windows Problem In VRPTW, a typical combinatorial searching problem, let G = ( N , A) be a directed graph where N = {0,1,..., n} is the node set, also represents the customer set in VRPTW, and

A = {(i, j ) : i, j ∈ N , i ≠ j} is the arc set, also represents the route from customer i to

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Study on Vehicle Routing Problem with Time Windows…

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customer j. Node 0 represents the depot and M = {1,..., n} denotes the set of nodes to be visited. Each arc (i,j) is associated to a travel cost cij ≥ 0 and a travel time t ij ≥ 0 . Each node i has a service request qi , a service time si and a time window [ei , l i ] . All vehicles have the same capacity Q and the same fixed cost h ≥ 0 . Assume that cij = t ij , ∀(i, j ) ∈ A and that the h is high enough to guarantee that minimizing the number of vehicles is the main objective. Without loss of generality, we may shrink the time windows by setting ei = max(e0 + t 0i , ei ) and li = min(l 0 − t i 0 , li ) . In fact, this removes portions which cannot be feasible used. Let us set xij = 1 if arc (i,j) is used, xij = 0 otherwise; pi represents the beginning of the service at node i; y i is the load of the vehicle leaving node i. The mathematical formulation of the VRPTW may then be defined as follows:

min ∑ hx0 j + j∈N

∑c

( i , j )∈ A

ij

xij

(4)

subject to

∑x j∈N

∑x i∈N

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if

if

ij

= 1 ∀i ∈ M

(5)

ij

= 1 ∀j ∈ M

(6)

xij = 1 ⇒ pi + si + t ij ≤ p j ∀(i, j ) ∈ A

(7)

ei ≤ pi ≤ li ∀i ∈ M

(8)

xij = 1 ⇒ y i + q j ≤ y j

∀(i, j ) ∈ A

(9)

qi ≤ y i ≤ Q ∀i ∈ M

(10)

xij ∈ {0,1} ∀(i, j ) ∈ A

(11)

The constraints (5) and (6) ensure that each customer be assigned exactly to a single route. The equations (7) and (8) represent time window constraints, while equations (9) and (10) represent capacity constraints. According to the constraint equations (11), the VRPTW has driven itself to be an integer programming problem. Therefore, a non-integer solution is infeasible for the VRPTW.

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Chun-Ta Lin

2.3. The Difficulty for Using Basic PSO to Solve the VRPTW Indeed, basic PSO algorithm use the velocity vector to drive the optimal process, and reflects both the experiential knowledge of the particle and socially exchanged information from the particle's neighborhood to update the position of particle; as a result, a near optimal solution can be found. But the random numbers provide a stochastic characteristic to the velocity updated of each particle in basic PSO. This fact makes the solution found by basic PSO algorithm become infeasible for combinatorial problem. This also makes the basic PSO become less efficient than the other heuristic algorithms applied in combinatorial search space. Meanwhile, a particle flies in this search space may locate outside the range of feasible solutions, so how to handle the particle flying within a determined boundary is another challenge to overcome. Therefore, in order to apply the PSO algorithm in VRPTW, we make some modifications on basic PSO to propose a new algorithm, named Predicting Particle Swarm Optimization (Predicting PSO). These modifications based on three new solution strategies in the Predicting PSO include: (a) the transformation of search space, (b) boundary constraints handling with periodical copies of search space, and (c) a forward predicting update is used on velocity equation are discussed in the next section.

2.4. The New Solution Strategies of Predicting Particle Swarm Optimization Strategy I: Search Space Transformation Let N = {0,1,..., n} be the node set, also the customer set, on the search space

S ⊆ R 2 and node 0 represents the depot. Each node i has a service request qi , a service Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

time si and a time window [ei , l i ] . Let D = {l i : ∀i ∈ N } and T : N → D be the ‘time transformation function’, then l i = T (i ), ∀i ∈ N represents a mapping from each node i to its latest possible service time, l i the due time of customer i. Let f be the ‘sorting function’ in increasing order of time, then, SL = f ( D ) ⊆ R forms a searching list on one-dimensional 1

search space with the length equal to L. Since f : D → SL is a 1-to-1 function, the compose function g = f D T and g : N → SL are 1-to-1 functions, too. As a result, the inverse function

g −1 : SL → N

existed.

That

is,

to

search

the

customer

i,

the

node i ∈ N ⊆ S ⊆ R , is similar to search g (i ) on the searching list SL ⊆ R ; in other 2

1

words, a solution on SL is equivalent to a solution on N (see the Figure 1). Therefore, technically using a ‘Round-off Operator’ to approach a solution to its nearest integer on SL, we can transfer the discrete search space SL to a quasi-continuous search space on R employing Predicting PSO algorithm.

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1

Study on Vehicle Routing Problem with Time Windows…

37

g = f DT

i∈N

T (i ) ∈ D

T

g −1 = ( f D T ) −1 = T

f

−1

D f

g (i ) ∈ SL

−1

Figure 1. The relationship between node set and searching list.

Strategy II: Boundary Constraint Handling Unlike other algorithms, such as genetic algorithms [8, 13], evolutionary strategies [15], and differential evolution [28], etc., the status of the t-th generation of each particle in swarm has direct effects on its status of the next generation. It is, of course, possible for a particle to overshoot the updating position out of boundary, mainly due to the velocity update. These results will violate the boundary constraints. Zhang et al. [30] propose a new boundary handling method, which is called as ‘Periodic Mode’. It provides an infinite search space for the flying of particles, which is composed of the periodic copies of original search space with same fitness landscape. The same idea used in Predicting PSO, we treat the searching list as a ‘Circle’ that means we periodically copy the searching list infinitely; then, the candidate customer can be searched through this searching list continuously (see Figure 2). Xi(t+1)

Vi(t+2)

Vi(t+1) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Xi(t+3)

Xi(t)

Xi(t+1) = mod(Xi(t) + Vi(t+1) , L)

Searching List with Length L

Xi(t+2) = mod(Xi(t+1) + Vi(t+2) , L) Xi(t+3) = mod(Xi(t+2) + Vi(t+3) , L) Xi(t+2)

Vi(t+3)

Figure 2: The Flying Path of Particle and Position Update

Figure 2. The flying path of particle and position update.

When the updating velocity has overshot the particle out of constrained boundary, the particle’s position can be updated through the ‘Modulus Operator’ and relocated on searching list. Then, as the position on searching list has been determined, the relative customer is also selected through transform mapping g

−1

(see Figure 1).

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38

Chun-Ta Lin

Strategy III: Forward Predicting Update on Velocity Update Equation Through the transformation of search space from node set N to the searching list SL, a candidate customer to be served can be searched from SL. Meanwhile, in VRPTW, a customer is restricted to be served by only one vehicle at one time. This make the length of searching list, L, dynamically change as the customer has been selected and removed from searching list SL. As a result, the particle’s velocity should be adjusted according to the change of searching space. Without loss of generality, we define Speed = L ÷ 2 as the maximum velocity can be used for each particle in Predicting PSO. Moreover, in order to increase the ability of exploiting, the formula of maximum velocity is further modified as Speed = L ÷ (2 + t ) where t is the current time status during the generation simulation. This dynamically modification on the velocity update in each time status make the particle have more global search ability at the beginning of the run while having more local search ability near the end of the run. In the initial solution search stage, since there is no any memory existed, the velocity update equation (3) is changed to:

vi (t + 1) = int( w × vi (t ) + (1 + rand () × Speed ))

(12)

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where 1 + rand () × Speed make sure that at least one position has been changed, rand() is a random function in the range [0,1], w is the inertia weight in the range of [0, 1], t is the current time status in the run, and int is the ‘Round-off Operator’ to make the velocity change near to the nearest integer. Since the searching space SL has been periodically copied, the particle can fly along the searching list circuitously and the position of the particle is updated accumulatively through generation to generation. Then, through the modulus operator, the updated position can point to somewhere on the searching list, SL _ X i (t + 1) ∈ SL . As results, the position update equation (2) is changed to:

xi (t + 1) = xi (t ) + vi (t + 1) and SL _ X i (t + 1) = mod( xi (t + 1), L)

(13)

where mod is a ‘Modulus Operator’ and mod( xi (t + 1), L) is equal to the residual of

xi (t + 1) divided by the length of searching list, L, and represents the updated position located on the searching list (see the Figure 2). After iteratively searching, a set of possible candidate customers, named CandidateSet, can be generated with equations (12), (13) based on time status set, TS, through generation to generation. Hence through the inverse transformation function g

−1

: SL → N , a set of candidate customer, S, can be generated

uniquely based on TS.

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Study on Vehicle Routing Problem with Time Windows…

39

g −1

j ∈V ⊆ S ⊆ N

g

SL _ X (t * ) ∈ SL

Modulus operator

xi (t * ) : t * ∈ TS

Figure 3. The relationship between customer j in selected route and its update position at time status t*.

After the feasibility test was implemented to obey the constraints of VRPTW, a feasible route, V, selected from S based on TS can be found, meanwhile, the experience memory of this selected route, {xi (t ) : t ∈ TS } , can be recorded according to their update positions based on TS (see Figure 3). Then, in the population search stage, the memory can be used to adjust the direction and velocity in next time status. But, basic PSO, like equation (3), the velocity update is based on the variance between current position and the position memorized by particle itself and the population from past experience only and without referring the memory of itself and the population to decide where the next position should be. In order to make the search more efficient, in Predicting PSO, we consider not only the variance adjustment between current position and the position in memory (the social and cognitive components) but also we use the memory to predict where the next position should be (the predicting component) to decide the direction and velocity in next time status. The comparison of velocity update between basic PSO and Predicting PSO has shown as Figure 4. pi (t + 1) □ Δ

pi (t + 1)

○ xi (t )

p g (t + 1)

xi (t ) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

p g (t ) ○

p g (t ) ○

□ p (t ) i

vi (t + 1)

Δ x (t + 1) i

Δ

□

Δ

○

p g (t + 1)

p g (t + 1) − xi (t )

pi (t ) □ p (t + 1) − x (t ) i i

Velocity update in Basic PSO

xi (t + 1)

vi (t + 1)

Velocity update in Predicting PSO

Figure 4. The velocity update comparison between basic PSO and predicting PSO.

In Figure 4, p g (t ) , p i (t ) , and xi (t ) are defined as the global position, the particle’s best position, and the current position of particle i at the time status t, respectively; vi (t + 1) and xi (t + 1) are the update velocity and update position at the next time status t+1. We can see that a forward predicting update on velocity at time t, used in Predicting PSO, will drive the particle more closely toward global position, p g (t + 1) from particle swarms, and local position, pi (t + 1) from particle i itself. This property enhances the ability more on the exploitation than on the exploration and makes the search more efficient than basic PSO. The velocity update equation, in Predicting PSO, based on equation (11) has been changed to:

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

40

Chun-Ta Lin

vi (t + 1) = Velocity at the time status t + Cognitive component update at time status t + Social component update at time status t. + Predicting update of cognitive component at time status (t+1) + Predicting update of social component at time status (t+1)

= int{w × vi (t ) + c1 × rand () × [( pi (t ) − xi (t )) + ( pi (t + 1) − xi (t ))] + c2 × rand () × [( p g (t ) − xi (t )) + ( p g (t + 1) − xi (t ))]}

(14)

The position update equation is written as

xi (t + 1) = xi (t ) + vi (t + 1) and SL _ X i (t + 1) = mod( xi (t + 1), L)

(15)

where w is the inertia weight and c1 and c 2 are the cognitive and social parameters, respectively, defined on equation (3).

pi (t + 1) xi (t )

Δ

□

xi (t + 1) ○ Δ

p g (t ) ○ pi (t ) □

p g (t + 1) − xi (t )

pi (t + 1) − xi (t )

xi (t + 1)

p g (t + 1)

xi (t )

p g (t ) ○

p g (t + 1)

○

Δ vi (t + 1)

vi (t + 1)

Velocity update in Predicting PSO (a) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Δ

Velocity update simplification in Predicting PSO

(b)

Figure 5. The velocity update further simplification in predicting PSO.

Since the global memory from population swarms always outperforms the local memory from particle itself, for simplification, we can remove the cognitive component and predicting update of cognitive component from equation (14) to obtain the equivalent result (see Figure 5(b)). Then, equation (14) can be further simplified as follows:

vi (t + 1) = Velocity at the time status t + Predicting component update at time status (t+1) + Social component update at time status t.

= int( w × vi (t ) + r1 × rand () × predicting (t + 1) + r2 × rand () × error (t )) where predicting component ≡ predicting (t + 1) = Pg (t + 1) − xi (t ) and social component ≡ error (t ) = Pg (t ) − xi (t )

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(16)

Study on Vehicle Routing Problem with Time Windows…

41

r1 , the predicting parameter, represents the degree of trusting on population memory and r2 , the social parameter is treated as an error adjustment on the time status t based on the population memory.

3. The Algorithm of Predicting PSO To solve the Vehicle Routing Problem with Time Windows (VRPTW), the objectives are not only to minimize the total traveling cost but also to minimize the total number of vehicles used in practice. There are three phases have been approached in Predicting PSO algorithm. The Predicting PSO used in the each phase of our algorithm is based on studies by described above the three new solution strategies.

Phase I: Candidate Customer Search in Initial Stage An initial solution can be searching through the search space by using velocity equation (12) and the position update equation (13), in this process, and a set of candidate customers is generated as CandidateSet without temporally considering the capacity constraint. Step 1: Sort the customers in search space based on their due time for service to form a searching list, SL.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Step 2: Calculate the length of searching list, L, to decide the maximum velocity can be used in each move; then, the Speed is calculated as Speed = L ÷ (2 + t ) . Step 3: Velocity Updating: vi (t + 1) = int( w × vi (t ) + (1 + rand () × Speed )) Step 4: Position Updating:

xi (t + 1) = xi (t ) + vi (t + 1) and SL _ X i (t + 1) = mod( xi (t + 1), L) , If SL _ X i (t + 1) ∈ CandidateSet , then goes to Step3 till a stop criterion is met; Otherwise, SL _ X i (t + 1) is added to CandidateSet, then, goes to Step 5. Step 5: Continue Step3 ~ Step4 until the stop criterion is met.

Phase II: Global Initial Solution Generated Through Feasibility Test When the CandidateSet is generated, the capacity constraint and time window constraint feasibility are tested to form a set of feasible routes for all possible combination of customers in this CandidateSet to maximize the loading amount of vehicle. Meanwhile, the transformed

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42

Chun-Ta Lin

positions on the searching list based on the optimal route in this stage are recorded as the global memory. Step 1: Transfer the position on searching list to its relative customer name from the CandidateSet. Step 2: Sort this CandidateSet based on customers’ due times for service to form a temporary searching list. Step 3: A set of feasible tours from the temporary searching list can be generated by inserting one customer at one time to the tour till violating the constraints: (3a) if capacity constraint test is violated, then the tour is completed and goes to Step 4; otherwise, goes to (3b). (3b) if there is any one time window of each customer in the tour is violated after inserting a new customer into this tour, then goes to Step 4; otherwise, goes to (3c). (3c) repeats the Step 3 till the stop criterion is met, then goes to Step 4. Step 4: Fitness Test for Finding Global Initial Solution If the objective function has been improved, then update the GlobalRoute by the new one and transfer the service order in the GlobalRoute to the position order in the searching list used in Step 2 to form a memory list; otherwise, repeats Step 3 ~ Step 4 with a new route till the stop criterion is met.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Step 5: Remove the customers of GlobalRoute from search space to form the UnservedSet.

Phase III. Population Search for Optimization After an initial feasible GlobalTour is generated, we can further improve this route by inserting another alternative of customers from search space based on the memory list to minimize the total traveling cost as follows: Step 1: Sort the customers based on the due time for service in UnservedSet to form a new search list and calculate the length of searching list, L, and Speed = L ÷ (2 + t ) . Step 2: Based on the memory list to search the other alternative customers in searching list through velocity updating equation (16) and position update equation (15). (2a) Velocity Updating:

vi (t + 1) = int( w × vi (t ) + c1 × rand () × predicting (t + 1) + c 2 × rand () × error (t )) If error (t ) does not exist in memory list, then the social component of equation (16) is changed to:

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Study on Vehicle Routing Problem with Time Windows…

error (t ) = 1 + rand () × Speed

43

(17)

If predicting (t + 1) does not exist in memory list, then, the predicting component of equation (16) is changed to:

predicting (t + 1) = 1 + rand () × Speed

(18)

Otherwise, goes to (2b). (2b) Position Updating:

xi (t + 1) = xi (t ) + vi (t + 1) and SL _ X i (t + 1) = mod( xi (t + 1), L) If SL _ X i (t + 1) ∈ GlobalTour , then goes to Step2; Otherwise, goes to step 3. Step 3: Transfer the position SL _ X i (t + 1) on searching list to its relative customer name from the UnservedSet.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Step 4: Insert the new customer selected from Step 3 to the original GlobalRoute and make the constraint feasibility test: (4a) if the capacity constraint feasibility test is violated, then goes to Step 2; otherwise, goes to (4b). (4b) if there is any one time window feasibility of each customer in the tour is violated after inserting this new customer into this tour, then goes to Step 2; otherwise, goes to Step 5. Step 5: Global test for population fitness If the objective function, minimize the total traveling cost, has been improved, then update the GlobalRoute by the new one and transfer the service order in the GlobalRoute to the position order in the searching list in Step 1 to form a new memory list and goes to Step 1; otherwise, repeats Step 2 ~ Step 4 till the stop criterion is met. Step 6: Remove the customers of GlobalRoute from search space and searching list, meanwhile, a new tour is created, if necessary. If search space is empty, then stop searching and a near optimal solution has been achieved; Otherwise, returns to Phase I.

4. Computational Experiment 4.1. Problem Data To analyze the performance of our proposed algorithm, an experiment was conducted over 56 VRPTW problem instances. The problems vary in fleet size, vehicle capacity, travel time,

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

44

Chun-Ta Lin

spatial and temporal customer distribution (position, time window density, time window width, and service time). Six VRPTW data sets R1,R2,C1,C2,RC1 and RC2 generated by Solomon in1987, which are commonly accepted as the main benchmark problems for VRPTW algorithm. They are 56 random problems made up 100 nodes. Vehicle capacity, customer time windows, service times and distribution in space and time vary so as to cover all configurations as thoroughly as possible. Thus, customers may be uniformly distributed (R1 and R2), clustered (C1 and C2), or a mix of the two patterns (RC1 and RC2). Problem sets R1,C1 and RC1 have a narrow scheduling horizon, tight time windows and a low vehicle capacity. Conversely, problem sets R2, C2 and RC2 have a large scheduling horizon, wide time windows and a high vehicle capacity. Some researchers have found out that setting a linearly decreasing inertia weight into basic of PSO, the performance of PSO has been greatly improved through experimental study on the benchmark problems of Schaffer’s F6 function [26, 27]. So, in Predicting PSO, the linearly decreasing inertia weight w in the range of [0, 1] is used in equations (12) and (16). Through experiment test with Solomon benchmark questions, r1 , the predicting parameter is set as a linear decreasing constant in the range of [1.0, 1.5], but the social parameter, r2 , is set as 2. Meanwhile, the stop criterion used in Predicting PSO is the maximum number of iterations and it is set as 200；The population size is 20 in numerical tests. Predicting PSO implemented in Matlab 7.0, without using any extra tuning technique, has been tested with the Solomon’s 56 VRPTW 100-customers and extension 200-customers problems using a Pentium III CPU, 846MHz computer.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

4.2. Computational Results The results compared with best known solution identified by heuristics are shown on Table 1. From Table 1, we can see the Predicting PSO is a very effective algorithm to achieve highquality solutions particularly on the RC1 problem set. From Table 1, there are 4 of 8 problems in the problem set RC1 have been improved both on the total traveling cost and the number of vehicles used. For R1 problem set, there are also 10 of 15 problems have been improved on the total traveling cost, but in the number of vehicles used have not been improved. Meanwhile, three of 200-customers benchmark problems, R1_2_1, R1_2_2, and R1_2_3, have been tested and resulted one improved solution. Since the Predicting PSO algorithm searches the solutions based on the time transformed search space, the updated positions of a particle are recorded according to the due time available for service by candidate customers and accumulated to form a route. Then the capacity constraint feasibility is examined and the travel time from customer to customer is calculated to obey the time window constraints of each customer in this route from beginning to the end. That is meant that the Predicting PSO algorithm considers the time feasibility first, the travel cost minimization second. According to this fact, while the targets are randomly distributed in the search space or clustered in separated subspaces, it seems that the Predicting PSO without associated another tuning method(s) is not so efficient to find the near optimal solutions.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Study on Vehicle Routing Problem with Time Windows…

45

Table 1. Comparison of Predicting PSO and Best Known Solutions Identified By Heuristics. HSPSO

Best-So-Far

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Test Problem R101 R102 R103 R104 R105 R106 R107 R108 R109 R110 R111 R112 C101 C102 C103 C104 C105 C106 C107 C108 C109 RC101 RC102 RC103 RC104 RC105 RC106 RC107 RC108

Improvement Tours

Cost

Tours

Cost

24 21 13 8 17 13 10 8 11 12 11 8 13 10 11 10 12 11 12 10 10 19 15 12 10 14 10 12 9

1201.82 1307.70 1749.61 1193.21 1318.21 1359.30 1476.82 1327.78 1661.44 1448.22 1362.71 1289.65 1046.12 1160.07 1048.22 1088.79 1021.11 1073.33 1025.81 933.33 1091.78 1087.4 1329.3 1370.08 1299.18 1301.3 1412.13 1569.02 1550.34

19 17 13 10 14 12 10 9 11 10 10 9 10 10 10 10 10 10 10 10 10 15 14 11 10 13 12 12 11

1645.79a 1486.12 a 1292.85 a 982.01 a 1377.11 a 1252.03 a 1113.69 a 964.38 a 1194.73 a 1125.04 a 1099.46 a 1003.73 a 828.94 a 828.94 a 828.06 a 824.78 a 828.94 a 828.94 a 828.94 a 828.94 a 828.94 a 1619.8 a 1457.4 a 1258 a 1135.4 a 1513.7 a 1401.2 a 1207.8 a 1114.2 a

-26.98%* -12.01%* 35.33% 21.51% -4.28%* 8.57% 32.61% 37.68% 39.06% 28.73% 23.94% 28.49% 26.20% 39.95% 26.59% 32.01% 23.18% 29.48% 23.75% 12.59% 31.71% -32.87%* -6.19%* 8.91% 14.42% -14.03%* 0.78% 29.91% 39.14%

a

Source: 'Branch-and-price algorithms for Vehicle Routing Problems', the dissertation of dott. Matteo Salani, PP95~96, Table 8.1 & 8.2, Anno accademico 2004/2005 b Source:”A synthesis of assignment and heuristic solutions for vehicle routing with time windows”, Table 5, Journal of the Operation Society 2004

So, for the R1 problem sets, customers are more randomly and widely distributed on the search space than they are in RC problem sets. The candidate customers selected from transformed searching list to form the feasible route might be widely distributed on the real search space, as results, the total traveling cost might increase. To minimize the total travel cost, more vehicles might be needed in solving the R1 problem sets. Moreover, the same results are happed in the 200-extension R1 problem test. For C1 problem sets, as customers are clustered distribution in the search space, the travel distances between two nodes of search space are more sensitive than the other problem sets.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Chun-Ta Lin

As results, the Predicting PSO seems inefficient to deal with the clustered problems simply. Maybe, an extra branching and grouping technique, like the branch-and-bound algorithm used in the other literatures, should be associated with Predicting PSO algorithm to deal with the C1 and R1 problems more efficient. From these pieces of evidence, the proposed Predicting PSO algorithm may do in the further research.

5. Conclusion

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

The purpose of this study was to create the Predicting Particle Swarm Optimization algorithm for solving vehicle routing problems with time windows. Among those meta-heuristic algorithms successfully applied in VRPTW, the basic Particle Swarm Optimization algorithm has never been applied in VRPTW. We propose three new solution strategies to modify the continuous properties of basic PSO to be more suitable for applying in VRPTW applications. The Predicting PSO algorithm based on the three new solution strategies and tested on Solomon’s benchmark problems without combined any tuning methods. The best results obtained with our algorithm were compared with the best results obtained by other metaheuristics providing excellent results. Generally it can be concluded that the results obtained with our algorithm are competitive with the results obtained with other approaches. The main contributions of this research include: (1) to provide a totally different concept of transforming the search space from multi-dimensions to one-dimension and (2) successfully transfer the combinatorial problem to continuous problem through boundary constraints handling. (3) Fully using the memory from swarm population, Predicting PSO algorithm not only can modify the variance between current status and memorial best status, but also can forward predict what the best position should be at next time status based on memory. This property has tremendously improved the efficiency of solution searching.

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[2]

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[5]

Angeline, P.J., “Evolutionary Optimization versus Particle Swarm Optimization: Philosophy and Performance Difference”, IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, May 4-9, 1998 Berger J., M. Salois and R. Begin, “A Hybrid Genetic Algorithm for the Vehicle Routing Problem with Time Windows”, Lecture Notes in Artificial Intelligence 1418, AI’98, Advances in Artificial Intelligence, Vancouver, Canada, pp114-127, 1998 Blanton J.L. and RL. Wainwright, “Multiple Vehicle Routing with Time and Capacity Constraints using Genetic Algorithm”, In: S. Forrest (ed), ‘Proceedings of the 5th International Conference on Genetic Algorithms’, pp 452-459 Morgan Kaufmann, San Francisco, 1993. CHEN Ai-ling, YANG Gen-ke, WU Zhi-ming, “Hybrid discrete particle swarm optimization algorithm for capacitated vehicle routing problem”, Journal of Zhejiang University SCIENCE A, Vol7(4),pp.607-614, 2006. Chiang WC. and RA. Russell, “A Reactive Tabu Search Metaheuristic for the Vehicle Routing Problem with Time Windows”, INFORMS Journal of Computing, Vol. 9, pp 417-430, 1997

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Study on Vehicle Routing Problem with Time Windows… [6]

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

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[21]

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Clerc, M., “Discrete Particle Swarm Optimization, illustrated by the Traveling Salesman Problem New Optimization Techniques in Engineering”, Springer, pp.219239, 2004. Cordeau JF, G. Desaulniers, MM. Solomon and F. Soumis, “The VRP with Time Windows”, In: P. Toth and D Vigo (eds), ‘The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications’, pp 157-194, SIAM, Philadelphia, 2001 Deb K., “An Efficient Constraint Handling Method for Genetic Algorithms”, Computer Methods in Applied Mechanics and Engineering, Vol. 186(2-4), pp 311-338, 2000 Desrosiers J, Y. Dumas, MM Solomon and F. Soumis, “Time Constrained Routing and Scheduling”, In: M. Ball, TL. Magnanti, CL. Monma and GL. Nemhouser (eds), ‘Network Routing, Handbook in Operations Research and Management Science’, Vol. 8, Elsevier, Amesterdam, pp 35-139, 1995. Eberhart RC., RW. Dobbins and P. Simpson, ‘Computational Intelligence PC Tools’, Boston: Academic Press, 1996 Eberhart RC. And J. Kennedy, “A New Optimizer Using Particle Swarm Theory”, Proceedings 6th International Symposium on Micro Machine and Human Science (Nagoya, Japan), IEEE Service Center, Piscataway, NJ, pp 39-43, 1995 Eberhart RC. And YH. Shi, “Comparison Between Genetic Algorithm and Particle Swarm Optimization”, 1998 Annual Conference on Evolutionary Programming, San Diego, 1998 Farmani R and J.A. Wright, “Self-adaptive Fitness Formulation for Constrained Optimization”, IEEE Trans. On Evolutionary Computation, Vol. 7(5), pp 445-455, 2003 Gendreau M, G Laporte and J. Potvin, “Vehicle Routing: Modern Heuristics. In: E. Aarts and JK. Lenstra (eds). ‘Local search in Combinatorial Optimization’, John Wiely & Son LTD, New York, 1997. Hamida S B and Schoenauer M. ASCHEA, “New Results Using Adaptive Segregational Constraint Handling”, Congress on Evolutionary Computation, Hawaii, USA, pp 884-889, 2002 Homberger J. and H. Gehring, “Two Evolutionary Metaheuristics for the Vehicle Routing Problem with Time Windows”, INFOR, Vol. 37, pp 297-318, 1999 Ioannou G and MN Kritikos,”A Synthesis of Assignment and Heuristic Solutions for Vehicle Routing with Time Windows”, Journal of the Operation Research, Vol. 55, pp 2-11, 2004. Kennedy J. and RC. Eberhart, “Particle Swarm Optimization”, Proceedings IEEE International Service Center, Piscataway, NJ, Vol.4, pp 1942-1948, 1995 Laporte G, “The Vehicle Routing Problem: An Overview of Exact and Approximate Algorithms”, European Journal of Operation Research, Vol. 59, pp 345-358, 1992. Montemann R., LM. Gambardella, AE. Rizzoli and AV. Donati, “A New Algorithm for a Dynamic Vehicle Routing Problem Based on Ant Colony System”, Technique Report IDSIA-23-02, IDSIA, November 2002 Onwubolu, G.C. “TRIBES application to the flow shop scheduling problem New Optimization Techniques in Engineering”, Springer, pp. 517-536, 2004.

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[22] Or I., “Traveling Saleman-Type Combinatorial Problem and Their Relation to the Logistics of Regional Blood Banking”. Ph.D. Thesis, Northwestern University, Evanston, USA, 1976 [23] OSman IH., “Metastrategy Simulated Annealing and Tabu search Algorithms for the Vehicle Routing Problem”, Annals of Operation Research, Vol. 41, pp 421-451, 1993 [24] Potvin JY. and JM. Roussear, “ An Exchange Heuristic for Routing Problems with Time Windows”, Journal of Operation Research Society, Vol. 46, pp 1433-1446, 1995 [25] Shen, Q., Jiang, J., Jiao, C.;, Shen, G. & Yu, R., “Modified particle swarm optimization algorithm for variable selection in MLR and PLS modeling: QSAR studies of antagonism of angiotensin II antagonists”, European Journal of Pharmaceutical Sciences, Vol. 22, pp.145-152, 2004. [26] Shi YH. and RC. Eberhart, “A Modified Particle Swarm Optimizer”, IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, May 4-9, 1998 [27] Shi YH. and RC. Eberhart, “ Parameter Selection in Particle Swarm Optimization”, 1998 Annual Conference on Evolutionary Programming, San Diego, 1998 [28] Storn R and K.V. Price, “Differential Evolution-A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces”, Journal of Global Optimization, Vol. 11, pp 341-359, 1997 [29] Tasgetiren, F., Liang, Y. & Sevkli, M., “Particle Swarm Optimization and Differential Evolution Algorithms for Single Machine Total Weighted Tardiness Problem”, Annals of Operations research, 2004. [30] Zhang WJ, XF. Xie and DC. Bi, “Handling Boundary Constraints for Numerical Optimization by Particle Swarm Flying in Periodic Search Space”, Congress on Evolutionary Computation (CEC), Oregon, USA, Vol 2, pp.2307-2311, 2004 [31] Zhu Qing, Limin Qian, Yingchun Li and Shanjun Zhu,” An Improved Particle Swarm Optimization Algorithm for Vehicle Routing Problem with Time Windows”, 2006 IEEE Congress on Evolutionary Computation, Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada, July 16-21, 2006.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 49-66

ISBN: 978-1-61668-527-0 © 2011 Nova Science Publishers, Inc.

Chapter 3

RELIABILITY OPTIMIZATION PROBLEMS USING ADAPTIVE GENETIC ALGORITHM AND IMPROVED PARTICLE SWARM OPTIMIZATION YoungSu Yun∗ Division of Business Administration, Chosun University, Dong-gu, Republic of Korea

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Abstract In this paper, several reliability optimization problems are considered to be optimized. For the optimization, a hybrid approach using adaptive genetic algorithm (aGA) and improved particle swarm optimization (iPSO) is proposed. For the aGA, an adaptive scheme is incorporated into genetic algorithm (GA) loop and it adaptively regulates crossover and mutation rates during genetic search process. For the iPSO, a conventional PSO is improved and it is applied to the hybrid approach. Therefore, the proposed hybrid approach takes an advantage of compensatory property of the aGA and the iPSO. For proving the performance of the proposed hybrid approach, conventional hybrid algorithms using GA and PSO are presented and their performances are compared with that of the proposed hybrid algorithm using several reliability optimization problems which have been often used in many conventional studies. Finally, efficiency of the proposed hybrid approach is proved.

Keywords: reliability optimization problem, adaptive genetic algorithm, improved particle swarm optimization, hybrid approach.

1. Introduction During the past few decades, reliability optimization problems, including redundancy allocation, reliability apportionment, and serial-parallel problems have been discussed by many researchers [3, 8, 13-14, 21-22, 25]. Furthermore, recent practical demand for process synthesis and optimization in highly reliable systems brings about the problem of extended ∗

E-mail address: [email protected]. Tel:+82 62 230 6243, Fax: +82 62 226 9664.

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system reliability design in order to locate optimal levels of component reliability and the number of redundant components in each sub-system. Unfortunately, optimizing these types of reliability systems is significantly difficult because they should consider complex search spaces. Recently, genetic algorithm (GA) has been proved to be a more effective approach for solving the reliability optimization problems [6, 19-20, 33, 35-36]. GA is basically simple, but its search ability to global optimal or near global optimal solutions is powerful. Furthermore, it is not fundamentally limited by restrictive assumptions about the search space. GA has been known to offer significant advantages over conventional algorithms. Its important characteristics include a population-based search and the principle of buildingblock combination. However, despite of the successful applications of GA to numerous reliability optimization problems, it has two major weaknesses in its application. One is the difficulty in the correct setting of genetic parameters such as crossover rate and mutation rate, and the other is to hybridize GA with conventional heuristics in order to the performance of GA is reinforced. First, the identification of the correct settings of parameters in GA is not an easy task. This is mainly because the performance of GA highly relies on the setting values of the parameters. Therefore, many studies have been performed in order to identify the correct setting values. [6, 17, 28, 31, 34] Gen and Cheng [6], in their book, surveyed various adaptive schemes using heuristics and fuzzy logic controllers (FLCs). Song et al. [28] used two FLCs for adaptive scheme: one for the rate of crossover operator and the other for the rate of mutation operator, which are considered as the input variables of GA and are taken as the output variables of the FLC. Yun and Gen [34] suggested an improved adaptive scheme using the FLC used in the ref. [28]. Mak et al. [17] adaptively controlled the rates of crossover and mutation operators with respect to the performance of GA operators in a manufacturing cell formulation problem. This heuristic scheme is based on the fact that it encourages well-performing operators to produce more efficient offspring, while reducing the chance for poorly performing operators to destroy the potential individuals, during genetic search process. Srinivas and Patnaik [29] and Wu et al. [31] also controlled the rates according to the various fitness values of the population at each generation. According to the previous studies mentioned above, Most of the studies recommend the use of adaptive scheme which can automatically regulate GA parameters such as crossover and mutation operators. Since keeping a balance between exploitation and exploration in genetic search process highly affects locating the optimal solution, it has been generally known that, during its search process, the algorithm both with a moderate and various increasing and decreasing trends in its parameter values is more efficient than the algorithm with rapid increasing or decreasing trends or the algorithm with a constant value. Therefore, much time for the correct setting of the genetic parameters can be saved, and the search ability of GA can be improved in finding a global optimal solution. Secondly, by hybridizing GA with other conventional heuristics, the hybrid algorithm can reinforce search ability rather than GA alone. In the hybrid methodologies using conventional heuristics, various studies have been performed [2, 15, 32]. Davis [2] suggested a theoretical algorithm for hybridization in order to offer the priori-knowledge to GA, Li and Jiang [15] proposed a new stochastic approach called the SA-GA-CA, which is based on the proper integration of a simulated annealing (SA), a GA, and a chemotaxis algorithm (CA) for solving

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Reliability Optimization Problems Using Adaptive Genetic Algorithm…

51

complex optimization problems. Yen et al. [32] described two approaches for incorporating a simplex method into a GA loop as an additional operator. They suggested an iterative hill climbing method which can be used for local search in GA. Recently, as a new type of hybrid algorithms, particle swarm optimization (PSO) was first proposed by Kennedy and Eberhart [11]. PSO follows from observations of social behaviors of animal, such as bird flocking and fish schooling. The theory of PSO describes a solution process in which each particle flies through the multidimensional search space while the particle’s velocity and position are constantly updated with respect to the best previous performance of the particle or of the particle’s neighbors, as well as the best performance of the particles in the entire population [10]. However, i) the PSO of Kennedy and Eberhart [11] was originally designed for global search and ii) some parameters had constant values. Therefore, PSO has a difficulty in exploring local search and in using various parameter values in the problems under consideration. Therefore, various methodologies to overcome the two weaknesses have been developed [1, 4, 5, 12, 23, 26] One is to consider a local search scheme in PSO search process so that its exploitation ability should be reinforced [4, 26-27]. Shi and Eberhart [26] introduced a new parameter, called inertia weight, into the original PSO [11] in order to reinforce the exploitation ability by local search. Simulations were performed to illustrate the influence of the inertia weight. As a result, the PSO with inertia weight in the range [0.9, 1.2] had a better performance than the original PSO. Shi and Eberhart [27] proposed an improved scheme of their previous study [26]. They used an improved inertia weight that is automatically changed according to current and maximum numbers of iterations. Eberhart and Shi [4] also improved the scheme used in the study of Shi and Eberhart [26]. They suggested that the scheme used in Shi and Eberhart [26] cannot predict whether exploitation and exploration will be better at any given time. Therefore, they proposed an improved scheme of the inertia weight that randomly varies within a given range. The other one uses adaptive schemes in PSO search process [5, 23]. Ratnaweera et al. [23] suggested that the constant values of C1 and C 2 used in original PSO [11] should be varied during its search process. They developed a new adaptive scheme, called time-varying acceleration coefficients, for automatically regulating the constant values. By the various experiment results, the suggested PSO with time-varying acceleration coefficients has shown to be significantly better performance than the original PSO. These approaches mentioned above for the hybridization of GA with other conventional heuristics usually use the complementary properties of GA and conventional heuristics in order to maintain a balance between exploitation and exploration of the hybridized approach. Therefore, recently hybridized approaches are more efficient and more robust than pure GAbased approaches or other conventional heuristics. Based on these contributions, this paper proposes a new hybrid approach with adaptive GA (aGA) and the improved PSO (iPSO). The proposed hybrid approach is used for effectively solving various reliability optimization problems. Section 2 shows a general type of reliability optimization problems. In Section 3, some concepts and logics of hybrid approach using iGA and iPSO are suggested. To compare with the efficiency of the proposed hybrid approach, numerical test results and analyses using some reliability optimization problems are presented in Section 4. Finally, Some conclusion and remarks are followed in Section 6.

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2. Reliability Optimization Problems Reliability optimization problems are usually decomposed into functional entities which compose units, subsystems, or components for the purpose of reliability. Combinatorial aspects of reliability analysis are connected to the components in serial, parallel or serialparallel. The objective of reliability optimization problems is to determine the optimum level of component reliability and the number of redundant components at each subsystem while meeting the goal with a maximum reliability or minimum cost. The optimal reliability allocation problem can be usually represented into the general form of nonlinear integer programming as follows [7]:

max .

s. t.

n

∑g j=1

ij

n

f ( x) = ∏ R j ( x j ) j =1

(x j ) ≤ bj,

i = 1, 2 , ..., m

x Lj ≤ x j ≤ x Uj : integer , j = 1, 2 , ..., n

(2-1)

(2-2)

(2-3)

where Rj(xj) is the j-th non-linear objective function represented a system reliability, gij(xj) is the j-th nonlinear function on the i-th constraint represented a system resource restraint, bi is the i-th right-hand side constant or available resource, x Lj and xUj are the lower and upper

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bounds for the integer decisions variable xj, respectively. The objective is to determine the number of redundant components at each subsystem with the known component reliability.

3. Hybrid Approach Using iGA and iPSO A hybrid approach using aGA and iPSO is proposed. For the aGA, an adaptive scheme is incorporated into genetic algorithm (GA) loop and it adaptively regulates the rates of crossover and mutation operators during genetic search process. For the iPSO, a conventional PSO is improved and it is applied to the hybrid approach. Therefore, the proposed hybrid approach takes an advantage of compensatory property of the aGA and the iPSO.

3.1. iGA Design The basic logic of adaptive scheme in GA is to enhance the performance of the search by adaptively regulating GA parameters during genetic search process. That is, whenever a new offspring is added to the population, a pointer is established for the genetic operator which generates the offspring. A check is then made to determine, if the fitness of the offspring is better or worse than its parents. The percentage of improvement or degradation is then

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Reliability Optimization Problems Using Adaptive Genetic Algorithm…

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recorded, and this record is reserved for later adjustments of the occurrence rates of GA parameters. For the aGA, adaptive crossover and mutation operators are developed. Therefore, the occurrence rates of the two operators are adaptively regulated during genetic search process. For this scheme, the concept of Mak et al. [17] is used for aGA. They employed the rate of the average fitness values of parent and offspring at each generation in order to construct adaptive crossover and mutation operators. The procedure of the adapting strategy is as shown in Figure 1. Procedure: Adaptive scheme for crossover and mutation operators begin if ( f par size (t ) / f off size (t )) ≥ 10% then −

−

pC (t + 1) = pC (t ) + 0.01 , p M (t + 1) = p M (t ) + 0.005 ;

if ( f par size (t ) / f off size (t )) ≤ 10% then −

−

pC (t + 1) = pC (t ) − 0.01 , p M (t + 1) = p M (t ) − 0.005 ;

if − 10% < ( f par size (t ) / f off size (t )) < 10% then −

−

pC (t + 1) = pC (t ) , p M (t + 1) = p M (t ) ;

end end Figure 1. Procedure of Adaptive scheme.

In Figure 1, f par size (t ) and f off size (t ) are respectively the average fitness values of parents −

−

and offspring at generation t. pC (t ) and p M (t ) are the rates of crossover and mutation Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

operators at generation t, respectively. In the cases of ( f par size (t ) / f off size (t )) ≥ 10% and −

−

( f par− size (t ) / f off − size (t )) ≤ 10% , the adjusted rates should not exceed the range [0, 1] for

pC (t + 1) and p M (t + 1) .

The above-stated procedure is evaluated in all generations during genetic search process, and the occurrence rates of crossover and random mutation operators are adaptively regulated according to the result of the procedure as shown in Figure 1.

3.2. iPSO Design Since all particles in PSO can be considered as members of the population during its searching process, hybridization with GA is easily implemented. The basic implementation procedure of PSO is as follows: Step 1. (Velocity of particle): the updating scheme of the velocity of each particle is as following. vki +1 = w ⋅ vki + C1 D1 (lbest k − xki ) + C 2 D2 ( gbest − xki )

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(3-1)

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YoungSu Yun

where w = 0.5 + (rand[0,1]/2.0), C1=C2=2.0, D1=D2=rand[0,1]. lbestk is the best fitness value at k-th iteration. gbest is the best fitness value at all iteration. v k is the velocity of the i-th particle at iteration k. x ki is the position of the i-th particle at iteration k. Step 2. (Position of particle): the updating scheme of the position of each particle is as following. xki +1 = xki + vk +1i

(3-2)

As mentioned in Section 1, the original PSO uses the w, C1 and C2 of constant values. The use of a constant value makes PSO implementation difficult, since they should be varied according to the problem under consideration. Therefore, the iPSO uses the adaptive scheme developed by Fornarelli and Giaquinto [5]. The adaptive scheme for w, C1 and C2 is as follows: MaxNo − k + w2 MaxNo

(3-3)

C1 = (C1 f − C1i ) ×

k + C1i MaxNo

(3-4)

C 2 = (C 2 f − C 2i ) ×

k + C 2i MaxNo

(3-5)

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w = ( w1 − w2 ) ×

where w1, w2, C1i , C1f , C2i and C2f are the initial and final values of inertia weight, cognitive and social parameters, respectively, k is the current iteration number and MaxNo is the maximum number of iterations.

3.3. Hybrid Approach During past several years, various types of hybrid approaches using GA and PSO have been developed [9-10, 16, 24]. For the detailed-analysis of conventional hybrid approaches, the hybrid approach (GA-PSO) using GA and PSO by Kao and Zahara [10] is analyzed here. In the paper, they used 4N individuals. Among them, the 2N individuals with superior fitness values are applied to GA search process and the remaining 2N individuals with inferior fitness values are used for PSO search process. During the search process of the GA-PSO, GA uses two operators with specific rates, i.e., the crossover with 100% rate and the mutation with 20% rate. For PSO, conventional updating scheme [4] is used and it regulates the velocity and position of particles in population. This procedure is continued until a termination criterion is satisfied. The schematic representation of the GA-PSO is shown in Figure 2.

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Reliability Optimization Problems Using Adaptive Genetic Algorithm…

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Figure 2. Schematic representation of the GA-PSO.

The unique characteristics of the GA-PSO are as follows: 1) In the GA procedure of Figure 2, 100% crossover rate is used, that is, all individuals are suffered from crossover operation. By the 100% rate, however, some respective individuals with good fitness values may be dropped during crossover operation. The search speed is also slower than that by using a portion of the rate. 2) In the PSO procedure of Figure 2, the parameters (w, C1 and C2) of the velocity of particles have a constant value. In general, the parameter values in PSO implementation should be varied according to the problem under consideration, since the proper use of the parameter values highly depends on each problem. The two weaknesses of the GA-PSO mentioned above are improved by the hybrid approach proposed in this paper. First, the proposed hybrid approach uses adaptive scheme to automatically regulate crossover and mutation rates. Therefore, the rates are dynamically changed during search process. Secondly, to prevent the drop of some individuals with respective fitness values during search process, the elitist strategy in enlarged sampling space [6] is used in selection. Third, the parameters (w, C1 and C2) values of the velocity of

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YoungSu Yun

particles in PSO procedure are varied depending on the equations (3-3) to (3-5) in each iteration. Using the above improved three approaches, the implementation procedure of the proposed hybrid approach is developed in the section 3.4

3.4. Overall Procedure The overall procedure of the proposed hybrid approach is developed here. The detailed implementation procedure is as follows: Step 1. (Initialization): Generate a population of 4N size for the problem under consideration. Step 2. (Fitness evaluation and ranking): Evaluate the fitness of each 4N individuals. Rank them on the basis of the fitness values. Step 3. (Selection): Select 4N individuals by using the elitist strategy in enlarged sampling space [6]. Step 4. (aGA and iPSO procedures)

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Step 4-1. (aGA Procedure): apply GA procedure using the 2N individuals with superior fitness values. (Crossover operator): uniform arithmetic crossover operator [18] is used. (Mutation operators): uniform mutation operator [18] is used. (Adaptive scheme): apply the procedure of the adaptive scheme shown in Figure 1. Step 4-2. (iPSO procedure): apply PSO operators (velocity and position of particles) using the 2N individuals with inferior fitness values. (Velocity of particle): the updating scheme of the velocity of each particle uses the equation (3-1). (Position of particle): the updating scheme of the position of each particle uses the equation (3-2). (Improved scheme): the parameters (w, C1 and C2) of the velocity of particles are varied by using equations (3-3) to (3-5). Step 5. (New offspring): new offspring resulting from aGA and iPSO procedures are produced. They all should satisfy constraints. Step 6. (Termination condition): if either a predefined-termination condition is satisfied or global optimal solution already known is located, then all steps are stopped. Otherwise, go Step 2. All the steps are briefly summarized in Figure 3.

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Figure 3. Flow chart of the proposed hybrid approach.

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4. Numerical Examples In this section, four types of reliability optimization problems with their global optimal solutions already known are presented in order to prove the effectiveness of the proposed hybrid approach. For various comparisons with the proposed hybrid approach, several conventional algorithms are presented as shown in Table 1. Table 1. Conventional algorithm for comparison. Notation GA a-GA PSO i-PSO GA-PSO

Description Conventional GA by Gen and Cheng [6] GA with the adaptive scheme of Yun and Gen [34] Conventional PSO by Kennedy and Eberhart [11] PSO with an improved scheme by Ratnaweera et al. [23] Hybrid algorithm developed by Kao and Zahara [10]

Each algorithm including the proposed hybrid approach (aGA-iPSO) using aGA and iPSO is compared with each other using various measures of performance as shown in Table 2.

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YoungSu Yun Table 2. Measures of performance.

Notation ANI NGS CPU

Description Average number of iterations until termination condition is satisfied. Total number of getting stuck at a local optimal solution Average CPU time (unit: Sec.)

For experimental comparison under a same condition, the parameters used in each algorithm are set as follows: maximum iteration number is 2,000, population size 20, crossover rate for the GA 0.5, and mutation rate for the GA 0.05. Altogether 20 iterations are executed to eliminate the randomness of the searches in each algorithm. The procedures of each algorithm are implemented in Visual Basic language under IBM-PC Pentium IV computer with 1.2Ghz CPU speed and 2GB RAM. In Table 2, the ANI is obtained after each algorithm reaches to the given total iteration numbers (in our case, 20 times). The NGS is the total number that each algorithm gets stuck at a local optimum after the given total iteration numbers. The CPU implies the average CPU time when each algorithm reaches to the given total iteration numbers.

4.1. Test Problems We present two types of test problems often used in reliability optimization design problems. One is the series-parallel systems with several subsystems. The other is a complex system in parallel.

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4.1.1. Test Problem 1 (T-1) Test problem 1 (T-1) has three cases in series-parallel system in which a specified number of subsystems is defined. For each subsystem, there are multiple component choices which can be used and selected in parallel. In the three cases, the costs, reliabilities and weights were already known. A representative N-stage series-parallel system is shown in Figure 4 [25]. Stage 1 ･ ･ ･ Stage 2 ･ ･ ･ Stage n

Figure 4. N-stage series-parallel system.

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Reliability Optimization Problems Using Adaptive Genetic Algorithm…

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1) Case 1 The Case 1 was proposed by Tillman et al. [30]. The mathematical model is as follows: max. f ( x) =

5

∏[1 − (1 − R )

xj

j

(4-1)

]

j =1

5

s. t. g1 ( x) = ∑V j x 2j ≤110

(4-2)

j =1

g 2 ( x) =

5

∑C [x j

j

j =1

g 3 ( x) =

1 + exp( x j )] ≤175 4

(4-3)

1 exp( x j )] ≤ 200 4

(4-4)

5

∑W [ x j

j

j =1

x j ≥ 0;

integer, j =1, 2, ⋅ ⋅ ⋅, 5

(4-5)

Table 3. Constant coefficients for Case 1. j

1 2 3 .80 .85 .90 7 7 5 7 8 8 1 2 3

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Wj Vj

4 .65 9 6 4

5 .75 4 9 2

The constant coefficients for the Case 1 are shown in Table 3. The optimal solution for the Case 1 was already known as the optimal value f(x) = 0.9045 with x = {3, 2, 2, 3, 3}, g1(x) = 83.0, g2(x) = 146.125, and g3(x) = 192.48. 2) Case 2 The Case 2 was also proposed by Tillman et al. [30] as another type of the Case 1. The mathematical model is as follows: max. f ( x) =

4

∏[1 − (1 − R )

xj

j

]

(4-6)

j =1

4

s. t. g1 ( x) = ∑ C j ⋅ x j ≤ 56 j =1

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(4-7)

60

YoungSu Yun g 2 ( x) =

4

∑W

j

(4-8)

⋅ x j ≤ 120

j =1

x j ≥ 0 ; integer, j =1, 2, 3, 4

(4-9)

Table 4. Constant coefficients for Case 2. J

1 2 3 4 .80 .70 .75 .85 1.2 2.3 3.4 4.5 5 4 8 7

Rj Cj Wj

The constant coefficients used in the Case 2 are shown in Table 4 and the optimal solution was already known as f(x) = 0.9975 with x = {5, 6, 5, 4}, g1(x) = 54.8, and g2(x) = 117.0. 3) Case 3 The Case 3 was taken from Rabi et al. [22]. The mathematical model is as follows: max. f ( x) =

15

∏[1 − (1 − R )

xj

j

]

(4-10)

j =1

15

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s. t. g1 ( x) = ∑ C j ⋅ x j ≤ 400

(4-11)

j =1

g 2 ( x) =

15

∑W

j

⋅ x j ≤ 414

(4-12)

j =1

x j ≥ 0 ; integer, j =1, 2, ⋅ ⋅ ⋅,15

(4-13)

Table 5. Constant coefficients for Case 3. j Rj Cj Wj

1 .90 5 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 .75 .65 .80 .85 .93 .78 .66 .78 .91 .79 .77 .67 .79 .67 4 9 7 7 5 6 9 5 5 6 7 9 8 6 9 6 7 8 8 9 6 8 8 9 7 6 5 7

Table 5 shows the constant coefficients used in the Case 3 and the optimal solution was already known as f(x) = 0.9456 with x = {3,4,5,3,3,2,4,5,4,3,3,4,5,5,5}, g1(x) = 392, and g2(x) = 414.

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4.1.2. Test Problem 2 (T-2) This problem is an optimal redundancy allocation problem with the 10 component complex system in which xj components at stage j are in parallel [13].

Figure 5. Redundancy allocation problem with 10 components.

Figure 5 shows such a complex system which is formulated mathematically as follows: max .

f ( x) = [ R4 ( x 4 ) ⋅ R5 ( x5 ) ⋅ Ψ1 ( x) + R4 ( x 4 ) ⋅ R 5 ( x5 ) ⋅ Ψ2 ( x) + R5 ( x5 ) ⋅ R 4 ( x4 ) ⋅ Ψ3 ( x) +

(4-14)

R 4 ( x 4 ) ⋅ R 5 ( x5 ) ⋅ Ψ4 ( x) ] ⋅ R8 ( x8 ) ⋅ R9 ( x9 ) ⋅ R10 ( x10 ) 10

s.t. g1 ( x) = ∑ (C j ⋅ x j ) ≤ 125

(4-15)

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j =1

g 2 ( x) =

10

∑ (W

j

(4-16)

⋅ x j ) ≤ 10000

j =1

x Lj ≤ x j ≤ xUj : integer, j =1,",10

(4-17)

Where Ψ1 ( x) = Π R j ( x j ) + Π R j ( x j ) + Π R j ( x j ) + j =1, 2,3

j =1,7

Π R j (x j ) −

j =6 , 7

Π

j =1, 2,3,7

j =3,6

R j (x j ) − Π

Π R j (x j ) − Π R j (x j ) +

j =1,6,7

j =3,6,7

j =1, 2,3,6

Π

R j (x j ) −

j =1, 2,3,6,7

R j (x j )

Ψ2 ( x) = Π R j ( x j ) + Π R j ( x j ) + Π R j ( x j ) − j =1, 2,3

j =1,7

Π

R j (x j ) − Π R j (x j )

j =1, 2,3,7

(4-18)

j = 6, 7

j =1, 6,7

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(4-19)

62

YoungSu Yun Ψ3 ( x) = Π R j ( x j ) + Π R j ( x j ) + Π R j ( x j ) − j =1, 2,3

j =3, 6

Π

R j (x j ) − Π R j (x j )

j =1, 2,3, 6

j =3,6, 7

Ψ4 ( x) = Π R j ( x j ) + Π R j ( x j ) − j =1, 2,3

(4-20)

j =6, 7

j =6 , 7

Π

j =1, 2,3,6, 7

(4-21)

R j (x j )

R j (x j ) = 1− R j (x j )

(4-22)

x

(4-23)

R j ( x j ) = (1 − R j ) j ,

j = 1,2,",10

Table 6. Constant coefficients for T-2. j

1 2 3 4 5 6 7 8 9 10 .75 .65 .78 .85 .80 .75 .70 .75 .71 .69 3 1 4 8 6 2 3 5 2 3 250 110 350 720 690 140 300 450 200 350 1 1 1 1 1 1 1 1 1 1 6 12 4 2 3 8 6 5 6 6

Rj Cj Wj

x Lj

xUj

Table 6 presents the constant coefficients for T-2, and the optimal solution was known as f(x) = 0.9966 with x = {1, 2, 1, 1, 1, 5, 5, 5, 6, 6}, g1(x) = 103, and g2(x) = 998.

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4.2. Test Results Experiments on the T-1 and T-2 have been carried out in order to compare the performances of the conventional algorithms and the aGA-iPSO. The computation results for the cases 1, 2 and 3 of T-1 and T-2 are shown in Tables 7 and 8. Table 7. Computational results for T-1. Case 1

ANI CPU NGS

GA

a-GA

PSO

i-PSO

890 0.23 0

780 0.12 0

1,101 0.21 0

988 0.17 0

Case 2 GAPSO 650 2.24 0

aGAiPSO 450 2.13 0

GA

a-GA

PSO

i-PSO

992 0.19 0

892 0.05 0

1,201 0.19 3

1,000 0.39 0

GAPSO 803 2.55 0

aGAiPSO 797 2.05 0

Table 8. Computational results for T-1 and T-2 Case 3 of T-1

ANI CPU NGS

T-2

GA

a-GA

PSO

i-PSO GA-PSO

2,000 15.40 20

1,850 10.39 8

2,000 18.90 20

2,000 16.08 20

1,243 24.80 15

aGAiPSO 890 22.50 3

GA

a-GA

PSO

i-PSO GA-PSO

2,000 3.79 20

2,000 4.25 20

2,000 3.77 20

2,000 5.19 20

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2,000 8.46 20

aGAiPSO 1,265 7.98 0

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Reliability Optimization Problems Using Adaptive Genetic Algorithm…

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For the case 1 of T-1 in Table 7, the proposed aGA-iPSO and the other algorithms (GA, a-GA, PSO, i-PSO and GA-PSO) located the optimal solution in all trials. Therefore, the results in terms of the NGS are same in all the algorithms. However, in terms of the ANI, the proposed aGA-iPSO is the quickest and the PSO is the slowest. In terms of the CPU, the GAPSO and the proposed aGA-iPSO had almost same results and they are slower than the other algorithms (cGA, aGA-1, aGA-2 and aGA-3), which means that the GA-PSO and the proposed aGA-iPSO used the 4N individuals, but the other algorithms used the N individuals. Similar results are also shown in the case 2 of T-1 in Table 7. In terms of the NGS, all the algorithms, except for the PSO, always located the optimal solution, which implies that the PSO does not well explore the whole search space. Therefore, the performance of the PSO in terms of the ANI is inferior to the other algorithms. A comparison of hybrid approach between the GA-PSO and the aGA-iPSO shows that the latter outperforms the former because the latter has the adaptive scheme in GA and the improved scheme in PSO. However, the search speeds in the GA-PSO and aGA-PSO significantly slower than the others. As a result summarized in Table 7, all the algorithms, except for the case of the PSO in the case 2, located the optimal solutions easily, and their performances were valuable and acceptable significantly in terms of ANI, CPU and NGS. This is mainly because the cases 1 and 2 have simple problem structures and small numbers of design variables and constraints. However, the case 3 of T-1 and T-2 have much more complicated design variables and constraints. The computational results using each algorithm are shown in Table 8. For the case 3 of T-1 in Table 8, the GA, PSO, and i-PSO do not locate the optimal solution even on one trial and the other algorithms have some mistakes in locating the optimal solutions. However, the result of the aGA-iPSO is considerately superior to those of the others. This means that the use of the adaptive scheme in GA and the improved scheme in PSO can make the aGA-iPSO more robust than the others. This result also mirrors that of the ANI, that is, the average number of iterations to the termination condition in the aGA-iPSO is significantly smaller than the others. In the comparison result of T-2, all the algorithms, except for the aGA-iPSO, do not located the optimal solution at all, even though the a-GA and the i-PSO have additional search abilities such as the adaptive and the improved schemes. However, the aGA-iPSO always locates the optimal solution in all the trials. By this result, the performances of the aGA-iPSO in terms of the ANI and NGS are the best. As a result summarized in Table 8, all the algorithms (GA, a-GA, PSO, i-PSO and GAPSO), except for the aGA-iPSO, have failed to locate the optimal solution, which means that the adaptive scheme used in the aGA-iPSO is well controlling the rates of crossover and mutation operators during the genetic search processes, and the improved scheme in the aGAiPSO also well explore the whole search space.

5. Conclusion There have been many various contributions to improve the performances of GA and PSO. Especially, adaptive schemes and hybridization concepts using other optimization algorithms or techniques have been great influence on GA and PSO performances. In this paper, a hybrid approach has been proposed to improve the performances of GA and PSO. For hybridization,

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YoungSu Yun

the adaptive scheme in GA and the improved scheme in PSO are used for the proposed hybrid approach. The proposed hybrid approach, called the aGA-iPSO, has applied to several reliability optimization problems in order to prove its efficiency. For various comparisons, several conventional algorithms (GA, a-GA, PSO, i-PSO, and GA-PSO) have been presented and compared with the proposed the aGA-iPSO using some measures of performance. Finally, various comparison results have shown that most of the performances of the proposed aGAiPSO are superior to the competing algorithms. For a future study, a more reinforced scheme such as parameter-free scheme should be inserted into the GA and PSO procedures.

References [1] [2] [3] [4] [5]

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[6] [7] [8] [9]

[10] [11] [12] [13]

Clerc, A. (1999). The Swarm and the Queen: Toward a Deterministic and Adaptive Particle Swarm Optimization. Proceedings of the Congress on Evolutionary Computation, pp. 1951-1957. Davis, L. (1991). Handbook of Genetic Algorithms. New York : Van Nostrand Reinhold. Dhingra, A. K. (1992). Optimal Apportionment of Reliability Redundancy in Series Systems Under Multiple Objectives. IEEE Transaction on Reliability, 41(4), 576-82. Eberhart, R. C. & Shi, Y. (2001). Tracking and Optimizing Dynamic Systems with Particle Swarms. Proceedings of the IEEE Congress on Evolutionary Computation, 9497. Fornarelli, G. & Giaquinto, A. (2009). Adaptive Particle Swarm Optimization for CNN Associative Memories Design. Neurocomputing, 72(16-18) , 3851-3862. Gen, M. & Cheng, R. (2000). Genetic Algorithms and Engineering Optimization. New York: John-Wiley & Sons. Gen, M. & Yun, Y. S. (2006). Soft Computing Approach for Reliability Optimization: State-of-the-Art Survey. Reliability Engineering and System Safety, 91(9), 1008-1026. Hikita, M. Y., Nakagawa, Y., Nakashima, K. & Narihisa, H. (1992). Reliability Optimization of System by a Surrogate-constraints Algorithm. IEEE Transaction on Reliability, 41(3), 473-80. Holden, N. P. & Freitas, A. A. (2007). A Hybrid PSO/ACO Algorithm for Classification. Proceedings of the 2007 Conference on Genetic and Evolutionary Computation. Kao. T-T. & Zahara, E. (2008). A Hybrid Genetic Algorithm and Particle Swarm Optimization for Multimodal Functions. Applied Soft Computing, 8, 849-857. Kennedy, J. & Eberhart, R. C. (1995). Particle Swarm Optimization. Proceedings on IEEE International Conference on Neural Networks, Piscataway, NJ, USA, 1942–1948. Kim, J. Y., Lee, H. S. & Park, J. H. (2007). A Modified Particle Swarm Optimization for Optimal Power Flow. Journal of Electrical Engineering and Technology, 2(4), 413419. Kuo, W., Prasad, V. R., Tillman, F. & Hwang, C. L. (2001). Optimal Reliability Design: Fundamentals and Applications. Cambridge University Press.

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[14] Kuo, W., Lin, H., Xu, Z. & Zhang, W. (1987). Reliability Optimization with the Lagrange Multiplier and Branch and Bound Technique. IEEE Transaction on Reliability, R-36(5), 624-30. [15] Li, B. & Jiang, W. (2000). A Novel Stochastic Optimization Algorithm. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 30(1), 193-198. [16] Li, C., Cao, C., Li, Y. & Yu, Y. (2007). Hybrid of Genetic Algorithm and Particle Swarm Optimization for Multicast QoS Routing. IEEE International Conference on Control and Automation, 2355-2359. [17] Mak, K. L., Wong, Y. S. & Wang, X. X. (2000). An Adaptive Genetic Algorithm for Manufacturing Cell Formation. International Journal of Manufacturing Technology, 16, 491-97. [18] Michalewicz, Z. (1994). Genetic Algorithms + Data Structures = Evolution Program, Second Extended Edition, Spring-Verlag. [19] Mohanta, D. K., Sadhu, P. K. & Chakrabarti, R. (2004). Fuzzy Reliability Evaluation of Captive Power Plant Maintenance Scheduling Incorporating Uncertain Forced Outage Rate and Load Representation. Electric Power Systems Research, 72, 73–84. [20] Mukuda, M., Yun, Y. S. & Gen, M. (2004). Reliability Optimization Problems Using Adaptive Hybrid Genetic Algorithms. Journal of Advanced Computational Intelligence and Intelligent Informatics, 8(4), 437-441. [21] Prasad, V. R. & Kuo, W. (2000). Reliability Optimization of Coherent Systems. IEEE Transactions on Reliability, 49(3), 323-330. [22] Rabi, V., Murty, B. S. N. & Reddy, P. J. (1997). Non-equilibrium Simulated Annealing Algorithm Applied to Reliability Optimization of Complex Systems. IEEE Transactions on Reliability, 46(2), 233-239. [23] Ratnaweera, A., Halgamuge S. K. & Watson, H. C. (2004). Self-Organizing Hierarchical Particle Swarm Optimizer with Time-Varying Acceleration Coefficients. IEEE Transactions on Evolutionary Computation, 8(3), 240-255. [24] Settles, M. & Soule, T. (2005). Breeding Swarms: a GA/PSO Hybrid. Proceedings of the 2005 Conference on Genetic and Evolutionary Computation. [25] Shelokar, P. S., Jayaraman, V. K. & Kulkarni, B. D. (2002). Ant Algorithm for Single and Multi-objective Reliability Optimization Problems. Quality and Reliability Engineering, 18, 497–514. [26] Shi, Y. & Eberhart, R. A. (1998). Modified Particle Swarm Optimizer. IEEE World Congress on Computational Intelligence, 69-73 [27] Shi, Y. & Eberhart, R. C. (1999). Empirical Study of Particle Swarm Optimization. Proceedings of the Congress on Evolutionary Computation. 3, 1945-1950. [28] Song, Y. H., Wang, G. S., Wang, P. T. & Johns, A. T. (1997). Environmental/Economic Dispatch Using Fuzzy Logic Controlled Genetic Algorithms. IEEE Proceedings on Generation, Transmission and Distribution, 144 (4), 377-382. [29] Srinvas, M. & Patnaik, L. M. (1994). Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms. IEEE Transaction on Systems, Man and Cybernetics, 24(4), 656-67. [30] Tillman, F. A., Hwang, C. L. & Kuo, W. (1980). Optimization of System Reliability, Marcel Dekker, New York. [31] Wu, Q. H., Cao, Y. J. & Wen, J. Y. (1998). Optimal Reactive Power Dispatch Using an Adaptive Genetic Algorithm. Electrical Power and Energy Systems, 20(8), 563-69.

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[32] Yen, J., Liao, J.C., Lee, B. J. & Randolph, D. (1998). A Hybrid Approach to Modeling Metabolic Systems Using a Genetic Algorithm and Simplex Method. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 28(2), 173-91. [33] You, P. S. & Chen, T. C. (2005). An Efficient Heuristic for Series–Parallel Redundant Reliability Problems. Computers and Operations Research,32(8), 2117-27. [34] Yun, Y. S. & Gen, M. (2003). Performance Analysis of Adaptive Genetic Algorithms with Fuzzy Logic and Heuristics. Fuzzy Optimization and Decision Making, 2(2), 16175. [35] Yun, W. Y. & Kim, J. W. (2004). Multi-level Redundancy Optimization in Series Systems. Computers and Industrial Engineering, 46, 337–46. [36] Zhao, R. & Liu, B. (2004). Redundancy Optimization Problems with Uncertainty of Combining Randomness and Fuzziness. European Journal of Operational Research, 157, 716–35.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 67-95

ISBN: 978-1-61668-527-0 © 2011 Nova Science Publishers, Inc.

Chapter 4

CONVERGENCE ISSUES IN PARTICLE SWARM OPTIMIZATION David Bradford Jr. and Chih-Cheng Hung Southern Polytechnic State University, Marietta, Georgia, USA

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Abstract Convergence. Optimal. Exploration. These three words, or variations thereof, weave their way into many an algorithms definition of functional success; without ‘exploration’ of the problem space there can be no ‘convergence’ upon some measured ‘optimal’ solution to the problem. Particle Swarm Optimization is not exempt. This chapter will present convergence issues that PSO must face due to its very structure as a population‐based swarm whose greatest desire is to move quickly and in tandem toward a favorable solution point. It will also present methods many PSO designers have used so far to address these issues and, as well, present a few solutions that other algorithms in the optimization family have tried and are trying today. Such topics as ‘how much exploration is too much’ (controlling population, velocity, and exploration) and ‘how does one determine when convergence is necessary’ (redefining a swarms search space) will be looked at, as will ‘what is a sub‐optimal solution’ (the local optima trap and the multiple‐optimal landscape) and ‘can local optima be overcome efficiently’ (methods for resuming exploration after an optimal success). These questions and their various answers, as implemented in the various PSO designs and hybridizations mentioned, will help illuminate the nature of not only PSO algorithms but, through inference, many evolution/ population based algorithms that explore through a problem space for one or more optimal solutions. Our conclusion will present an analysis of those included PSO solutions (as applied to the convergence issues experienced) and attempt to classify their level of success in the problem domain being explored.

Introduction What is convergence? Convergence will be defined as “the approach toward a definite value” [Convergence]. Inherent in that definition is a concept of exploration when given that ‘to approach’ will mean movement from one point to another and such movement occurs from one known point to another point. This exploration (point to point movement) will be defined

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David Bradford Jr. and Chih-Cheng Hung

as “the act of searching or traveling a terrain for the purpose of discovery” [Exploration]. The terms ‘purpose of discovery’ and ‘act of searching or traveling’ are important for both convergence and exploration as they together imply a goal to achieve and a means of evaluating if the goal is yet met. If the goal is determined to be unreached then further exploration and searching is necessary, importantly, aiming toward that goal – this is optimization. Optimization will thus be defined as “choosing the best element from some set of available alternatives” [Optimization]. As can be inferred by the definition just given, a ‘best element’ must be an evaluation performed between two or more elements to decide which eventually fullfills the given purpose of being the best element in the pursuit of the goal. Another important point about convergence is not only the existence in space of a goal to achieve but that there must be ‘a thing’ not at the given goal which must engage in the act of searching for, with the purpose of achieving, that goal. This particle must evaluate its own position as compared to the goal, adjust its position toward the goal, and repeat, until eventually achieving the goal; it must optimize and explore in order to converge upon its destination. In fact, the very acts of exploring and optimizing now can be said to be equivalent to the earlier stated definition of convergence – ‘the approach toward a definite value’ restated as ‘a particle’s exploration and optimization toward its goal.’ A “particle swarm” then takes advantage of a multitude of particles’ ability to explore and optimize toward the given goal, toward convergence, by allowing the particles to communicate amongst themselves their individual solutions and then orient their own exploration and optimization toward the best solution region yet discovered by any one member of the swarm. This communication between particles is not, for a particle swarm, an instant commandment to abandon its solution – it is tempered by a particles ‘desire∗’ to search in the region of its own best solution as well as the region of the swarms best solution, in effect causing each particle to search the regions between its best solution and the swarms’ best for ever-better solutions. Communication speeds the process of convergence by allowing particles to abandon any less than optimal search areas for any more promising search areas. Given enough time, because of this communication, eventually all particles will converge (approach a definite value), to some degree, around the ‘best solution’ (the best definite value) found by any one particle.

Managing Population and Exploration Population counts (total number of particles initialized) are usually an educated guess, depending on the researchers past experience with PSO and the problem being solved, with counts varying consistently. “There is no pat answer as to the best population size to initialize a particle swarm” [Kennedy & Eberhart, 2001]. Having an effective number of particles initialized in problem space is crucial, however, for the effectiveness of the PSO technique, per: “As one particle discovers a local optimum, it becomes the “best” in its neighbors’ neighborhoods, and they too are attracted to the optimal region. As they move toward the new ∗ This desire is the result of a calculated stochastic choice.

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Convergence Issues in Particle Swarm Optimization

69

optimum, their search may uncover new regions that are even better, and they may end up attracting the first particle toward their best positions, and so on.” [Kennedy & Eberhart, 2001]

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At creation of the PSO algorithm, regardless the population count decided upon, comes the decision of the particle swarms’ topology through which its communication options are governed – to use either circle or wheel topologies. Circle, aka ring, (aka local best – see Figure 4.1) allows a particle to communicate only with its closest neighbors to determine movement∗ direction. Wheel, aka star, (aka global best – see Figure 4.2) permits communication of all particles with one ‘common’ particle in order to determine movement direction. Implicitly important for either topologic choice is each particle’s memory of its own best position/solution achieved in pursuit of the swarms’ best solution. Communication between particles therefore influences the swarms’ movement (as each particle changes position/solution) and ultimately its convergence.

Figure 4.1. a)Topologic communication in Figure 4.1 is via Circle/Ring/Local Best in that each black dot is a particle’s best and the grey dots the particle’s neighborhoods’ local best – each particle is connected to the swarms and if one ‘local best’ is better than another then the particles will ‘abandon’ the neighborhood solution that is of lesser fitness in favor of the one that represents the best. b) [Poli, et al., 2006] performed extermal interruption of this type of topographic communiction so that particle 1 may in fact be influenced by the best solution found by particles 2, 3, and 4.

Calculating a particles position/solution using topography/communication is given as:(and it is understood that this applies for each particle of the swarm in each dimension of the problem): (Adapted from [Kennedy & Eberhart, 2001] and [Suganthan, 2007]) V ← V(t-1) + ILW * (Pbest – X(t-1)) + SIW * (Tchoice – X(t-1))

(4.2.1)

X(t) ← X(t-1) + V

(4.2.2)

∗ The calculation of a solution to the given problem, with the understanding that all solutions possible exist as ‘points’ in a problem ‘landscape’ through which the particle ‘travels’ and thus movement occurs. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

70

David Bradford Jr. and Chih-Cheng Hung Where for (4.2.1): •

• •

V denotes the velocity+ of the particle (and inherently the direction of particle travel, being toward its own recent best or the swarm global best – V is usually managed by an equation to check if its value is greater than some maximum allowable value such as “ if V > Vmax or if V < - Vmax then V = either Vmax or - Vmax” (paraphrased)). V(t-1) denotes the ‘most recent velocity’ and is equivalent to the current velocity prior to this calculations outcome. ILW denotes the individual learning weight which is a randomly generated number between 0 and 1and serves to influence the particle to search more in the region of its own best solution the closer that random number is to 1.

Maximum Dispersion Measure

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Maximum Allowed Variance – if a particle best and global best are stable (converged) so that the search region of that particle is not changing (the diversity measure is fluctuating only slightly) and the diversity measure is greater than the MAV then the application of a constrictive/restrictive measure is necessary to force the particle(s) to another ‘tighter’ or ‘less dispersed’ level of convergence.

Minimum Diversity Measure

Figure 4.2. a) Topologic communication in Figure 4.2 is via Wheel/Star/Global Best in that each black dot is a particles best and the grey dot the swarms’ global best – each particle is connected to the swarms’ global best and orients is exploration after it is updated. b) The dispersion of the swarm is its two most distant particles’ separated distance. Diversity of solution is measured between a particle best solution and the topologic choices’ best solution. c) The ‘stable’ particle best and global best does not have to be one such that the particle best is in the same vicinity as the global best – a ‘stable search region can cover a large elliptical region of the problem domain.

Pbest denotes the individual particles ‘current’ best calculated position/solution. •

X(t-1) denotes the individual particles ‘most recent solution’ and is equivalent to its current solution/position in the problem domain prior to this calculations outcome.

+ Implicitly, the metaphoric distance between solutions in the landscape as calculated by a particle and controlled by the values within the velocity equation; the greater the metaphoric distance allowed a particle between solution points in the landscape then the greater the particles velocity is said to be. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Convergence Issues in Particle Swarm Optimization •

•

71

SIW denotes the social influence weight which is a randomly generated number between 0 and 1and serves to influence the particle to search more in the region of the neighborhood best solution the closer that random number is to 1. Tchoice denotes the topologic/communication choice for the swarm, either: ƒ Lbest is usually the symbolic representation of the circle/ring/local topography and denotes the neighborhoods best solution at this time-step ƒ Gbest is usually the symbolic representation of the wheel/start/global topography and denotes the best solution of the swarm at this time-step.

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And for (4.2.2): • “X(t) ← X(t-1) + V” denotes the movement of the particle from its previous time-step solution to its new and current time-step solution as restricted/influenced by V. Managing exploration of the swarm, as can be seen, is therefore dependent upon a particles calculated value to the problem given (at any given time step) and whether the solution search region explored is its best or the topologic choices best. The calculation of that ‘best’ is a static calculation (performed each iteration) and is, as presented by [Poli, et al., 2006], “an unnatural feature of PSOs” since “in natural populations, in financial markets, in other distributed computational systems, etc., there are always external influences” [Poli, et al., 2006]. They present a modification to the ring topology, affecting exploration that serves to reduce “the ability of individuals to perceive the state of their neighborhood” [Poli, et al., 2006] and allows the possibility of “interactions with individuals other than the neighborhood best” [Poli, et al., 2006] so that “leadership change can happen even if an individual with a better state has not been found in the neighborhood” [Poli, et al., 2006].Their formula, X*new = αX*s + (1- α)x*old, using a “probabilistic update rule” (γ) to determine when, and for which particle, a predefined constant external value (termed Xext) acts upon would then allow any particle X*new to be updated thusly:

or

X*new = αX*s + (1- α)x*old with probability γ

(4.2.3)

X*new = α Xext + (1- α)x*old with probability 1 – γ

(4.2.4)

(Their formula can be restated using the definitions given earlier as: X(t) ← SIW * Lbest + (1 – SIW) * X(t-1) where the topologic choice, in this case Lbest, is replaceable by their value Xext in the same probabilistic fashion.) They tested ring neighborhoods (for a population initialization count of 80) having radii “in the set {1,2,3,4,5,10,20,40} thereby going from the most extreme form of local interaction to a gbest-type of topology” [Poli, et al., 2006] where the external input probabilities per individual γ is an element in {0,0.01,0.03,0.05,0.1,0.2,0.5,1}. Results were given that pointed toward “how the swarm consensus can be deceived away from the optimal state and how exogenous sources of influence can break symmetries, modify natural search biases and even

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lead the PSO to completely ignore fitness” [Poli, et al., 2006] (see Figure 4.1). The effect on convergence is understandably a diffusive result where convergence is not guaranteed. Another method of managing exploration and affecting the population of the swarm, rather than influencing a particle’s solution calculation, is presented by [Wei, Guangbin, & Dong, 2008] with the introduction of ‘cloning-replacement’ and mutation within a wheel topology (named Elite PSO with mutation, EPSOM). To paraphrase the article, this strategy takes a percentage of particles of the swarm, to be deemed those with ‘elite’ or ‘good’ fitness, and, after a certain number of iterations, replaces the percentage of particles which are not ‘elite’ with copies of the so determined ‘good’ particles; because this cloning reduces diversity of the swarm there is the necessity for the addition of a mutation operator to prevent premature convergence on local optima. Their mutation operation equation: P'g = Pg * (1 + 0.5 * η)

(4.2.5)

where η is a random number between 0 and 1, is implemented with a probability of occurrence being “a random number in the range of [0.1, 0.3].” (Their equation can be restated using the definitions given earlier as:

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Gbest = Gbest * (1 + 0.5 * SIW)).

(4.2.6)

They simulated Rosenbrock, Rastrigin, and Griewank functions (Rosenbrock is unimodal∗ with a relatively smooth problem domain surface. The Griewank problem domain surface is bumpy with a gradual slope providing many slight local optima. The Rastrigin problem domain surface “features many steep gradients toward local optima” [Kennedy & Eberhart, 2001] - concerning its topologic communication, Eberhart/Kennedy write, “Wheels perform better than circles on Rastrigin ... might be that the buffering effect of communication through a ‘hub’ slows the population’s attraction toward the population best, preserving diversity and preventing premature convergence on local optima.” [Kennedy & Eberhart, 2001]) EPSOM results point to improved “individual quality of the swarm and accelerations the convergence” [Wei, Guangbin, & Dong, 2008]. The addition of a mutation factor by EPSOM allows better transition on complex problem domain surfaces, such as Rastrigin, precisely because there are many steep local optima nearby for the particles to ‘jump’ onto via mutation and explore. Convergence, with constriction around a smaller and smaller goal, within EPSOM is guaranteed due to the clone-replace step that constantly works to restrict the swarm as a whole to the smaller and smaller search region of the ‘elite.’ A defined stopping condition is necessary (and presumed to be a maximum number of iterations). Convergence of the particles of the swarm would understandably be swifter than non-cloned variations. As stated earlier, population initialization counts are at best an educated guess. Since PSO does not change the population count after initialization there are ample opportunities, for convergence, to modify the population using methods from other algorithms, such as EPSOM has done with a cloning and mutation factor (which is similar and certainly inspired ∗ In this chapter Unimodal means that a problem has only one solution or global optimum, conceived as a peak on the fitness landscape. Paraphrased from [Kennedy & Eberhart, 2001]

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by evolutionary algorithms). Because the population is responsible for solution-generation within the problem domain it is important to have a population large enough to sample the solutions available early in the execution of the algorithm, before restriction of the allowable search region occurs due to the pull on the swarm members toward that region of global best (see Figure 4.3). The change in ‘velocity’ is one control on the explorative ability of the swarm, another being the ability of particles to locate a set of ‘bests’ in a search region – as shown by [Poli, et al., 2006], disruption of the swarms’ ability to locate those ‘bests’ disrupts its convergence toward a best solution. When is the end of exploration?

1 B 2

4

3

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A

A

Implosive Convergence – forces focus on an ever decreasing search region using limited information without allowing exploitation due to natural particle stabilization. Due to forced constriction of the search region prior to allowing the particles the ability to search a region fully, implosive convergence is likely (for problems having multiple local optima or multimodal solutions) to settle on less than optimal answers. In the diagram the numbers designate the ever-shrinking search region delineated by the circles (the constriction stages), the two lettered dots two possible solutions. With implosive constriction the best solution, A, is never found because the particles did not ‘calculate the solution’ before search region (2) shrank and thus B is presented as the optimal solution. It may be possible to offset this difficulty by ‘over-populating’ the swarm (in hopes more particles give a better chance to calculate the solution), but, at a computational cost.

Perturbation – If, in our example for Implosive Convergence, a particle calculated a personal best solution, the black dot near A, with an allowable search region of the ‘solid-lined circle’ then the best potential solution of A is outside its ability to calculate. However, perturbation works to expand the particles allowable search region (and forces the particle to exploit it) which, in this given case, expands (the dashed-line circle) to include the best potential solution thus giving the particle the chance to calculate that solution A (with hopefully enough time allotted to do so before any restrictive calculation takes place). (The illustration here does not show nor link the search regions to whatever/wherever the current global best solution is in this example problem domain.)

Figure 4.1. The search region, as illustrated above, constricts from circle 1 to circle 2 to circle 3 to circle 4 – without enough time to search each region the optimal solution of that region (and perhaps of the problem domain entirely) could be overlooked; however, with an increase in population it may be possible to sample more of each search region prior to shrinkage.

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Managing Velocity and Search Region Exploration (without mutation and cloning as used by EPSOM) is further managed by a velocity maximum which restricts the particle to a particular search region1 of the problem domain (see Figure 4.4) around its and the swarms’ best solutions. Since a swarm’s exploration is dependent upon its particles’ velocity in the problem domain and the resultant search region, a too restrictive velocity maximum can affect a swarms’ eventual convergence with respect to the given problem and ‘trap’ the swarm in local optima. Convergence of the swarm should be looked at, as done so in [Olorunda & Engelbrecht, 2008], as the swarms’ state of diversity. “The diversity of a particle swarm optimization algorithm can be defined, simply, as the degree of dispersion of the particles in the swarm” [Olorunda & Engelbrecht, 2008] (see Figure 4.2). This is an extremely important point in the discussion of exploration, optimization, and convergence. The initial search region is the problem domain as a whole for the particles of the swarm, while the given problem itself as a mathematic equation allows the particles to calculate a solution at a particular time-step, best or not. Particles will be pulled to the region of the best solution found (by the very nature of PSO communication), however, their ‘degree of dispersion’ around their best solutions will not constrict unless some function or equation changes their maximum searchable region to be smaller. The maximum searchable region is that distance beyond its best and the global best (but inclusive of the two) which is delineated by the velocity maximum described. Convergence at this ‘maximum searchable region’ can thus be looked at as stages somewhat similar to Russian-style ‘wooden nesting dolls’ (where there is a large hollow doll that you can open and find inside it another smaller doll, which you can open and find inside it another smaller doll, etc). Initial convergence of the particle swarm can be likened to the swarm attaining a degree of dispersion simulated by the largest nesting-doll-stage (when there is not a built in constriction factor to force the swarm ever ‘inward’ toward each next-smaller doll). This convergence level implies stable global best and stable particle best solution values are present and thus the search region becomes ‘fixed’ by the velocity maximum. (Illustrated and discussed in Figure 4.2.) Modification of the swarms’ velocity, and thus the particle’s search region within which it is allowed to search, leading toward convergence takes many forms; one such already presented is the EPSOM discussion earlier where the search region shrinks by means of increasing population in the region around the elite particles at the cost of particles in the less elite region. Another method for redefining the search region is the use of an inertia weight. It can be applied to the particle (using the previous formula (4.2.1)) as an adjustment that affects its velocity, adding W: V ← W * V(t-1) + ILW * (Pbest – X(t-1)) + SIW * (Tchoice – X(t-1))

(4.3.1)

Where W is defined as ((adapted) from [Kennedy & Eberhart, 2001]): W ← (Wstart – Wend) * ((Smax – iter) / Smax ) + Wend

1

(4.3.2)

The range of solutions in the problem domain that any particle is allowed to ‘sample’ from – if a solution is outside of this range then the ‘velocity’ of the particle is restrained and the solution adjusted accordingly.

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Problem Domain as defined

Global Best Solution– external to particles Particle Best Solution – particle memory; not to be confused with the particle itself! Search Region, the dashed line around the two dots and determined by velocity; this limits the region in the problem domain that the particle can explore. Larger velocity allows a larger search region ‘around the bests’. The pattern indicates just some of the possible positions the particle can sample – if any sampled positions are better than the ‘particle memory best’ then that position would become the particle best (a ‘new’ black dot replacing the current black dot, thus affecting the layout/orientation of the search region) – if a particle best is better than the global best then that would change the global best and affect the search region too. If a particle best and a global best are ‘stable’ (limited fluctuations) through a number of iterations then the term applied in this chapter is ‘converged’. Exploitation takes place only during this ‘converged’ level.

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Dispersion Measure of just this particle best and the global best.

Diversity Measure between b Figure 4.3. The dotted lined oval is a defined search region (as constrained by the velocity maximum) in which the particle is allowed to calculate, and is the location of the particle’s best solution and the swarms’ best solution.

And (4.3.2) is defined: • • • •

Wstart is defined as the initial value of the inertia weight (a value such as 0.9) Wend is defined as the ending value of the inertia weight (a value such as 0.4) Smax is defined as the maximum number of allowable time steps (iterations) iter is defined as the current time step (iteration) number

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In this equation (4.3.1) there is no need for a velocity maximum to limit the particles allowed search region. The power of W is that it is expected to decrease over time, thus restricting the particles allowable search region as the algorithms iterations (time steps) proceed; if W did not change then its effect would be similar to the velocity maximum previously mentioned (which serves only to keep the particles in the swarm at a particular ‘convergence level/depth’ as illustrated by the nesting dolls metaphor). So as the ‘inertia weight value’ linearly decreases (with each time step) the particles search region gets smaller, encouraging ever tighter convergence ‘levels’ – in fact, by its very design, every time step creates a new and more restricted/constricted search space for the swarm. In [Chen, Min, Jia, & Huang, 2006] the inertia weight W is modified to become ‘self active’ (meaning “the inertia weight is updated according to the convergence rate of the search process related to the optimized function” [Chen, Min, Jia, & Huang, 2006]), to be known herein as SAPSO. The convergence rate in SAPSO is assumed to be “the average fitness and current progress step of the whole swarm” [Chen, Min, Jia, & Huang, 2006] respectively A(t) (the current average fitness of the swarm as a whole, at each time step) and Δ(t) (the current progress step), where t is the current time step of the algorithm, with Δ(t) as: Δ(t) ← A(t) –A(t)(t-1)

(4.3.3)

This value of Δ(t) is then used in (4.3.2) (the calculation of inertia weight), replacing the Smax calculation phrase with “eΔ(t)/Δ(t-1)” and Wstart and Wend become fixed values (still within the same probable values given in (4.3.2)) of Wmax and Wmin expressed:

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W ← (Wmax – Wmin) * eΔ(t)/Δ(t-1) + Wmin

(4.3.4)

SAPSO results are presented, as executed over the functions for Sphere, Rosenbrock, Griewank, and Rastrigin (and as compared to the ‘plain’ inertia weighted PSO), which proclaim it to have “significantly faster convergence” [Chen, Min, Jia, & Huang, 2006] Convergence is moderated (so that the possibility of becoming trapped in local optima is reduced) because at the beginning of the iterations … Δ(t) is far less than Δ(t-1), so W is close to Wmax which therefore causes the SAPSO algorithm to behave similarly, initially, to (4.3.2). It’s as the swarm as a whole begins to pull its members toward a ‘best’ solution that SAPSO takes over, causing a rapid constriction of the dispersion of the particles (see the discussion in Figure 4.3) and thus a rapid convergence upon a definitely selected value. Also able to affect a swarms’ dispersion/diversity and thus constrict the convergence of the swarm by narrowing its search space is a method using constriction coefficients, first expounded by Clerc and Kennedy in 2000. The simplest constriction coefficient, “Type 1”, modifies the inertia weight formula by applying a formula not just to the previous velocity but to the entire right side of the equation, thusly: V ← χ * (V(t-1) + ILW * (Pbest – X(t-1)) + SIW * (Tchoice – X(t-1)))

(4.3.5)

Where χ is defined as ((adapted) from [Kennedy & Eberhart, 2001]): χ ← 2k / ( | 2 – ILW – √(SIW2 – (4 * ILW)) | )

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(4.3.6)

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and k is defined as a value between 0 and 1 with the understanding that (ILW+SIW)>4. As pointed out by [Kennedy & Eberhart, 2001]: “This constriction method results in particle convergence over time; that is, the amplitude of the individual particle’s oscillations decreases as it focuses on a previous best point” [Kennedy & Eberhart, 2001] (where they are making use of the conclusion by Ozcan/Mohan that “the particle does not ‘fly’ through the search space, but rather ‘surfs’ it on sine waves” when they mention amplitude and oscillations). This type of solution does not require a velocity maximum to control convergence depth since “if the swarm is still exploring various optima and a particle’s own previous best is in a different region from the neighborhoods previous best … then the particles cannot converge … until members of a neighborhood begin to cluster in the same optimal region” [Kennedy & Eberhart, 2001]. In the above methods of constriction, the dispersion of the particles in the problem domain, is modified by the adjustment (of some type of weight/constriction which itself may be dynamic in nature) of either the first term of the velocity equation or the entire equation. [Latiff & Tokhi, 2009] include the diversity of the particles in their modification of the inertia weight. They also modify the particles own ILW so that, as the particles converge toward the global best, the particle is less likely to include its own previous best in its velocity calculation and thus forces the particle toward the swarm best. This formula is dependent upon a fixed number of iterations that, as the number of iterations reaches its maximum, forces the swarm to converge on any global best discovered; the equation is: SF ← .05 * (spread + deviation) •

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•

(4.3.7)

Spread is defined as the “the maximum distance between particles in the best and worst positions with respect to the function fitness” whose value is thus taken to be “the distribution of the particles in the search space” [Latiff & Tokhi, 2009]. Deviation is defined as the “distance of average particle position from the global best particle” whose value is thus taken to be “the deviation of particles from the global best position” [Latiff & Tokhi, 2009].

This is then used to modify the inertia weight (along with the current iteration count (iter) and the maximum number of iterations the algorithm is allowed to perform (iter_max)) accordingly: W ← exp( -iter /(SF * iter_max)

(4.3.8)

And, as mentioned the particle has its ILW modified as the algorithm reaches is maximum, thus allowing for all particles to converge on the global best; the calculation is: ILW ← 2 * (1 – (iter / iteration_max)

(4.3.9)

As is evident (and presented by [Latiff & Tokhi, 2009]) the ILW and the SIW are initially 2, with the ILW value being modified as indicated/presented above). This requires no velocity maximum for the same reason the original inertia weight PSO modification does not precisely because inertia weight is the method used to control swarm dispersion during the

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iterations allowable the swarm. It is the modification of the ILW that improves/increases the speed at which swarm dispersion/diversity decreases (and thus convergence depth increases). [Latiff & Tokhi, 2009] take advantage of work done by Ratnaweera, Halfamuge, and Watson in “Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients” (in which these three values decrease linearly: the inertia weight, the ILW, and the SIW, still making use of the same iteration limitations as presented) and incorporates the ideas of swarm diversity/dispersion as presented in [Olorunda & Engelbrecht, 2008]. These methods, as presented, lead to a metaphor of convergence not as a ‘nested doll’ but more as a balloon which, at initialization is fully inflated (at its maximum inflated diameter symbolizing the problem domain as a whole) but, as the algorithm iterates, releases air from the balloon which decreases its diameter (which thus constricts/restricts the search region as well as particle velocity), slowly shrinking at first but shrinking at an ever increasing rate until the balloon is deflated and the algorithm is forced to stop (reached its maximum iteration count and minimum search region within the problem domain)( see the discussion in Figure 4.3 about ‘implosive convergence’). Returning to [Olorunda & Engelbrecht, 2008], their effort to quantify exploration/exploitation∗ raises the question here of ‘what is exploration versus exploitation’ and what does one or the other mean for convergence. When the swarm’s diversity is at its greatest is initialization and thus the swarms’ first exploration mode is at that stage. The particles will self-restrict their search region as their own best solution and the global best solution ‘move’ about the current problem domain, exploring it. Exploration’s outcome is the particle swarm stabilizing its members around a certain pair of solution ‘points’, indefinitely sampling values around those solution points, which in fact is exploitation (remembering that without the influence of some constriction/restriction measure to affect the velocity maximum and thus the search region the particles would then be dispersed at the ‘largest nesting doll’ dispersion/convergence level described earlier and discussed in Figure 4.4). With the change or addition of some constriction/restriction measure (a change in velocity, or some other calculation, which redefines the search region), when the swarm is at a particular dispersion stage/level, or ‘nested-doll’, and is disrupted by that new constriction of its search region, then there again begins exploration - up until that moment when particles begin stabilizing and thus the start of exploitation (which has again optimized the swarm around certain solution points). Exploitation so defined is an optimization/sampling of the particles around two solution points in the problem domain and is exemplified by the nesteddoll, or staged/levels, of convergence. However, continual optimization (through continuous recalculation/shrinking of the search region) is continual convergence which is continual exploration leaving no time for exploitation of a particular ‘level’ or ‘stage’ of solution. This is because as the search region continually shrinks there are solutions that are not sampled and, as the shrinking/constriction continues, cannot be sampled because they too quickly become solutions outside of the allowable search region – such continuous recalculation of the search region does not allow exploitation of any of the solutions discovered, instead forcing group convergence. In [Maeda, Matsushita, Miyoshi, & Hikawa, 2009] the idea of “simultaneous perturbation” (from work by J.C. Spall) is presented, which replaces the ‘previous particle ∗ Defined in this chapter as “developed or used to greatest advantage”, from exploit as to “make good use of” (Wordnet Search).

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velocity’ with a formula to force exploitation of a solution area discovered in the search region. The revision takes the following form:

V ← χ * ((α * γ) + ILW * (Pbest – X(t-1)) + SIW * (Gbest – X(t-1)))

(4.3.10)

where α is a positive constant and γ: γ ← (f(X(t-1) + pvt) – f(X(t-1))) / pvti

(4.3.11)

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where pv “denote a perturbation vector and its i-th element that is randomly generated under the following condition” [Maeda, Matsushita, Miyoshi, & Hikawa, 2009]: E(st) = 0

(4.3.12)

E(st1i * st2j) = δij * δt1t2

(4.3.13)

“where E denotes the expectation δij is the Kronecker’s delta. Then, γ is an estimator of the gradient of the function” [Maeda, Matsushita, Miyoshi, & Hikawa, 2009]. Introducing the simultaneous perturbation factor in the place of the previous velocity forces the particle to remain in the region of its best solution longer, exploiting the search region of that particular solution much more thoroughly; remember that a particle velocity allows a particle to cycle about its best solution by exceeding that solution by its ‘maximum velocity’ factor – simultaneous perturbation reigns in the velocity of the particle thus limiting its allowable search region to that of the local solution it has found (instead of including the global best too, it excludes it). This forced exploitation of a solution region is an attempt to overcome the possibility of missing a solution even better than the one found based on the idea that, to quote: “since the particle swarm optimization itself does not have a capability searching the neighbor of the present position, and it may miss the optimal point by the present point” because “the best individual means the best candidate for a global optimal …” and “… a possibility that the particle is close to the global optimal is high” [Maeda, Matsushita, Miyoshi, & Hikawa, 2009] (See the discussion in Figure 4.3 about perturbation.) This type of exploitation allows refining a best solution found in a local region by a particle to become the most likely ‘best’ best solution of that region. If a better solution is found by another particle in another region then, as the particle exploits that solution to find the ‘best’ best solution of that region, the swarm is free to move in its standard fashion toward that solution. Therefore, those regions of most promise are more thoroughly exploited by shrinking the search region of each individual particle around its own best solution region without affecting the swarm search region as a whole (while maintaining standard PSO communication/pull toward the global best). Convergence to the ‘best’ best solution is more probable since each particle-best is more thoroughly exploited. This returns us to the initial definitions presented in the introduction to this chapter for convergence (approach toward a definite value), exploration (the act of searching), and optimization (choosing the best element from some set of available alternatives). With the added ideas of dispersion/diversity and search region restriction/constriction added to those definitions we now can define the swarm as having two means of convergence: a ‘continual approach toward a definite value (constantly shrinking the search region)’ or a ‘staged

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approach to a definite value (where the problem domain shrinks only to a point)’, where the staged approach allows exploitation and the continual approach is solely exploration. How do these types of convergence work for suboptimal/local solutions, escaping those solutions if necessary, and for multimodal solutions?

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Managing Optima and Resuming Exploration Some problems having local optima have already been discussed, such as Rastrigin. The PSO algorithm is known to have difficulties in convergence in these types of problems precisely because it, as a swarm, desires only to focus on the best solution thus far found. In a ‘bumpy’ landscape the best solution so far found may not be the best solution of the problem domain. Another unique feature of a problem domain may be the existence of multiple optimal solutions (the square root of 25 has answers of 5 and -5, for example); in these cases it may be possible for the PSO algorithm to converge on only one of the optima if a global best solution for the swarm is discovered by a single particle in advance of any other particle discovering another equal global best optima (though [Kennedy & Eberhart, 2001] have said concerning multi modal problem domains “particles that are halfway between two optima have twice as many good directions to turn, not twice as many opportunities to fail” [Kennedy & Eberhart, 2001] – their statement implies that two particles have already achieved the global answers and each pulls a number of neighboring particles toward it; in the event only one particle has found a global best the likelihood of the swarm subsequently finding another global best decreases greatly as time elapses). Many methods of delaying convergence, or for re-expansion of the swarm, have been presented in research. Once such, which addresses the issue of local optima and re-expansion of the swarm, is ‘Catfish’ by [Chuang, Tsai, & Yang, 2008] using wheel topology and inertia weight constriction; “The catfish effect can be summed up by the statement that an active few members of a group can stimulate the performance of a group as a whole” [Chuang, Tsai, & Yang, 2008]. The ‘catfish’ method implements a time measure applied to the swarms’ global best such that if the global best does not change by a predefined amount through a number of iterations then the catfish method destroys 10% of the swarm (those with the worst fitness) and creates new particles “randomly positioned at extreme points of the search space” [Chuang, Tsai, & Yang, 2008] in the hopes that the new particles will find a better global solution that in turn pulls the rest of the swarm away from the local optima in which it has become trapped. This is somewhat similar to EPSOM in that members of the swarm are destroyed and replaced (without EPSOM’s mutation because the new particles created via catfish are not based on nor within the current swarm search region), and similar too in that a stopping condition is necessary, presumably some number of iterations. ‘Catfish’ method presents results that indicate “introduction of new individuals into a group (catfish particles) has a significant effect on the swarm” [Chuang, Tsai, & Yang, 2008] and that their method “not only accelerates the search, but also improves the solution quality, especially when Rastrigin and Griewank benchmark functions are used” [Chuang, Tsai, & Yang, 2008]. The presented method of re-expanding the swarm when there is belief that the solution presented by the swarm is a local optima and not the global optima can be seen to actually be a method of introducing another swarm altogether into the problem domain (a smaller swarm that shares the ‘originating swarm’s global best value) and allowing it to

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propagate through the solutions it finds thus influencing the original swarm only if a better global best is presented. Another manner of escaping local optima is modifying the particles velocity. In the abstract for [Fan & Jiang, 2008] is presented the idea of adding a chaotic search value to velocity so that particles “are accelerated to overstep the local extrema” and thus occurs ‘reexpansion’ of the search region (and the increased possibility of locating another, better, solution). For multiple optimal solutions (multimodal) one method of providing a means to effectively explore the problem domain is presented by SMPSO (speed-constrained Multiobjective PSO) [Nebro, Durillo, Garcia-Nieto, Coello Coello, Luna, & Alba, 2009] and the use of a ‘leaders archive’ (using the wheel topology with an inertia weight - additionally, SMPSO uses ‘turbulence’ aka mutation, and also a further restriction on particle velocity). The restriction on particle velocity is presented as useful in keeping the particles searching within the solutions so far found (instead of allowing a wider range of solution values than is necessary) and is (where upper_limit and lower_limit are the prescribed allowable ranges): Delta ← (upper_limit – lower_limit)/2

(4.1)

and is employed to limit velocity (once it is normally calculated with its inertia weight) by:

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V ← Delta (if V>Delta) or V ← -Delta (if V 0, such The symbol wk = O(z k ) indicates that there exist an index k k k k k ¯ that the real sequences {w } and {z } satisfy |w | ≤ a|z |, for any k > k. In Section 2.-3. we describe a generalized PSO iteration. Then, the Sections 4.-5. introduce both the theory and the motivations for our modification of PSO iteration. Finally, in Sections 6.-7. we describe some new algorithms and we carry out the related convergence analysis.

2.

A Generalized Scheme for PSO

PSO solves (1.1) by iteratively generating subsequences of points in IR n , which possibly approach a solution. At the current step of any subsequence, the next point both depends on the position of the current point in the subsequence, and the information of f (x) provided by the other subsequences. In particular, we use the subscript j to indicate the subsequence, while the superscript k indicates the iterate in the subsequence. We preliminarily consider the following PSO iteration for any k ≥ 0 (see [1]): h i vjk+1 = χ wk vjk + cj rj ⊗ (pkj − xkj ) + cg rg ⊗ (pkg − xkj ) , (2.1) = xkj + vjk+1 , xk+1 j

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where j = 1, ..., P is used to indicate the j-th particle (i.e. the j-th subsequence of points), P is a finite integer, and the vectors vjk and xkj are n-real vectors, which respectively represent the speed (i.e. the search direction) and the position of the j-th particle at step k. With rj ⊗ (pkj − xkj ) (similarly with rg ⊗ (pkg − xkj )) we indicate that every entry of the vector (pkj − xkj ) must be multiplied by a different value of the coefficient rj . Finally, the n-real vectors pkj and pkg satisfy the condition f (pkj ) ≤ f (x`j ),

for any ` ≤ k, pkj ∈ {x`j }

f (pkg ) ≤ f (x`j ),

for any ` ≤ k and j = 1, . . . , P, pkg ∈ {x`j },

(2.2)

while χ, wk , cj , rj , cg , rg are positive bounded coefficients. Observe that pkj represents the ‘best position’ in the j-th subsequence, while pkg is the ‘best position’ among all the subsequences. We recall that the choice of the coefficients is often problem dependent, though several standard values for them were proposed in the literature [3, 20, 27]. Anyway, general rules for assessing the coefficients in (2.1) are still under investigation. Of course relations (2.1) include the case where either the inertia coefficient wk or the constriction coefficient χ are used. Moreover, without loss of generality we assume that rj and rg are uniformly distributed random parameters with rj ∈ [0, 1] and rg ∈ [0, 1]. One possible generalization of (2.1) is obtained by assuming that possibly the speed vjk+1 depends on the P vectors pkh − xkj (see also [16]), h = 1, . . ., P , and not only on the vectors pkj − xkj , pkg − xkj [1]. The resulting new iteration for the j-th particle is given for

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any k = 0, 1, ... by "

vjk+1 = χkj wjk vjk +

P X

#

ch,j rh,j (pkh − xkj ) , (2.3)

h=1

= xkj + vjk+1 . xk+1 j Observe that in (2.3) the coefficients ch,j and rh,j both depend on the particle j and the remaining particles (h). It can be readily seen [2] that for each particle j, assuming that rh,j is the same for all the entries of (pkh − xkj ), χkj = χj and wjk = wj , for any k ≥ 0, the iteration (2.3) is equivalent to the discrete stationary (time-invariant) system 

 χj wj I  Xj (k + 1) =    χj wj I

−

P X

χj ch,j rh,j I h=1 ! P X 1− χj ch,j rh,j I





P X

     Xj (k) +  h=1 P   X  

h=1

χj ch,j rh,j pkh χj ch,j rh,j pkh



  ,  

h=1

(2.4) where



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Xj (k) = 

vjk xkj



 ∈ IR2n ,

k ≥ 0.

(2.5)

We observe that the sequence {Xj (k)} identifies the trajectory of the j-th particle in the state space IR2n , and it can be split into the free response XjL (k) and the forced response XjF (k) (see also [18]). In other words, for any k ≥ 0, Xj (k) may be rewritten according with (2.6) Xj (k) = XjL (k) + XjF (k), where XjL (k) = Φj (k)Xj (0),

XjF (k) =

k−1 X

Hj (k − τ )Uj (τ ),

(2.7)

τ =0

and (after a few calculations -see also [1]) 

 χj wj I  Φj (k) =    χj wj I

−

P X

χj ch,j rh,j I !

h=1 P X

1−

χj ch,j rh,j

k

I

   ,  

h=1

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(2.8)

Globally Convergent Modifications of Particle Swarm Optimization... 

 χj wj I  Hj (k − τ ) =    χj wj I 

−

P X

χj ch,j rh,j I h=1 ! P X 1− χj ch,j rh,j I

101

k−τ −1     

,

(2.9)

h=1 P X

  h=1 Uj (τ ) =  P  X 

χj ch,j rh,j pτh χj ch,j rh,j pτh



  .  

(2.10)

h=1

We remark that unlike XjF (k), the free response XjL (k) in (2.6)-(2.7) only depends on the initial point Xj (0), and not on the vector pτh , τ ≥ 0. The latter observation will be largely used to carry out our results. In particular, the next section will be devoted to report some relevant analysis on the free response XjL (k) (see also [3]).

3.

Issues on the Parameters Assessment in PSO

It is well known (see for instance [18]) that if the j-th trajectory {Xj (k)} in (2.6) is nondiverging, it satisfies the condition lim Xj (k) = lim XjF (k),

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k→∞

k→∞

j = 1, . . ., P ;

i.e. the free response XjL (k) is bounded away from zero only for finite values of the index k. Moreover, from (2.8) we have Φj (k) = Φj (1)k , for any k ≥ 0, and it was proved [2] that the 2n eigenvalues of the unsymmetric matrix Φj (1) are real. In particular, by setting for the sake of simplicity in (2.8) aj = χj wj ,

ωj =

P X

χj ch,j rh,j ,

(3.1)

h=1

we can prove [1] that the matrix Φj (1) has at most the two distinct eigenvalues λj1 and λj2 with  1/2 1 − ωj + aj − (1 − ωj + aj )2 − 4aj λj1 = 2 (3.2)  1/2 2 1 − ωj + aj + (1 − ωj + aj ) − 4aj . λj2 = 2 In addition, each of them has algebraic multiplicity n. A necessary (but in general not sufficient) condition for the j-th trajectory {Xj (k)} to be non-diverging, is provided by the following result (see also [18]), which imposes some conditions on the coefficients of PSO iteration. Proposition 1 Consider the PSO iteration (2.3). For any j ∈ {1, . . ., P } and any k ≥ 0, let rj,h be the same for all the entries of the vector (pkh − xkj ), with χkj = χj and wjk = wj .

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Suppose that for any particle j ∈ {1, . . . , P } the eigenvalues λj1 and λj2 in (3.2) satisfy the conditions |λj1| < 1 (3.3) |λj2| < 1. Then, for any j the sequence {XjL (k)} satisfies limk→∞ XjL (k) = 0. The condition (3.3) is also a necessary condition for the trajectory {Xj (k)} to be non-diverging.  We highlight that most of the typical settings for PSO parameters proposed in the literature (see e.g. [3, 27]), satisfy the condition (3.3). In the light of the results in Proposition 1, a couple of issues still arise, which deserve further consideration. 1. The hypotheses in Proposition 1 neither ensure that for a fixed j the sequence {Xj (k)} is converging, nor they guarantee that {Xj (k)} admits limit points. I.e., for a fixed j there may be diverging subsequences of {Xj (k)} even if (3.3) holds. 2. Suppose that the sequence {Xj (k)} converges for k → ∞, i.e. {Xj (k)} → Xj∗ with (see (2.5))  ∗  vj ∗  . Xj = ∗ xj Then, x∗j may fail to be a local minimum of f (x), i.e. the property f (x∗j ) ≤ f (x),

∀x s.t. kx − x∗j k ≤ ,  > 0,

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may not be satisfied. Observe that the first issue was addressed and partially investigated in [13, 21, 23]. Here we focus on the second issue above. On this purpose, we claim that under mild assumptions if the function f (x) is continuously differentiable, it is possible to modify the PSO iteration in such a way that the sequence {x11 , . . . , x1P , . . . , xk1 , . . . , xkP } admits stationary limit points for f (x), i.e. either of the following properties holds



lim inf k→∞ ∇f (xkj ) = 0 (3.4)

k limk→∞ ∇f (xj ) = 0. The next sections deal with the latter claim, which will be proved theoretically. We will give evidence that the satisfaction of condition (3.4) may be met at the expense of a reasonably larger computational cost, i.e. an increase of the number of function evaluations.

4.

Our Optimization Framework

As described in the Introduction, in the last decades many design optimization and simulation-based optimization problems have claimed for more effective and robust methods, which do not explicitly use derivatives. Meanwhile, the advances on parallel and Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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distributed computing have considerably helped to reduce the impact of the strong computational burden of challenging problems. The combination of the latter two trends has yielded a mature field of research, where efficient algorithms show both a complete convergence analysis and noteworthy performance: namely direct search methods. In the latter class we include (see [22, 13]) all the optimization methods which do not use derivatives and are simply based on “the ranks of a countable set of function values”. In particular, we deal with iterative methods in the subclass of Generating Set Search (GSS), where at each iteration a suitable set of search directions generating a cone is considered, in order to guarantee a decrease of the objective function. The pattern search methods [22] and the derivative-free methods [4, 7] are both included in GSS. The first provide convergence analysis by enforcing at each iteration a simple decrease of the objective function f (x), on suitable geometric patterns. On the other hand, the second group imposes a sufficient decrease of the objective function by relying on a local model of f (x). We highlight that the schemes described do not encompass several heuristics, which are often broadly used in the literature (see [13] and the cited references). The local convergence analysis of GSS methods may be fruitfully combined with other techniques, in order to provide globally convergent algorithms. On this purpose, examples of combined methods where evolutionary strategies and GSS schemes yield globally convergent algorithms, can be found in [8, 9, 10, 24, 25]. In particular, in the last two references PSO is combined with a pattern search framework, in order to provide methods converging to stationary points. Here, we similarly want to combine a PSO-based evolutionary scheme with a linesearch-based derivative-free algorithm, in order to provide a unified convergence analysis yielding (3.4). The latter approach is motivated by the promising performance of derivative-free methods when combined with a linesearch technique [14]. We also remark that a PSO-based approach combined with a trust-region framework was already proposed in the literature [24, 25], in order to provide methods converging to stationary points. In this section we consider the solution of the problem (1.1), by means of a modified PSO scheme, combined with a derivative-free globally convergent algorithm, based on a linesearch strategy. We study in particular some convergence properties of the sequences {xkj }, j = 1, . . ., P , under very mild assumptions on f (x). Moreover, we propose four algorithms for continuously differentiable functions, whose distinguishing feature is the generation of sequences of points, which admit stationary limit points for f (x). To the latter purpose, here we also impose additional conditions on the sequences of coefficients {χkj }, {wjk }, {ch,j }, {rh,j } in (2.3). We highlight that in accordance with formulae (2.6)-(2.7), here we impose in our analysis some conservative conditions on the forced response XjF (k) of the particle j. In particular, in order to solve (1.1) with PSO, we have to guarantee that suitable PSO parameters exist such that the sequences {xkj }, for any j, admit limit points. On this guideline, as remarked above, the Proposition 1 provides only necessary conditions to guarantee the existance of limit points for the sequences {xkj }, j = 1, . . . , P . To sum up, in order to carry on a convergence analysis for PSO, in this section we focus on two main ingredients. First we provide conditions on the PSO iteration, in such a way that a bounded and closed set L0 exists which satisfies {xkj } ⊂ L0 , for any j and k (so that for any fixed j the sequence {xkj } admits limit points in L0 ). Then, recalling that PSO is a

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heuristics and therefore the sequence {pkg } may not converge to a stationary point of f (x) on L0 , we modify as slightly as possible the PSO iteration so that either of the stationarity conditions holds lim inf k→∞ k∇f (pkg )k = 0 limk→∞ k∇f (pkg )k = 0. The resulting algorithms are modified PSO schemes, which guarantee that under mild assumptions at least a subsequence of the points {xkj } converges to a stationary point (that is possibly a minimum point) of f (x).

5.

Preliminary Theoretical Results

We are concerned with analyzing the derivative-free approach in [14], which is based on a local model of the objective function. The proposals in [14] draw their inspiration from the idea of combining the pure pattern search and derivative-free approaches. Indeed, in [14] a suitable pattern of search directions is first identified, as in pattern search methods. Then, a one-dimensional linesearch is possibly performed along these directions, as in derivativefree schemes. We report here some mild (simplified) conditions, which will be considered for generating search directions in modified PSO algorithms (see [14]).

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Proposition 2 Let f : IR n → IR, with f ∈ C 1 (IRn ). Suppose that for any k the points in the sequence {xk } are bounded. Let for any k the directions {dkj }, j = 1, . . . , n + 1, be bounded and satisfy one of the following two conditions: (a) the directions dk1 , . . . , dkn+1 form a positively spanning set of IRn , i.e. for any w ∈ IRn , there exist n + 1 coefficients ηjk ≥ 0, j = 1, . . ., n + 1, such that P k k w = n+1 j=1 ηj dj ; (b) the directions dk1 , . . ., dkn are uniformly linearly independent. Moreover, the bounded direction dkn+1 satisfies  k  2n X w1 − w`k ρk` dkn+1 = , (5.1) ¯k ξ ` `=1 where – the sequences {ρk` }, ` = 1, . . . , 2n, are bounded, with ρk` ≥ 0 and ρk2n ≥ ρ > 0, for all k; k } – given the vectors zjk ∈ IRn , j = 1, . . . , n, for any k the vectors {w1k , . . ., w2n in (5.1) are defined by  k  h = 1, 3, 5, . . ., 2n − 1,  zbh/2c+1 k (5.2) wh =   z k + ξ k dk h = 2, 4, 6, . . ., 2n, h/2 h/2 h/2 ξjk > 0

j = 1, . . . , n,

(5.3)

lim ξjk = 0

j = 1, . . . , n,

(5.4)

k→∞

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105

and ∗ the points {whk } are reordered and (possibly) relabelled in such a way that k k ) ≤ f (w2n ); f (w1k ) ≤ f (w2k ) ≤ · · · ≤ f (w2n−1

∗ there exist constants c1, c2 > 0 such that the sequences {zjk } and {ξjk } satisfy max {ξjk }

j=1,...,n

min {ξjk }

≤ c1,

(5.5)

j=1,...,n

kzjk − xk k ≤ c2ξjk ,

j = 1, . . ., n;

(5.6)

∗ the sequences {ξ¯`k }, ` = 1, . . . , 2n in (5.1), satisfy the condition min {ξjk } ≤ ξ¯`k ≤

j=1,...,n

max {ξjk }.

(5.7)

j=1,...,n

Then, the following stationarity condition holds for the function f (x) k

lim k∇f (x )k = 0

k→∞

if and only if

lim

k→∞

n+1 X

min

n

0, ∇f (xk )T dkj

o

= 0. (5.8)

j=1

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 Observe that considering the sequence {xk } in (5.8), Proposition 2 suggests that it is possible to provide necessary and sufficient conditions of stationarity. In particular, this can be accomplished by simply exploiting at any point of the sequence {xk } the objective function (through its directional derivatives), along the directions dk1 , . . . , dkn+1. Furthermore (see [14]), in Table 1 we report a derivative-free method for unconstrained minimization, which uses the results of Proposition 2, to generate sequences with stationary limit points. A full convergence analysis was developed for the Algorithm DF-0a and the following conclusion was proved (see [14] Proposition 5.1) Proposition 3 Suppose the directions dk1 , . . . , dkn+1 satisfy the Proposition 2. Consider the sequence {xk } generated by the Algorithm DF-0a and let the level set L0 = {x ∈ IR n : f (x) ≤ f (x0 )} be compact. Then we have lim inf k∇f (xk )k = 0. k→∞

(5.9) 

Observe that the condition (5.9) is met only asymptotically; nevertheless, in the practical application of Algorithm DF-0a, a stopping condition occurs when α ¯k at Steps 2 and 3 becomes too small. We also note that at Step 4 we can possibly choose xk+1 ≡ y k , since the convergence analysis does not require f (xk+1 ) < f (y k ). Furthermore relation (5.9) may be consistently strengthened by adopting a different strategy at Step 2 of the Algorithm DF-0a. In particular, we highlight that at Step 2 just one direction of sufficient decrease

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Emilio F. Campana, Giovanni Fasano and Daniele Peri Table 1. The derivative-free Algorithm DF-0a (see [14]).

Step 0.

Set k = 0; choose x0 ∈ IR n , set α ¯ 0 > 0, γ > 0, θ ∈ (0, 1).

Step 1.

If there exists y k ∈ IRn such that f (y k ) ≤ f (xk ) − γ α ¯ k , then go to Step 4.

Step 2.

If there exists j ∈ {1, . . ., n + 1} and an αk ≥ α ¯k such that f (xk + αk dkj ) ≤ f (xk ) − γ(αk )2 , ¯k+1 = αk and go to Step 4. then set y k = xk + αk dkj , set α

Step 3.

¯k and y k = xk . Set α ¯k+1 = θα

Step 4.

Find xk+1 such that f (xk+1 ) ≤ f (y k ), set k = k + 1 and go to Step 1.

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for the objective function is sought. On the contrary, if we modify (reinforce) Step 2 and consider an exploitation of f (x) along all the directions in the set {dk1 , . . . , dkn+1}, we obtain the Algorithm DF-0b in Table 2. The following proposition was also proved in [14] and provides a stronger result with respect to Proposition 3. Proposition 4 Suppose the directions dk1 , . . . , dkn+1 satisfy the Proposition 2. Consider the sequence {xk } generated by the Algorithm DF-0b and let the level set L0 = {x ∈ IR n : f (x) ≤ f (x0 )} be compact. Then we have lim k∇f (xk )k = 0.

k→∞

(5.10) 

Observe that the stronger result is obtained at the cost of a more expensive Step 2, where the linesearch procedure and the cyclic use of all the directions dk1 , . . . , dkn+1 , may increase the number of function evaluations required to detect the stationary point. The use of the procedure LINESEARCH() is aimed to determine the smallest possible steplength αkj , such that a sufficient decrease of f (x) is guaranteed. We recall that the Propositions 3 and 4 provide only local convergence properties for the objective function f (x), similarly to any gradient method for continuously differentiable functions. In other words, starting from any initial point x0 ∈ IRn , as long as the set L0 is compact, a stationary point which is possibly only a local minimum is asymptotically approached.

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Table 2. The derivative-free Algorithm DF-0b in [14].

Step 0.

Set k = 0. Choose x0 ∈ IRn and α ¯ 0j > 0, j = 1, . . ., n + 1, γ > 0, δ ∈ (0, 1), θ ∈ (0, 1).

Step 1.

Set j = 1 and y1k = xk .

Step 2.

If f (yjk + α ¯ kj dkj ) ≤ f (yjk ) − γ(α ¯kj )2 then ¯kj , yjk , dkj , γ, δ) and set α ¯k+1 = αkj ; compute αkj by LINESEARCH(α j else set αkj = 0, and α ¯k+1 = θα ¯kj . j k Set yj+1 = yjk + αkj dkj .

Step 3.

If j < n + 1 then set j = j + 1 and go to Step 2.

Step 4.

k ), set k = k + 1 and go to Step 1. Find xk+1 such that f (xk+1 ) ≤ f (yn+1

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LINESEARCH(α ¯kj , yjk , dkj , γ, δ): n o Compute the steplength αkj = min α ¯ kj /δ h , h = 0, 1, . . . such that f (yjk + αkj dkj ) ≤ f (xk ) − γ(αkj )2,

f

6.

αkj k d yjk + δ j

!



≥ max f (yjk + αkj dkj ), f (yjk ) − γ

αkj δ

!2  .

New Algorithms

Now we want to couple the PSO scheme described in Section 2. with the algorithms in Tables 1 and 2, in order to possibly obtain new methods endowed with both local convergence properties and global strategies of exploration. In particular, we use the heuristic exploration of PSO to provide a global information on f (x), then a derivative-free scheme is used to enforce the local convergence towards stationary points. On this guideline, the most obvious way to couple PSO and derivative-free schemes

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ˆ ≥ 0. Then, is by performing the PSO iteration in Section 2. up to the finite iteration k we could apply either Algorithm DF-0a or Algorithm DF-0b after setting the initial point ˆ (see Tables 1 and 2) x0 = pkg . I.e., the local convergence is carried on starting from the ˆ Unfortunately, the latter strategy is a blind sebest point detected by PSO up to step k. quential application of two different algorithms, which does not join the advantages of the two approaches. On the contrary, we want to consider at once both the exploitation (the local strategy) and the exploration (the global strategy) of the objective function, at any step of a new scheme. More explicitly we consider a PSO-type scheme, which provides the investigation of a global minimum over IRn , while retaining the asymptotic convergence properties of a (local) derivative-free technique. We propose at first the Algorithm DF-1a in Table 3. It is a derivative-free method which uses for any iteration k the directions dk1 , . . . , dkn+1 , described in Proposition 2. We can prove the following result (see also [14]). Proposition 5 Consider the Algorithm DF-1a. Suppose the directions dk1 , . . . , dkn+1 and the sequences {zjk }, j = 1, . . . , n, satisfy the hypotheses of Proposition 2. Let the level set L0 = {x ∈ IR n : f (x) ≤ f (x0)} be compact. Then, the Algorithm DF-1a generates the sequence of points {xk } such that lim inf k∇f (xk )k = 0.

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k→∞

(6.1)

Proof. Observe that the Algorithm DF-1a and the Algorithm DF-0a differ only at Step 1 and Step 4. In particular, observe that the Algorithm DF-1a is obtained from the Algorithm DF-0a, where the vectors y k at Step 1 and xk+1 at Step 4 are computed by means of a PSO method. Thus, the convergence properties of the sequence generated by the Algorithm DF-1a follow straightforwardly from Proposition 3.  We remark that the Steps 1 and 4 of the Algorithm DF-1a include the global exploration by using PSO. On the other hand the Steps 2 and 3 are substantially the same of the Algorithm DF-0a. In a very similar fashion we can also couple the Algorithm DF-0b with a PSO method. Consequently, a conclusion as in Proposition 4 trivially holds for the resulting scheme (i.e. the condition (6.1) is reinforced giving condition (5.10)). Another proposal to join PSO-type schemes and the linesearch-based derivative-free technique in Table 1, is the Algorithm DF-2a reported in Tables 4-5. At Step k the Algorithm DF-2a exploits the function f (x) in a neighborhood of the point xk , along the directions dk1 , . . . , dkn+1 . Then, the new point xk+1 is generated in such a way that possibly a sufficient decrease of the objective function is obtained. Observe that at Step 0 of the Algorithm DF-2a, a suitable set of n+1 search directions is used, which meet the conditions of Proposition 2. In particular, according with the guidelines of Proposition 2, for any k we freely set the n uniformly linearly independent directions dk1 , . . . , dkn, which exploit local information on the function. Then, the direction dkn+1 is generated by the resulting application of the modified PSO scheme in Table 5, which provides global information on the objective function. Observe that in particular, we apply here a PSO-based method with exactly n particles (nevertheless the results can be extended readily to the case of P > n

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Table 3. The derivative-free Algorithm DF-1a.

Data.

Set k = 0; choose x0 ∈ IRn and zj0 ,vj0 ∈ IRn , j = 1, . . . , P . Set α ¯0 > 0, γ > 0, θ ∈ (0, 1).

Step 1.

Set hk ≥ 1 integer. Apply hk PSO iterations considering the P particles with respective initial velocities and positions vjk and zjk , j = 1, . . . , P . Set y k = argmin1≤j≤P, `≤hk {f (zj` )}. If f (y k ) ≤ f (xk ) − γ α ¯ k , then set vjk = vjk+hk and zjk = zjk+hk , and go to Step 4.

Step 2.

If there exists j ∈ {1, . . ., n + 1} and an αk ≥ α ¯k such that f (xk + αk dkj ) ≤ f (xk ) − γ(αk )2 ,

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then set y k = xk + αk dkj , α ¯k+1 = αk and go to Step 4. Step 3.

¯k and y k = xk . Set α ¯k+1 = θα

Step 4.

Set qk ≥ 1 integer. Apply qk PSO iterations considering the P particles with respective initial velocities and positions vjk and zjk , j = 1, . . . , P . Set xk+1 = argmin1≤j≤P, `≤qk {f (zj` )}; if xk+1 satisfies f (xk+1 ) ≤ f (y k ), then set k = k + 1 and go to Step 1.

particles). Finally, we remark that at Step 4 of Algorithm DF-2a if f (xk+1 ) > f (y k ), then the PSO-based scheme (7.1)-(7.1) is substantially ineffective to improve the current iterate yk .

7.

How to Generate Search Directions for Global Convergence

In order to prove the global convergence properties of the overall method in Tables 4-5, we remark that a modified PSO scheme must be adopted. In particular we focus on the algorithm in (7.1)-(7.1), which is supposed to include (at least) n particles. In this scheme we note that the position of the particles is (partially) affected by the choice of the directions dk1 , . . . , dkn (see relation (7.1)). Moreover, assuming that for any k the directions dk1 , . . . , dkn are assigned, in Table 5 we generate the direction dkn+1 by applying the PSO scheme (7.1)(7.1) and using the sequence {zjk }. Observe that in formulae (7.1)-(7.1) the projection PB(c,ρ)(·) onto the convex compact set B(c, ρ) is introduced, with c ∈ IR n , ρ > 0, in order to guarantee that the direction

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Emilio F. Campana, Giovanni Fasano and Daniele Peri Table 4. The derivative-free Algorithm DF-2a.

Data.

Set k = 0, choose zj0 ∈ IR n , j = 1, . . ., n. Set x0 = argmax1≤j≤n {f (zj0)}. Let α ¯ 0 > 0, β1 > 0, β2 > 0, γ1 > 0, c2 > 0, θ ∈ (0, 1).

Step 0.

Set ξ1k = · · · = ξnk = β1/(k + 1)β2 . Either set dk1 , . . ., dkn+1 as in (b) of Proposition 2, or compute dk1 , . . . , dkn as in (b) of Proposition 2 and dkn+1 by using procedure PSO-gen(k; dk1 , . . . , dkn; z1k , . . . , znk ; ξ1k , . . . , ξnk ).

Step 1.

If there exists y k ∈ IRn such that f (y k ) ≤ f (xk ) − γ1α ¯ k , then go to Step 4.

Step 2.

¯k such that If there exists j ∈ {1, . . ., n + 1} and an αk ≥ α f (xk + αk dkj ) ≤ f (xk ) − γ1 (αk )2,

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then set y k = xk + αk dkj , α ¯k+1 = αk and go to Step 4. Step 3.

¯k and y k = xk . Set α ¯k+1 = θα

Step 4.

Let xk+1 and zjk+1 , j = 1, . . ., n, (possibly) satisfy (7.1)-(7.1). If f (xk+1 ) ≤ f (y k ) then go to Step 0, else set xk+1 = y k and choose zjk+1 ∈ IRn , j = 1, . . . , n. Set k = k + 1, and go to Step 0.

dkn+1 is bounded. For any step k and any particle j we have (see also (2.1) and (2.3), where without loss of generality we have replaced ‘ ⊗’ with a simple multiplication)   n h i  X   k+1 k k k k k k k  v = χ w P (v ) + c r p − P (z )  B(c,ρ) j h,j h,j h j j B(c,ρ) j j    h=1,h6=g h ii k k k+1 k r x − P (z ) , +c  B(c,ρ) g,j g,j j        k+1 zj = PB(c,ρ)(zjk ) + vjk+1 , where

   pkh = argmin`≤k f PB(c,ρ)(zh` ) ,

h = 1, . . . , n,

     xk+1 = argmin`≤k, h=1,...,n f PB(c,ρ) (zh` ) , f PB(c,ρ)(zh` + ξh` d`h ) ,

(7.1)

and PB(c,ρ) (y) indicates the orthogonal projection of vector y ∈ IR n onto the compact and convex set B(c, ρ). Due to the computational burden which may be involved in the Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Table 5. The procedure PSO-gen(k; dk1 , . . . , dkn; z1k , . . . , znk ; ξ1k , . . . , ξnk ).

Data: k; dk1 , . . . , dkn; z1k , . . . , znk ; ξ1k , . . . , ξnk ; ρ > 0, β1 > 0, β2 > 0, c ∈ IRn . k Step k: Compute the vectors w1k and w2n as

 w1k = argmin1≤j≤n f(zjk ), f(zjk + ξjk dkj ) ,  k w2n = argmax1≤j≤n f(zjk ), f(zjk + ξjk dkj ) . Compute the direction dkn+1 as dkn+1 =

(k + 1)β2 k k (w1 − w2n ). β1

projection over a convex set, we suggest to choose B(c, ρ) among the following possibilities (see Figure 1):

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• ρ ∈ IR and B(c, ρ) = {x ∈ IRn : kx − ck2 ≤ ρ, ρ > 0}, which implies that for any y ∈ IRn ,   y if ky − ck2 ≤ ρ,   PB(c,ρ)(y) = y−c   otherwise.  c+ρ ky − ck2 • ρ ∈ IR and B(c, ρ) = {x ∈ IR n : |xi − ci | ≤ ρ, i = 1, . . . , n, ρ > 0}, which implies that for any y ∈ IR n and any i = 1, . . . , n  if |yi − ci | ≤ ρ,  yi   PB(c,ρ)(y) i =  otherwise. ci + ρ sgn(yi − ci ) • ρ ∈ IRn and B(c, ρ) = {x ∈ IRn : |xi − ci | ≤ ρi , ρi > 0, i = 1, . . . , n}, which implies that for any y ∈ IRn and any i = 1, . . . , n  if |yi − ci | ≤ ρi ,  yi   PB(c,ρ)(y) i =  otherwise. ci + ρi sgn(yi − ci ) Now, let us consider the following assumption in order to prove the convergence results for the Algorithm DF-2a. Assumption 7.1 Consider the modified PSO scheme (7.1)-(7.1). Suppose the condition (3.3) holds. Let in (7.1) the coefficients ξjk , j = 1, . . . , n, k ≥ 0, be chosen as in Proposition 2. Let L0 = {x ∈ IRn : f (x) ≤ f (x0 )} be compact and let the convex set Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Figure 1. Projections over the compact convex set B(c, ρ).

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B(c, ρ) in (7.1)-(7.1) satisfy B(c, ρ) ⊇ L0 . Assume that in iteration (7.1) the sequences k } satisfy {χkj }, {wjk }, {ckh,j }, {rh,j (1)

χkj wjk = O(ξjk+1 ),

j = 1, . . ., n;

(2)

k = O(ξjk+1 ), χkj ckh,j rh,j

j = 1, . . ., n, h = 1, . . . , n, h 6= g;

(3)

k = 1 + O(ξjk+1 ). χkj ckg,j rg,j

j = 1, . . ., n.

 Note that the conditions (1), (2) and (3) in Assumption 7.1 can be readily fulfilled. Thus, the assumption on the coefficients {ξjk } is not particularly restrictive. On the other hand, also the assumption B(c, ρ) ⊇ L0 = {x ∈ IRn : f (x) ≤ f (x0)} is not strong, since our method is tailored for applications where physical bounds on the unknowns are usually easy to determine. Now, we are ready to prove the following result, which ensures that under mild assumptions we can define a globally convergent modification of the PSO scheme in (7.1)-(7.1). Proposition 6 Suppose f : IR n → IR is continuously differentiable and consider the Algorithm DF-2a. Let the level set L0 = {x ∈ IR n : f (x) ≤ f (x0 )} be compact. Assume that for any step k the directions dk1 , . . . , dkn are uniformly linearly independent and bounded, and let the Assumption 7.1 hold. If the direction dkn+1 in Algorithm DF-2a is generated by

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the procedure PSO-gen(·), then the Algorithm DF-2a generates the sequence {xk } such that (a)

(b)

and lim inf k∇f (xk )k = 0. {xk } ⊂ L0 k→∞  k k inf kzj − x k = 0, j = 1, . . ., n;   lim k→∞

  lim inf k(z k + ξ k dk ) − xk k = 0, j = 1, . . . , n. j j j

(7.2)

(7.3)

k→∞

Proof As regards (a), by the hypotheses, for any k the directions dk1 , . . . , dkn are bounded and uniformly linearly independent. Furthermore, consider the Proposition 2, along with the Step 0 and Step 4 of the Algorithm DF-2a. For any k, with the settings c1 = 1 ξjk = ξ¯`k =

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β1 (k + 1)β2 k k w2j = zj + ξjk dkj k w2j−1 = zjk k w1k , . . . , w2n k ρ` = 0 ρk2n = 1

j = 1, . . ., n, ` = 1, . . . , 2n, β1 > 0, β2 > 0, j = 1, . . ., n, j = 1, . . ., n, k ), are renamed and relabelled so that f (w1k ) ≤ · · · ≤ f (w2n ` = 1, . . . , 2n − 1,

the sequences {ξjk }, {ξ¯lk } and {ρk` } satisfy (5.3), (5.4), (5.5), (5.7) of Proposition 2. Morek computed by the procedure PSO-gen(·) are bounded, inasmuch over, the vectors w1k , w2n k as by (7.1)-(7.1) zj is bounded and dkj is bounded by the hypothesis, for any j. Now we prove that for any k both (5.6) holds and also the direction dkn+1 , generated by the procedure PSO-gen(·), is bounded. Thus, by the above choice of the coefficients ρk` , ` = 1, . . ., 2n, relation (5.1) becomes 1 k ). dkn+1 = ¯k (w1k − w2n ξ2n Now, by the choice of ρk` and ξ¯`k , for any index j the triangular inequality yields kdkn+1 k ≤

k k k kw1k − w2n kw1k − w2n kw1k − xk k kw2n k k − xk k = ≤ + , k ξjk ξjk ξjk ξ¯2n

(7.4)

where xk is the current iterate in the sequence {xk }, generated in (7.1) by the Algorithm DF-2a. Then, two cases have to be analyzed. k = z k ), or w k = z k +ξ k dk for some j (similarly Either w1k = zjk for some j (similarly if w2n j 1 j j j k = z k + ξ k dk ). In the first case we have from Assumption 7.1, formulae (7.1)-(7.1) if w2n j j j

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and the boundedness of the convex set B(c, ρ) kw1k − xk k = kzjk − xk k

= PB(c,ρ)(zjk−1 ) + O(ξjk )PB(c,ρ)(vjk−1 ) + n X

h i k−1 O(ξjk ) pk−1 − P (z ) B(c,ρ) j h

h=1,h6=g

h

ih i k−1 k k x − PB(c,ρ)(zj ) − x

O(ξjk )

+ 1+



≤ PB(c,ρ)(zjk−1 ) + O(ξjk ) + xk − PB(c,ρ)(zjk−1 ) − xk ≤ c2ξjk , c2 > 0,

(7.5)

where the first inequality follows from the relation O(ξjk )PB(c,ρ)(vjk−1 )

n X

+

h i k−1 O(ξjk ) pk−1 − P (z ) B(c,ρ) j h

h=1,h6=g

h i +O(ξjk ) xk − PB(c,ρ)(zjk−1 ) = O(ξjk ).

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k = zjk + ξjk dkj ), we have Otherwise, when w1k = zjk + ξjk dkj for some j (similarly if w2n





k k k k k k k k k k kw1 − x k = kzj + ξj dj − x k = zj − x + ξj dj ≤ c2ξjk , c2 > (7.6) 0,

where the last inequality follows from the boundedness of dkj and relation (7.5). Therefore, from (7.4)-(7.6) the direction dkn+1 is bounded. Moreover, from (7.5)-(7.6) it is readily seen that for any j we have kzjk − xk k ≤ c2 ξjk , i.e. (5.6) holds. Finally, by (7.1) the vector xk+1 is bounded for any k. Moreover, from the definition of x0 we have zj0 ∈ L0 , j = 1, . . ., n, and consequently from (7.1) {xk } ⊂ L0 . Indeed, either at Step 4 of Algorithm DF-2a the vector xk+1 is computed by (7.1)-(7.1), or it is xk+1 = y k . Since the directions dk1 , . . . , dkn+1 satisfy Proposition 2, the results of Proposition 3 hold, i.e. the Algorithm DF-2a yields the condition (7.2). As regards (b), the result follows directly by considering the relations (7.2) and (7.5)(7.6).  Remark 7.1 A straightforward set of uniformly linearly independent directions dk1 , . . . , dkn , to be used in Proposition 6, is obtained by setting dkj = ej ,

j = 1, . . ., n,

where ej is the j-th unit vector. We highlight that for any k at Step 2 of the Algorithm DF-2a, the sufficient decrease of f (x) is checked along the n + 1 directions dk1 , . . . , dkn+1 . Anyway, the first direction which satisfies the test is chosen and the cyclic check stops. Consequently, in order to use as frequently as possible the direction dkn+1 (generated by the procedure PSO-gen(·)), to update the point y k at Step 2 of Algorithm DF-2a, the search over j ∈ {1, . . ., n + 1} should preferably be started with j = n + 1.

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Remark 7.2 Observe that by (7.1) the computational cost per iteration k of the procedure PSO-gen(·) amounts to 2n function evaluations. Furthermore, item (b) of Proposition 6 shows that eventually all the particles will cluster around the stationary point detected. On the guidelines of Proposition 4, we aim at extending the results of Proposition 6 so that any subsequence of the sequence {xk } possibly converges to a stationary point. In particular, the Algorithm DF-2b in Table 6 meets the latter requirement and the following proposition holds. Proposition 7 Suppose f : IR n → IR is continuously differentiable and consider the Algorithm DF-2b, let the level set L0 = {x ∈ IRn : f (x) ≤ f (x0)} be compact and Assumption 7.1 hold. Let for any step k the directions dk1 , . . . , dkn be uniformly linearly independent and bounded. If the Algorithm DF-2b is applied, where dkn+1 is generated by the procedure PSO-gen, then we have (a)

(b)

{xk } ⊂ L0 and lim k∇f (xk )k = 0. k→∞  lim kzjk − xk k = 0, j = 1, . . . , n   k→∞

  lim k(z k + ξ k dk ) − xk k = 0, j = 1, . . . , n j j j

(7.7)

(7.8)

k→∞

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Proof The proof trivially follows by observing the correspondence of the Algorithms DF-0a and DF-0b, with the Algorithms DF-2a and DF-2b. Thus, the results of Propositions 4 and 6 yield (7.7)-(7.8). 

8.

Conclusions

In this paper we have considered four different globally convergent modifications of the PSO iteration, applied for the solution of unconstrained global optimization problems. We have proved in Propositions 5 and 6 that under mild assumptions, at least a subsequence of the iterates produced by our modified PSO methods converges to a stationary point, which is possibly a minimum point. This is a relatively strong result, if we consider that by no means the standard PSO iteration [12] can guarantee the convergence towards stationary points. In addition, the latter result is in our knowledge among the first schemes (see also [24, 25]) where a modified PSO scheme is proved to be globally convergent, i.e. either (7.2) or (7.7) holds. Moreover, this accomplishment contributes to fill the gap between the theory and the numerical performance of PSO based methods. Our conclusions have also been reinforced in Proposition 7, where any subsequence generated by the modified PSO iteration was proved to converge to a stationary point. We highlight that this stronger result implies an additional computational burden. However, the latter additional cost may be consistently reduced according with the indication reported in the Remark 7.1. Finally, considering the wide range of applications which require the use of efficient derivative-free algorithms, we guess that a new paper will be necessary to describe further

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Emilio F. Campana, Giovanni Fasano and Daniele Peri Table 6. The derivative-free Algorithm DF-2b.

Data.

Set k = 0, choose zj0 ∈ IRn , set x0 = argmax1≤j≤n{f(zj0 )}, j = 1, . . . , n. Let α ¯ 0 > 0, j = 1, . . . , n, β1 > 0, β2 > 0, γ > 0, c2 > 0, θ ∈ (0, 1), δ ∈ (0, 1).

Step 0.

Set ξ1k = · · · = ξnk = β1 /(k + 1)β2 . Either set dk1 , . . ., dkn+1 as in (b) of Proposition 2, or compute dk1 , . . . , dkn as in (b) of Proposition 2 and dkn+1 by means of the procedure PSO-gen(k; dk1 , . . . , dkn; z1k , . . . , znk ; ξ1k , . . . , ξnk ).

Step 1.

Set j = 1 and y1k = xk .

Step 2.

If f(yjk + α ¯ kj dkj ) ≤ f(yjk ) − γ(¯ αkj )2 then compute αkj by LINESEARCH(¯ αkj, yjk , dkj , γ, δ) and set α ¯ k+1 = αkj ; j

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k else set αkj = 0, and α ¯ k+1 = θα ¯ kj . Set yj+1 = yjk + αkj dkj . j

Step 3.

If j < n + 1 set j = j + 1 and go to Step 2.

Step 4.

Let xk+1 and zjk+1, j = 1, . . . , n, (possibly) satisfy (7.1)-(7.1). k k ) then go to Step 0, else set xk+1 = yn+1 and choose If f(xk+1 ) ≤ f(yn+1 n k+1 zj ∈ IR , j = 1, . . . , n. Set k = k + 1, and go to Step 0.

theoretical results, along with the numerical tests. In particular, the removal of the continuous differentiability assumption for the objective function f (x), seems the natural extension of the theory described here. On this purpose, we need to include in our approach several results from non-smooth analysis. On the other hand, the extension of our approach to bounded and linearly constrained problems, is another topic of great interest.

Acknowledgments E.F.Campana and D.Peri would also like to thank the support of the US Office of Naval Research through Dr. Ki-Han KIM (NICOP project 00014-0810957). G.Fasano wishes to thank the INSEAN research program “VISIR”, and Programma PRIN 20079PLLN7 “Nonlinear Optimization, Variational Inequalities, and Equilibrium Problems ”.

References [1] E.F. Campana, G. Fasano, A. Pinto, Dynamic analysis for the selection of parameters and initial population, in Particle Swarm Optimization, Accepted for publication on Journal of Global Optimization, 2009.

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[2] E.F. Campana, G. Fasano, A. Pinto, Dynamic system analysis and initial particles position in Particle Swarm Optimization, IEEE Swarm Intelligence Symposium 2006, Indianapolis 12-14 May 2006. [3] M. Clerc, J. Kennedy, The Particle Swarm - Explosion, Stability, and Convergence in a Multidimensional Complex Space, IEEE Transactions on Evolutionary Computation , Vol. 6, No. 1, 2002. [4] R. De Leone, M. Gaudioso, and L. Grippo, Stopping criteria for linesearch methods without derivatives, Mathematical Programming , No. 30, pp. 285-300, 1984. [5] P.C. Fourie, A.A. Groenwold, Particle Swarms in Size and Shape Optimization, Proceedings of the International Workshop on Multidisciplinary Design Optimization, Pretoria, South Africa, August 7-10, 2000, pp. 97-106 . [6] A. Griewank, Evaluating Derivatives, Philadelphia, 2000.

SIAM Frontieres in applied mathematics,

[7] U.M. Garcia-Palomares and J.F. Rodriguez, New sequential and parallel derivativefree algorithms for unconstrained minimization, SIAM Journal of Optimization, No. 13, pp. 79-96. [8] W.E. Hart, Evolutionary Pattern Search Algorithms , Technical Report SAND95-2293, Sandia National Laboratories, Albuquerque, NM, 1995.

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[9] W.E. Hart, A stationary point convergence theory for evolutionary algorithms, Foundations of Genetic Algorithms 4, Morgan Kaufmann, San Francisco, CA, 1996, pp. 325-342. [10] W.E. Hart, A generalized stationary point convergence theory for evolutionary algorithms, Proceedings of the International Conference on Genetic Algorithms, Morgan Kaufmann, San Francisco, CA, 1997, pp. 127-134. [11] J. Haslinger, R.A.E. M¨ akinen Introduction to Shape Optimization, SIAM Advances in Design and Control, Philadelphia, 2003. [12] J. Kennedy, R.C. Eberhart, Particle swarm optimization, Proceedings of the 1995 IEEE International Conference on Neural Networks (Perth, Australia) , IEEE Service Center, Piscataway, NJ, IV: 1942-1948, 1995. [13] T.G. Kolda, R.M. Lewis, V. Torczon, Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods SIAM Review Vol. 45, No. 3, pp. 385482. [14] S. Lucidi, M. Sciandrone, On the global convergence of derivative-free methods for unconstrained optimization, SIAM Journal of Optimization , No. 13, pp. 97-116, 2002. [15] B. Mohammadi, O. Pironneau, Applied Shape Optimization for Fluids, Clarendon Press, Oxford, 2001. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[16] R. Mendes, Population Topologies and Their Influence in Particle Swarm Performance, PhD Dissertation, University of Minho, Departamento de Informtica Escola de Engenharia Universidade do Minho, 2004. [17] J.D. Pinter, Global Optimization in Action. Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, Kluwer Academic Publishers, The Netherlands, 1996. [18] P.E. Sarachik, Principles of linear systems, Cambridge University Press, 1997. [19] Y. Shi, R. Eberhart, Parameter Selection in Particle Swarm Optimization, The seventh Annual Conference on Evolutionary Computation , 1945-1950, 1998. [20] J.F. Schutte, A.A. Groenwold, A Study of Global Optimization Using particle Swarms, Journal of Global Optimization , No. 31, pp. 93-108, 2005. [21] I. C. Trelea, The Particle Swarm Optimization Algorithm: Convergence Analysis and Parameter Selection Information Processing Letters , vol. 85, pp. 317-325, 2003. [22] M.W. Trosset I know it when I see it: Toward a definition of direct search methods, SIAG/OPT Views-and-News: A Forum for the SIAM Activity Group on Optimization , No. 9, pp. 7-10, 1997. [23] F. Van den Bergh, A. P. Engelbrecht, A Study of Particle Swarm Optimization Particle Trajectories Information Sciences, vol. 176, pp. 937-971, 2006.

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[24] A. I. F. Vaz, L. N. Vicente, A particle swarm pattern search method for bound constrained global optimization Journal of Global Optimization , vol. 39, pp. 197-219, 2007. [25] A. I. F. Vaz, L. N. Vicente, PSwarm: A hybrid solver for linearly constrained global derivative-free optimization Optimization Methods and Software, vol. 24, pp. 669685, 2009. [26] G. Venter, J. Sobieszczanski-Sobieski, Multidisciplinary optimization of a transport aircraft wing using particle swarm optimization Structural and Multidisciplinary Optimization, vol. 26, No. 1-2, pp. 121-131, 2004. [27] Y.L. Zheng, L.H. Ma, L.Y. Zhang, J.X. Qian, On the convergence analysis and parameter selection in particle swarm optimization, Proceedings of the Second International Conference on Machine Learning and Cybernetics, Xi’an, 2-5 November 2003.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 119-126

ISBN 978-1-61668-527-0 c 2011 Nova Science Publishers, Inc.

Chapter 6

N ONLINEAR 0-1 P ROGRAMMING THROUGH PARTICLE S WARM O PTIMIZATION U SING D ECODING A LGORITHMS Takeshi Matsui∗ and Masatoshi Sakawa Graduate School of Engineering, Hiroshima University

PACS 87.55.de Keywords: nonlinear 0-1 programming, decoding, particle swarm optimization.

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AMS Subject Classification: 87.55.de.

1.

Introduction

In general, actual various decision making situations are formulated as large scale mathematical programming problems with many decision variables and constraints. In mathematical programming problems, for the programming problems that decision variables are 0 or 1, we can theoretically get strict solution by application of dynamic programming. In particular, for nonlinear 0-1 programming problems, there are not general strict solution method or approximate solution method, such as branch and bound method in case of linear 0-1 programming problems. In such a case, a solution method depended on property in problems is proposed. Thus, Sakawa [7] proposed genetic algorithms with double string representation based on updating basepoint solution using continuous relaxed as general approximate solution method for nonlinear 0-1 programming problems. In recent years, a particle swarm optimization (PSO) method was proposed by Kennedy et al. [2] and has attracted considerable attention as one of promising optimization methods with higher speed and higher accuracy tha those of existing solution methods. Hence, Kato et al. [1] proposed revised particle swarm optimization (rPSO) method solving drawbacks that it is not directly applicable to constrained problems and its liable to stopping around local solutions and showed its effectivity. Moreover we expanded revised particle swarm optimization ∗

E-mail address: [email protected]

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method for application to nonlinear integer programming problems (rPSONLIP) [4] and showed more efficien y than genetic algorithm. And we expanded rPSO for application to nonlinear 0-1 programming problems (rPSONLZOP [5], rPSOmNLZOP [6]) and showed more efficien y than genetic algorithm and QUADZOP. However, in those researches, rPSONLZOP and rPSOmNLZOP method needs many computational time as same as genetic algorithm. In this research, we focus on nonlinear 0-1 programming problems and propose higher approximation solution method based on particle swarm optimization using the decoding algorithm than those methods.

2.

Nonlinear 0-1 Programming Problems

In this research, we consider general nonlinear programming problem with constraints as follows:  minimize f (x) = cx  subject to g(x) = Ax − b ≤ 0 (2.1)  n x ∈ {0, 1}

where c = (c1 , . . . , cn ) is an n-dimensional row vector; x = (x1 , . . . , xn )T is an ndimensional column vector of 0-1 decision variables; A = [aij ], i = 1, . . . , m, j = 1, . . . , n, is an m × n coefficien matrix; and b = (b1 , . . . , bm )T is an m-dimensional column vector.

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3.

Particle Swarm Optimization

Particle swarm optimization [2] method is based on the social behavior that a population of individuals adapts to its environment by returning to promising regions that were previously discovered [3]. This adaptation to the environment is a stochastic process that depends on both the memory of each individual, called particle, and the knowledge gained by the population, called swarm. In the numerical implementation of this simplifie social model, each particle has three attributes: the position vector in the search space, the current direction vector, the best position in its track and the best position of the swarm. The process can be outlined as follows. Step 1: Generate the initial swarm involving N particles at random. Step 2: Calculate the new direction vector for each particle based on its attributes. Step 3: Calculate the new search position of each particle from the current search position and its new direction vector. Step 4: If the termination condition is satisfied stop. Otherwise, go to Step 2. To be more specific the new direction vector of the i-th particle at time t, v t+1 i , is calculated by the following scheme introduced by Shi and Eberhart [8]. v t+1 := ω t v ti + c1 R1t (pti − xti ) + c2 R2t (ptg − xti ) i

(3.1)

In eq.(3.1), R1t and R2t are random numbers between 0 and 1, pti is the best position of the i-th particle in its track at time t and ptg is the best position of the swarm at time t. There are three parameters such as the inertia of the particle ω t , and two parameters c1 , c2 . Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Then, the new position of the i-th particle at time t, xt+1 i , is calculated from eq.(3.2). (3.2)

xt+1 := xti + v t+1 i i

where xti is the current position of the i-th particle at time t. After the i-th particle calculates the next search direction vector v t+1 by eq.(3.1) in consideration of the current search i direction vector v ti , the direction vector going from the current search position xti to the best search position in its track pti and the direction vector going from the current search position xti to the best search position of the swarm ptg , it moves from the current position calculated by eq.(3.2). In general, the parameter ω t is xti to the next search position xt+1 i set to large values in the early stage for global search, while it is set to small values in the late stage for local search. For example, it is determined as: ω t := ω 0 −

t · (ω 0 − ω Tmax ) 0.75 · Tmax

(3.3)

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where t is the current time, Tmax is the maximal value of time, ω 0 is the initial value of ω t and ω Tmax is the fina value of ω t . The search procedure of PSO is shown in Fig. 1. If the next search position of the t i-th particle at time t, xt+1 i , is better than the best search position in its track at time t, pi , t+1 t+1 t+1 t i.e., f (xi ) ≤ f (pi ), the best search position in its track is updated as pi := xi . t+1 t+1 t Otherwise, it is updated as pi := pi . Similarly, if pi is better than the best position of t the swarm, ptg , i.e., f (pt+1 i ) ≤ f (pg ), then the best search position of the swarm is updated t+1 := pt . as pt+1 := pt+1 g g i . Otherwise, it is updated as pg

pgt v it+1 pit

xit

xit+1 v it

Figure 1. Movement of an individual. Such the PSO method has two drawbacks. One is that particles stop at the best search position of the swarm and they cannot easily escape from the local solution since the move direction vector v t+1 calculated by eq.(3.1) always includes the direction vector to the best i search position of the swarm. The other is that the next search position of a particle is not always feasible for constrained problems. Thus, Kato et al. [1] proposed revised particle swarm optimization method solving drawbacks that it is not directly applicable to constrained problems and its liable to stop-

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Takeshi Matsui and Masatoshi Sakawa Generate the initial swarm.

Stop. YES

Move each particle from the current position to the next one.

NO

t = T max ?

Calculate the objective function value for the current position of each particle.

Update the best search position in its track for each particle.

Update the best search position of the swarm.

Figure 2. The basic algorithm of PSO.

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ping around local solutions and showed its effectivity. Moreover we expanded revised particle swarm optimization method for application to nonlinear integer programming problems (rPSONLIP) and showed more efficien y than genetic algorithm [4]. Furthermore we expanded the PSO method for nonlinear 0-1 programming problems (rPSONLZOP) and showed more efficien y incorporating new move scheme [6]. However the rPSONLZOP method needs many computational time than the rPSONLIP method. Thus, we revised the rPSONLZOP method incorporating memory structures [5]. However, in that research, we showed reducing computational time but turned worse on accuracy. In this research, we propose high speed and efficien y particle swarm optimization method using decoding for nonlinear 0-1 programming problems.

4.

Decoding Algorithm Using a Reference Solution with Backtracking and Individual Modification

In this research, we incorporate the decoding algorithm proposed by Sakawa [7]. In the algorithm, n, j, s(j), gs(j) , xs(j) and Ls(j) denote length of a string, a position in a string, an index of a varibale, the value of the element, 0-1 value of a variable with index s(j) decoded from a string, and a s(j)-th column vector of the coefficien matrix A. By using the feasible solution x0 thus obtained, the decoding algorithm using a reference solution with backtracking and individual modificatio can be described as follows: Step 1: Set j = 1, l = 0 and sum = 0. Step 2: If gs(j) = 0, set j := j + 1 and go to Step 4. If gs(j) = 1, let sum = sum + Ls(j) and go to Step 3. Step 3: If sum ≤ b, let l := j and j := j + 1, and go to Step 4. Otherwise, let j := j + 1 and go to Step 4. Step 4: If j > n, go to Step 5. Otherwise, return to Step 2. Step 5: If l > 0, let x∗s(j) := gs(j) for all j such that 1 ≤ j ≤ l and x∗s(j) := 0 for all j such

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that l + 1 ≤ j ≤ n, and go to Step let x∗ := x0 and go to Step 6. Pn 6. Otherwise, Step 6: Let j := 1 and sum := k=1 Ls(k) x∗s(k) . Step 7: If gs(j) = x∗s(j) , let xs(j) := gs(j) and j := j + 1, and go to Step 9. If gs(j) 6= x∗s(j) , then go to Step 8. Step 8: 1) If gs(j) = 1 and sum + Ls(j) ≤ b, let xs(j) := 1, sum := sum + Ls(j) and j := j + 1. Here, if there exists at least one negative element in Ls(j) , then go to Substep 1 for backtracking and individual modification If not, go to Step 9. If gs(j) = 1 and sum + Ls(j) > b, let xs(j) := 2 and j := j + 1, and go to Step 9. 2) If gs(j) = 0 and sum - Ls(j) ≤ b, let xs(j) := 0, sum := sum - Ls(j) and j := j + 1. Here, if there exists at least one positive element in Ls(j) , then go to Substep 1 for backtracking and individual modification If not, go to Step 9. If gs(j) = 0 and sum - Ls(j) > b, let xs(j) := 2 and j := j + 1, and go to Step 9. Substeps for backtracking and individual modificatio Substep 1: Set h := 1. Substep 2: If xs(h) = 2, go to Substep 3. Otherwise, let h := h + 1 and go to Substep 4. Substep 3: 1) If gs(j) = 1 and sum + Ls(h) ≤ b, let xs(h) := 1, sum := sum + Ls(h) and h := h + 1. Then, interchange (s(j), gs(j) )T with (s(h), gs(h) )T . If there exists at least one negative element in Ls(h) , then return to Substep 1. If not, go to Substep 4. 2) If gs(h) = 0 and sum - Ls(h) ≤ b, let xs(h) := 0, sum := sum - Ls(h) and h := h + 1. Then, interchange (s(j), gs(j) )T with (s(h), gs(h) )T . If there exists at least one positive element in Ls(h) , then return to Substep 1. If not, go to Substep 4. If gs(h) = 0 and sum - Ls(h) > b, let h := h + 1 and go to Substep 4. Substep 4: If h ≥ j, then go to Step 9. Otherwise, return to Substep 2. Step 9: If j > n, let h := 1 and go to Step 10. Otherwise, return to Step 7. Step 10: If xs(h) = 2, let xs(h) := x∗s(h) and h := h + 1, and go to Step 11. Otherwise, let h := h + 1 and go to Step 11. Step 11: If h > n, stop. Otherwise, return to Step 10.

5.

The Procedure of Revised PSO Using the Decoding Algorithm

The procedure of the revised PSO proposed in this research summarized as follows and shown in Fig. 3. Step 1: Generate feasible initial search positions and decode each particle mentioned in Section 4.. In addition, let the initial search position of each particle, x0i , be the initial best position of the particle in its track, p0i , and let the best position among x0i , i = 1, . . . , N be the initial best position of the swarm, p0g . Go to Step 2. Step 2: Calculate the value of ω t by eq.(3.3). For each particle, using the information of pti and ptg , determine the direction vector v t+1 to the next search position xt+1 by the modifie i i move schemes. Next, move it to the next search position by eq.(3.2) and go to Step 3. Step 3: If the evaluation function value f (xt+1 i ) is better than the evaluation function value for the best search position of the particle in its track, pti , update the best search position of t+1 the particle in its track as pt+1 := xt+1 := pti and go to Step 4. i i . If not, let pi

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124

Takeshi Matsui and Masatoshi Sakawa (KPFCHGCUKDNGUQNWVQPCPF WUGCUDCUGRQKPVUQNWVKQPr.

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Figure 3. The algorithm of proposed PSO. Step 4: If the minimum of f (xt+1 i ), i = 1, . . . , N is better than the evaluation function value for the current best search position of the swarm, ptg , update the best search position t+1 := pt+1 and go to Step 5. of the swarm as pt+1 := xt+1 g g imin . Otherwise, let pg Step 5: Finish if t = Tmax (the maximal value of time). Otherwise, let t := t + 1 and return to Step 2.

6.

Numerical Examples

In order to show the efficien y of the propsed PSO (rPSODANLZOP), we apply genetic algorithm (GANLZOP) [7] and the proposed PSO (rPSODANLZOP) to nonlinear 0-1 programming problems. The results obtained by these two methods are shown in Table 1. In these experiments, we set the swarm size N = 100, the maximal search generation number Tmax = 100, the inertia weight initial value ω 0 = 1.2, the inertia weight last one ω Tmax = 0.1, weight parameters c1 = c2 = 2CR1t , R2t are uniform random number in the interval [0, 1]. From Table 1, in the application of rPSODANLZOP, we can get better solutions in the sense of average value, the difference between best and worst than those obtained by

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Table 1. Results for a problem with n = 30 and m = 5 method best average worst time (sec)

rPSODANLIP (proposed) −126 −120.6 −104 9.9189

GANLZOP −126 −114.0 −104 13.198

GANLZOP [7]. Therefore, it is indicated that the proposed PSO (rPSODANLIP) is superior to GANLZOP for nonlinear 0-1 programming problems.

7.

Conclusion

In this research, focusing on nonlinear 0-1 programming problems, we proposed new particle swarm optimization using genetic operators. In order to apply nonlinear 0-1 programming problems, we incorporated a new method for generating initial search points, the revision of move methods and decoding algorithms. And we showed the eff ciency of the proposed particle swarm optimization method by comparing it with an existing method through the application of them into the numerical examples.

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References [1] K. Kato, T. Matsui, M. Sakawa and K. Morihara: An approximate solution method based on particle swarm optimization for nonlinear programming problems (in Japanese). Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, 20, 3, 399–409. (2008) [2] J. Kennedy and R.C. Eberhart: Particle swarm optimization . Proceedings of IEEE International Conference on Neural Networks, pp. 1942–1948. (1995) [3] J. Kennedy and W. M. Spears: Matching algorithms to problems: an experimental test of the particle swarm and some genetic algorithms on the multimodal problem generator. Proceedings of IEEE Int. Conf. Evolutionary Computation (1998). [4] T. Matsui, K. Kato, M. Sakawa, T. Uno and K. Matsumoto: Particle Swarm Optimization for Nonlinear Integer Programming Problems. Proceedings of International MultiConference of Engineers and Computer Scientists 2008, pp. 1874–1877. (2008) [5] T. Matsui, M. Sakawa, K. Kato: Particle Swarm Optmization using Memory Structures for Nonlinear 0-1 Programming Problems. Proceedings of The 11th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, pp. 57–62. (2008) [6] T. Matsui, M. Sakawa, K. Kato, T. Uno: Particle Swarm Optmization for Nonlinear 0-1 Programming Problems. Proceedings of The IEEE International Conference on Systems, Man, and Cybernetics 2008, pp. 168–173. (2008) Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[7] M. Sakawa: Genetic Algorithms and Fuzzy Multiobjective Optimization. Kluwer Academic Publishers (2001)

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[8] Y.H. Shi, R.C. Eberhart: A modifie particle swarm optimizer. Proceddings of IEEE International Conference on Evolutionary Computation, pp.69–73. (1998)

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 127-167

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Chapter 7

C OMPARATIVE S TUDY OF D IFFERENT A PPROACHES TO PARTICLE S WARM O PTIMIZATION IN T HEORY AND P RACTICE Stefanie Thiem1,2 and J¨org L¨assig3 1 Institut f¨ur Physik, Technische Universit¨at Chemnitz, 09107 Chemnitz, Germany 2 Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK 3 International Computer Science Institute, 1947 Center Street, Berkeley, CA 94704, USA

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Abstract A research area, which in particular has gained increasing attention in recent years, investigates the application of biological concepts to various optimization tasks in science and technology. Technically, global extrema of objective functions in a ddimensional discrete or continuous space have to be determined or approximated, which is a standard problem with an ample number of applications. In this chapter the particle swarm optimization paradigm is described in different variations of the underlying equations of motion and studied comparatively in theory and empirically on the basis of selected optimization problems. To facilitate this task, the different variants are described in a general scheme for optimization algorithms. Then different side constraints as initial conditions, boundary conditions of the search space and velocity restrictions are investigated in detail. This is followed by an efficien y comparison of swarm optimization to a selection of other global optimization heuristics, such as various single state iterative search methods (simulated annealing, threshold accepting, great deluge algorithm, basin hopping, etc.), ensemble based algorithms (ensemble-based simulated annealing and threshold accepting), and evolutionary approaches. In specific the application of these algorithms to combinatorial problems and standard benchmark functions in high-dimensional search spaces is examined. Further, we show how particle swarm optimization can be effectively used for the optimization of so-called singlewarehouse multi-retailer systems to optimize the ordering strategy and the transportation resources. Optimization tasks of this class are very common in economy as e.g. in the manufacturing industry and for package delivery services.

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1.

Stefanie Thiem and J¨org L¨assig

Introduction

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In recent years several biological motivated optimization algorithms as Genetic Algorithms [1], Artificial Immune Systems [2] and Ant Colony Optimization [3] have been proposed. Another possibility is the usage of the dynamics of swarms to fin solutions to optimization problems. This algorithm was invented by Kennedy and Eberhart in 1995 and is referred to as Particle Swarm Optimization (PSO) [4]. The basic idea of PSO is the application of swarm behavior to different optimization problems. Originally, the algorithm was only introduced for problems with a continuous solution space but, meanwhile, also the application to combinatorial problems is possible [5, 6]. However, the latter is not considered in this chapter. In addition, many different versions and additional heuristics have been introduced during the last years. In this chapter we focus on four of them. After introducing the algorithms, further heuristics and different boundary conditions are discussed in the next section. Then in Sec. 3. a performance comparison is accomplished based on a set of continuous benchmark functions by considering various aspects. Besides the overall performance of the different methods, we study the convergence behavior and the scaling behavior with increasing dimensionality and population size. Different aspects are also visualized by plots. After comparing the different PSO variants, in Sec. 4. also other optimization heuristics are taken into account to compare the performance of the PSO approaches to them. More specifi Simulated Annealing [7], Threshold Accepting [8], their ensemble-based variants [9], Basin Hopping [10] and Evolution Strategies [11] are taken into account. They are combined with different move classes in the continuous domain, based on different distribution functions for the generation of new solutions.

2.

The Particle Swarm Optimization Approach

The basic idea of the PSO algorithm is that a swarm of individuals can share information about the global so far best solution and additionally each individual has an internal memory to store its best so far solution. The movement of each particle is then given by a tradeoff between its current velocity (velocity component), a movement in the direction of its local best solution (cognitive component) and the global optimum (social component).

2.1.

The Algorithm

To compare different variants of this approach later on we introduce a special notation. States will be specifie by Greek letters α, β, etc. Since the algorithm works with more than one individual, the states are distinguished by subscripts, i.e. we write αi , and all states together are written as a vector in bold letters α. Bringing this notation and the previously described ideas together we obtain a standard pattern as shown in Algorithm 1. The single parts of the algorithm are now explained in more detail. We begin with the different variables in the algorithm. ◦ The current state vector for all particles is α. In PSO the single state αi is equivalent to the position ri of the individual i in the solution space.

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Algorithm 1 Particle Swarm Optimization Input: problem P and a solution space S Output: solution γ ∈ S 1: α ← compute initial states(a1 , a2 , . . . , am ) 2: β ← α 3: for all particles i do 4: if ( γ = null or f (βi ) < f (γ) ) then 5: γ ← βi 6: δ ← 0 7: repeat 8: α, δ ← compute new state(α, β, γ, δ, w, c1 , c2 ) 9: for all particles i do 10: if (f (αi ) < f (βi ) ) then 11: βi ← αi 12: if (f (βi ) < f (γ) ) then 13: γ ← βi 14: until ( termination condition(d1 , d2 , . . . , dp ) ) 15: return γ

◦ The vector β is the vector of the local best states of each particle, i.e. the component βi is equivalent to the best so far position li of individual i. ◦ The global best state is given by γ, which is equivalent to the best solution g of the complete swarm during the optimization run.

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◦ The vector δ is not a solution but contains the current velocity vector for each particle. This vector can have very different forms depending on the optimization problem. ◦ The algorithm parameters w, c1 and c2 are user define and can influenc the performance of the algorithm. The parameter c1 is the cognitive parameter and weights the influenc of the local solution βi in the construction of a new solution. In the same way the social parameter c2 weights the influenc of the so far best solution of the swarm γ. The weight factor w is responsible for the influenc of the velocity vector. We will briefl describe the three different functions included in this algorithm. ◦ The method compute initial states(a1 , a2 , . . . , am ) computes an initial state for each particle i. It can depend on an arbitrary number of variables a1 , a2 , . . . , am . ◦ The method compute new state(α, β, γ, δ, w, c1 , c2 ) is a method to compute a new state and velocity for each of the particles i. It depends on the parameters α, β, γ, δ, w, c1 , c2 and there are many different internal realizations possible. The details for several realizations for the continuous domain are given in Sec. 2.2.1. to 2.2.4.. ◦ The function termination condition(d1 , d2 , . . . , dp ) returns true when the convergence criteria are fulfille and can depend on an arbitrary number of variables d1 , d2 , . . . , dp . Regarding the convergence criteria we have several options briefl discussed here: ◦ Maximum number of evaluations: With this option we perform the algorithm until a user define number of objective function evaluations is reached. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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◦ Sufficient good approximation of the optimum: In this case the loop is executed until the algorithm has found a solution f (γ) which differs from the problem’s best solution fbest by less than a given percentage, i.e. |f (γ) − fbest | < ε.1 ◦ No improvement in the last iterations: The algorithm terminates when it has not found any improvements during the last iterations. The number of iterations can be specifie by the user. ◦ Contraction of the swarm: This option is a special option only for swarm optimization, whereas the three others are also applicable for other algorithms. With this option we will stop the algorithm when the mean displacement of the swarm is reduced to a user define value. This is possible because the swarm contracts itself near the so far global best solution.

2.2.

Move Classes for Particle Swarm Optimization

So far we have not discussed how new states are constructed by the method compute new states(α, β, γ, δ, w, c1 , c2 ). The original version of the PSO algorithm, which was presented by Kennedy and Eberhart in 1995 [4], realized it as shown in Listing 1. The matrix vx is aquivalent to our vector δ, where δi represents the velocity vi of the individual i. The matrix presentx represents the currents state α, pbest the local best states β and the vector gbest is given by γ describing the overall best so far discovered state. Listing 1: Original velocity and position update rule

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vx [ ] [ ] = vx [ ] [ ] + 2∗ r a n d ( ) ∗ ( p b e s t [ ] [ ] − p r e s e n t x [ ] [ ] ) + 2∗ r a n d ( ) ∗ ( g b e s t [] − p r e s e n t x [ ] [ ] )

Later on scaling factors c1 and c2 replaced the constant factors 2 and Shi and Eberhart introduced a weighting factor for the velocity, receiving better results than for the original version [12]. Because the meaning of the term rand() is not unique, several different variants were proposed during the years or even new suggestions for the update rules were introduced. Thus, in the following section we will present four different versions of PSO and compare their performance. The firs version interprets the equations given by Kennedy and Eberhart [4] in Listing 1 in the way that the rand() expression equals two scalar random numbers r1 and r2 which are multiplied with each vector component (see Sec. 2.2.1.). The other interpretation would be to think of random diagonal matrices R1 and R2 which weight each component of the velocity vectors differently (see Sec. 2.2.2.). Both methods have been applied in the past: for instance the firs one was used in [13, 14] and the latter one in [4, 15]. 2.2.1.

Variant 1

The firs version referred to as PSO1 is given by the velocity and position update rule

vt+1 = w · vti + c1 · r1 · lti − rti + c2 · r2 · gt − rti (2.1a) i rt+1 = rti + vti i

,

1

(2.1b)

Using the option of sufficien approximation, we have to know the exact value of the global optimum, which is not given for real world problems. Hence, this option is only useful for performance analysis tasks. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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where r1 and r2 are uniform random numbers in [0, 1) which weight the importance of the global best solution g to the local best solution li of particle i on the way of findin a new position rti . The algorithmic realization is quite simple and shown in Algorithm 2. The position and the velocity components for the different particles i and dimension d are written as subscripts, i.e. vid is the dth component of the velocity vector of particle i. The meaning of the velocity equations is also shown in Fig. 1, where the new position directly follows from the addition of the three vectors, namely the velocity component, the cognitive component and the social component. Each vector in Equ. (2.1) is weighted by a scaler, which changes only the length of the vectors and not their directions. Depending on the choices of the parameters w, c1 and c2 the influenc of each component can be adapted. The new position follows then from the superposition of the three vectors as it is also visualized in Fig. 1.

Figure 1. Visualization of the iterative solution update in PSO1 according to Equ. (2.1). Algorithm 2 Variant 1 of position / velocity update rule Input: position αi ≡ ri , velocity δi ≡ vi and local best position βi ≡ li for each particle i global best position γ ≡ g and cognitive parameter c1 , social parameter c2 and weight w Output: α, with αi ∈ S, the solution space, and δ 1: for all particles i do 2: r1 ← get uniform random number(0, 1) 3: r2 ← get uniform random number(0, 1) 4: for all dimensions d do 5: vid ← w · vid + c1 · r1 · (lid − xid ) + c2 · r2 · (gd − xid ) 6: xid ← xid + vid 7: return α ≡ {ri }, δ ≡ {vi }

For this velocity update rule Wilke showed that the search space of the particles is restricted to line search if no new global or local best solution is found. This is a disadvantage because the swarm individuals are not able to explore a sufficientl large area of the problem space [16]. We will see that this behavior leads to a worse performance later on in the experimental Sec. 3., but for the moment we will consider the second variant.

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132 2.2.2.

Stefanie Thiem and J¨org L¨assig Variant 2

As already mentioned in the previous section the original version of the algorithm is written ambiguously. The second velocity update rule PSO2 is given by 

 vt+1 = w · vti + c1 · R1 · lti − rti + c2 · R2 · gt − rti , (2.2) i

where R1 and R2 are two diagonal matrices. The elements are uniform random numbers in [0, 1). On the firs view this seems quite complicated but looking at the pseudo code it reveals to be only a minor change to the algorithm variant PSO1. Here the computation of the random numbers is included in the inner loop as can be seen in Algorithm 3. Algorithm 3 Variant 2 of position / velocity update rule

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Input: position αi ≡ xi , velocity δi ≡ vi and local best position βi ≡ li for each particle i global best position γ ≡ g and cognitive parameter c1 and social parameter c2 Output: α, with αi ∈ S, the solution space, and δ 1: c ← c1 + c2 √ 2: w ← 2/ 2 − c − c2 − 4c 3: for all particles i do 4: for all dimensions d do 5: r1 ← get uniform random number(0, 1) 6: r2 ← get uniform random number(0, 1) 7: vid ← w · (vid + c1 · r1 · (lid − xid ) + c2 · r2 · (gd − xid )) 8: xid ← xid + vid 9: return α ≡ {ri }, δ ≡ {vi }

An interpretation of the expression R1 (li − ri ) is that each vector component of the vector (li − ri ) is weighted by a random number. Analogously, the vector (g − ri ) is treated. Since every component is multiplied with a different random number the vectors are not only changed in length but also perturbed from their original direction. Hence, this version does no longer show the disadvantage of version 1, namely the collapse of the search to line search [16]. In 2002 Clerc and Kennedy analyzed the PSO algorithm theoretically and proposed a new version within this concept given by the velocity update rule (2.2) with a weight factor w=

|2 − c −

2 √

c2 − 4c|

and c = c1 + c2 .

(2.3)

This algorithm was extensively studied and an optimal parameter setup was determined by Carlisle and Dozier with c1 = 2.8 and c2 = 1.3 [17]. 2.2.3.

Variant 3

Variant 3 of the velocity update rule is based on the firs variant described in Sec. 2.2.1.. The problem of the version PSO1 is that the movements of the particles collapse to line searchers and thus can only cover a small area of the search domain. To overcome this problem the two vectors (li − ri ) and (g − ri ) are perturbed by multiplying them with two independent random rotation matrices Q1 and Q2 [16]. Thus, the velocity equation is now

(2.4) = w · vti + c1 · r1 · Q1 · lti − rti + c2 · r2 · Q2 · gt − rti . vt+1 i

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A remaining problem is the construction of such a random rotation matrix Q. A rotation matrix has to be orthogonal, meaning that the row vectors are orthonormal to each other. Thus, also the relation QT = Q−1 is valid and this property is necessary to achieve the invariance of the rotation, because QQ−1 = I and therefore a rotation by ϕ and a succeeding rotation by an angle −ϕ results in the original vector. A method to construct such a rotation matrix is to use the exponential of a matrix by applying the series method described in [18] 1 1 1 W + W 2 + W 3 + ... , (2.5) 1! 2! 3!

ϕπ T . The expression A − AT represents an antisymmetric where W = 180 ◦ A − A matrix, i.e. the matrix elements follow the relation aij = −aji . The antisymmetric part of a matrix contains only rotational gradients, i.e. changes of the original system due to a rotation. The matrix Q consists now of a unit matrix I transforming the original system into itself and the asymmetric matrix W including the rotational changes of the original system. To obtain a random rotation we compute the random matrix A with aij being a random number in [−0.5, 0.5]. For our purposes it is sufficien to use only the firs two terms of the Taylor series expansion of Equ. (2.5) because we only have small perturbations, i.e.
ϕπ T Q=I +W =I + A − A . (2.6) 180◦ Q = eW = I +

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Algorithm 4 Variant 3 of position / velocity update rule Input: position αi ≡ ri , velocity δi ≡ vi and local best position βi ≡ li for each particle i global best position γ ≡ g and cognitive parameter c1 , social parameter c2 , weight w and angle ϕ Output: α, with αi ∈ S, the solution space, and δ 1: for all particles i do 2: r1 ← get uniform random number(0, 1) 3: r2 ← get uniform random number(0, 1) 4: for all dimension l do 5: for all dimension k do 6: a1lk ← get uniform random number(−0.5, 0.5) 7: a2lk ← get uniform random number(−0.5, 0.5) 8: for all dimension l do 9: for all dimension k do 2 1 ←0 ← 0, qlk 10: qlk 11: if l = k then 1 2 12: qlk ← 1, qlk ←1
ϕπ 1 1 1 1 13: qlk ← qlk + 180 ◦ alk − akl
ϕπ 2 2 2 2 14: qlk ← qlk + 180◦ alk − akl 15: for all dimensions d do 16: vid ← w · vid 17: for all dimensions l do 1 2 18: vid ← vid + c1 · r1 · qdl · (lid − xid ) + c2 · r2 · qdl · (gd − xid ) 19: xid ← xid + vid 20: return α ≡ {ri }, δ ≡ {vi }

The resulting pseudo code is shown in Algorithm 4. From this we can see that the computation of the rotational matrices is very expensive because we have to compute 2D2 N Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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additional random numbers compared to variant PSO1, where d is the problems dimension and N is the number of particles. So when looking at the performance later on we have to keep this fact in mind.

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2.2.4.

Variant 4

One disadvantage of the previously introduced variants is their fast convergence behavior, which often prevents the swarm individuals from escaping from local minima in the optimization process. In particular, a known result is that the particles quickly converge to a small area of the search space with a rate of shrinkage following a scaling law for spherical symmetric functions [19]. An idea to overcome this premature convergence is the introduction of a repulsive force among the individuals. The approach of Repulsive PSO uses an attraction to the global minimum and a repulsive force depending linearly on the distance of the current particle and a randomly chosen other particle [20]. However, a main disadvantage is that the repulsive force increases with distance, which seems to be unnatural, and the performance improvement was found to be minor. Another version introduces a repulsive force according to the Coulomb law as suggested by Blackwell et al. and is hence named Charged PSO (CPSO) [21]. Besides strongly positive effects as good results for dynamic search spaces one faces now the problem to encompass fina convergence if applied to static problems. We proposed a solutions to this problem. The resulting algorithm uses a dynamic charge reduction over time definin particle groups which are iteratively merged, reducing the overall charge of the system gradually during the optimization run and shows superior results in complex state spaces or high dimensionality [22]. The CPSO approach is based on the PSO2 version, where an acceleration term is added to the equations of motion, which are given by 

 vit+1 = w vit + c1 · R1 · lti − rti + c2 · R2 · gt − rti + ai . (2.7) The acceleration of a particle i is then determined additively from the repulsive force term between particle i and k by X ai = ai (k) . (2.8) k

While in the classical model ai = 0, in CPSO the repulsive term is based on the well-known Coulomb law [21], which is also similar to the commonly used gravitational force in swarm robotics [23]. The easiest choice of the force is Qi Qk (ri − rk ) . (2.9) ai (k) = |ri − rk |3

In this application the charge product Qi Qk is define to be −1 for particles of different groups and zero otherwise. The idea now is to modify the number of repelling particles with time by reorganizing particles in groups. On algorithm start, the group number G0 equals the number of particles P and the behavior is just like in CPSO. If we sum up all pairs which repel each other to a variable S (charge sum), we have     G0 P P2 − P S0 = = = . (2.10) 2 2 2

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(a) Deterministic CPSO

135

(b) Random CPSO

Figure 2. Visualization of possible grouping and reorganization schemes for CPSO. Table 1. Random grouping for a swarm with 10 particles using integer division. Number of Merges

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0 1 2 3 4 5 6 7 8 9

Charge sum S

Group belongings of the particles 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0

2 1 0 0 0 0 0 0 0 0

3 1 1 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0

5 2 1 1 1 0 0 0 0 0

6 3 2 1 1 1 0 0 0 0

7 3 2 1 1 1 1 0 0 0

8 4 2 2 1 1 1 1 0 0

9 4 3 2 1 1 1 1 1 0

45 40 36 32 25 24 21 16 9 0

The subscripts of Si count the number of reorganizations, and with each reorganization the number of groups is reduced gradually. A straightforward approach is for example to unify pairs of groups in each iteration. Finally after n iterations the dynamic reduction of the number of groups eventually conducts convergence to classical PSO2 and all particles end up in one single group, i.e. Sn = 0 and Gn = 1. This guaranties an excellent convergence of the algorithm. A large variety of merging schemes are possible. A highly efficien one is based on integer division and was named Random CPSO because the particles can change the group belonging over time instead of new groups being constructed purely by collapsing two or more groups (Deterministic CPSO), see Fig. 2. An example of the group assignment for Random CPSO is shown in Tab. 1 and the pseudo code is shown in Algorithm 5. The number of merging iterations in random grouping is P − 1, caused intrinsically by the behavior of the integer division. In this case the number of groups in each iteration is not reduced exactly by one (cp. Tab. 1). In the firs iterations the number of groups decreases faster but afterwards the group number keeps constant over many iterations, which results in a quasi constant merging speed, i.e. an almost constant decrease of the charge sum of the swarm. For the performance comparisons in Sec. 4. the merging speed is chosen in a way that after 50% (i.e. p = 0.5) of the allowed number of objective function evaluations all particles are collapsed into one single group. Having introduced four different realizations of compute new state(α, β, γ, δ, w, c1 , c2 ) we will shortly consider different heuristics

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Algorithm 5 Variant 4 of position / velocity update rule Input: position αi ≡ ri , velocity δi ≡ vi and local best position βi ≡ li for each particle i, global best position γ ≡ g, cognitive parameter c1 , social parameter c2 , weight w and group parameter t Output: α, with αi ∈ S, the solution space, and δ, t 1: c ← c1 + c2 √ 2: w ← 2/|2 − c − c2 − 4c| 3: for all particles i do 4: ai ← 0 5: for k = i + t to k < i + P with k = k + t do {equivalent to i div t 6= k div t} 6: ai ← ai + ai (k mod P ) {see Equ. (2.9)} 7: for all dimensions d do 8: r1 ← get uniform random number(0, 1) 9: r2 ← get uniform random number(0, 1) 10: vid ← w · (vid + c1 · r1 · (lid − xid ) + c2 · r2 · (gd − xid ) + aid ) 11: xid ← xid + vid 12: if t < N ∧ fi ed iterations reached() then 13: t←t+1 14: return α ≡ {ri }, δ ≡ {vi }, t

and boundary conditions in the next sections.

2.3.

Heuristics

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In literature one can fin the application of several heuristics to PSO in order to further improve the performance [16]. In the following we will introduce several commonly used heuristics and later on we will test some of them whether they are beneficia to the algorithms. Random Initial Velocities: Each particle will get a random initial velocity, whereas the velocity components are chosen to be within 10% of the search space size. This option increases the diversity of the particles because now they not only differ in their initial position but also in their initial velocities. On the other hand, a random initialization does not consider any characteristics of the search space and the initial velocities are forgotten during an optimization run due to the damping by the weight factor w. Thus, it is questionable whether it is an improvement of the algorithm. Maximum Velocity Restriction: This is a commonly used heuristic [24] and is introduced to overcome instabilities of the algorithm in case of high velocities. Basically there are two options: either we can restrict each velocity component, i.e. |vid | ≤ vmax , or we re√ strict the total velocity vector of each particle i to a given length, i.e. |vi | = vi vi < vmax . The firs option, namely the restriction of the single velocity components, does not only change the length of the velocity vector but also its direction and thus introduces an additional diversity into the algorithm. Another advantage is its simple implementation. The second option only changes the length of the velocity vector and therefore leads to a slower convergence but also to the minimization of instabilities. The impact of both options on a vector is shown in Fig. 3.

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(a) Restriction of vector length

137

(b) Restriction of component size

Figure 3. Visualization of the effect of different maximum velocity restrictions on the length and direction of the velocity vector.

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Wilke showed that the application of the firs velocity restriction option improves variant PSO1 mainly due to the additional diversity which prevents the particles from collapsing to line searchers [16]. For PSO2 Carlisle and Dozier pointed out that a fi e tuning of vmax can help to improve the performance but rather suggested to abandon this option and to introduce adequate boundary conditions, i.e. the velocity of the particle is set to zero at the boundary [17]. More details on the influenc of boundary conditions are given in Sec. 2.4.. Minimum Velocity Restriction: We can also restrict the particles’ velocities by introducing a minimal velocity in order to prevent the algorithm from early convergence. However, it is questionable whether the algorithm can converge in general. Because upon convergence the swarm contracts to a small region in search space and is able the perform a fin exploration of this region. In this case also the velocities of the individuals become smaller and smaller. But with the application of the minimum velocity restriction, it will not be possible to obtain this behavior. A possible implementation of these velocity restriction rules, namely the minimum and maximum velocity restriction applied to single vector components, is shown in Algorithm 6. Algorithm 6 Realization of velocity restriction rules Input: velocity vi for each particle i, minimum velocity vmin and maximum velocity vmax Output: new velocity value vi for each particle 1: for all particles i do 2: for all dimensions d do 3: if |vid | < vmin then 4: vid ← sgn (vid ) · vmin 5: if |vid | > vmax then 6: vid ← sgn (vid ) · vmax 7: return δ ≡ {vi }

Craziness: This heuristic places particles with a certain probability p randomly in the search space in the course of the algorithm. A probability p = 1 would reduce the particles Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Stefanie Thiem and J¨org L¨assig

to random walkers and for p = 0 we will get the original behavior of PSO. This heuristic can be compared to the mutation operator in Genetic Algorithms because both introduce additional diversity in the algorithm and can help to overcome local minima. Unfortunately, we have to point out that we will encounter the same problems like for the minimum velocity restriction, namely the failing of the fina contraction of the swarm at the best found solution. Similar results are also reported by Wilke, who shows that this heuristic improves variant PSO1 due to the introduction of more diversity but fails to improve PSO2 [16].

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Adaptive Weight Factor: Zheng et al. suggested to adapt the weight factor w during an optimization run starting at a weight w = 0.4 and iteratively increasing it to w = 0.9 [25]. They argued that small weights increase the search abilities of the algorithm and large weight values help to stabilize the algorithm. Therefore, this setup allows a good exploration of the search domain in the beginning with w = 0.4 and by slowly increasing this value the algorithm converges. This method is applicable for PSO variants 1 and 3. In the other two variants the weight factor is directly given by the cognitive and social parameters and thus makes the update of w difficult Neighborhood: The neighborhood define which individuals can share information about the global best solution among each other. Up to now we have assumed a global neighborhood, i.e. the information about the global optimum is distributed instantaneously among the complete swarm. Arbitrary other neighborhoods are possible. For instance the restriction to nearest neighbors, where the information can only be passed to the nearest neighbors and thus needs time to be distributed in the whole swarm. This leads to the emergence of groups within the swarm which can search different regions of the search space [26]. The complexity of the neighborhood can by very different, starting from the easy choices of global and nearest neighborhoods up to complex network relations like preferential attachment or Erd˝os-Renyi networks [26, 27]. Our test suite includes the firs three heuristics and the weight factor can be easily controlled by the user [28]. Regarding the neighborhood we would like to point out that Carlisle and Dozier reported the best performance for a global neighborhood [17]. For this reason, the huge amount of possible other neighborhood relations and also the simplicity of implementing a global neighbor relation, we restrict our consideration to the global neighborhood in this chapter.

2.4.

Boundary Conditions

Also the boundary conditions can have an influenc on the algorithm performance. Therefore, we will describe three different versions of possible boundary conditions, introduced by Robinson and Rahmat-Samii [29]. For a better understanding of their nature we visualize the variants in Fig. 4 and shortly explain them in the following. Invisible Boundaries: In the case of invisible boundary conditions we do not change anything on the particles’ position or velocity at the boundaries. Consequently, the particle

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(a) Invisible

(b) Absorbing

139

(c) Reflectin

Figure 4. Visualization of different boundary conditions for a particle movement in a twodimensional search space.

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is allowed to enter forbidden regions, but we only will evaluate the objective function in the feasible search space. This kind of boundary condition is possible because the PSO algorithm will automatically pull the particle back into the feasible search space due to the attraction of an individual to the global optimum and its local optimum, which obviously are within the search domain. Absorbing Boundaries: When the particle reaches the search domain boundaries, absorbing boundary conditions lead to the absorbtion of all velocity components which would bring the individual out of the search domain. We can formulate this mathematically by   xid + vid < xmin xmin xid = xid + vid xmin ≤ xid + vid ≤ xmax , (2.11)   xmax xid + vid > xmax

with predefine boundary values xmin and xmax for each component. The velocities are adapted by ( vid xmin ≤ xid + vid ≤ xmax vid = , (2.12) 0 otherwise where the equation for vid depends on the used PSO variant. Reflecting Boundaries: With reflectin boundary conditions the particle is reflecte at the search domain boundaries by changing the sign of the responsible velocity components and not allowing the particle to enter the forbidden region by restricting its position to the minimum respectively maximum value. The corresponding equation for the position is equivalent to (2.11) and the velocity equation is ( vid xmin ≤ xid + vid ≤ xmax vid = , (2.13) −vid otherwise where again vid depends on the used PSO variant.

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3.

Stefanie Thiem and J¨org L¨assig

Performance Comparison of Particle Swarm Optimization Approaches

In this section we compare the different PSO variants regarding their performance for continuous benchmark functions in order to determine the method with the best performance for this class of problems. In order to compare different algorithms we have to introduce algorithm quality measures. All in all the solution quality of an optimization heuristic can be measured by a number of different measures [30, 31]. Most common are Q1) The mean fina objective value should be as small as possible. Q2) The fina probability of findin the ground state should be as large as possible. Q3) The expected number of visits to the ground state should be as large as possible. Q4) The probability of visiting the ground state during the optimization run should be as large as possible.

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Q5) The mean fina best-so-far objective value should be as small as possible. From the theoretical point of view it would be necessary to show that an algorithm performs better than another algorithm for each of these measures. However, in this work we predominantly focus on empirical studies. In this case choice Q2 is problematic because for large problem instances we almost never obtain the optimal solution. This would make it difficul to compare the algorithm, since all of them will have a probability of fi ding the ground state near or equal to zero. Therefore, we almost do not obtain any statistic information from these measures for the comparison of the performance. Similar considerations rule out the quality measures Q3 and Q4. Except from the fact that these three measures do not deliver enough statistical information, they are also hard to determine in experimental studies. For these reasons we will concentrate on the measures Q1 and Q5, where the last option is the most common choice. We also determine how often the global optimum is obtained in a given number of runs. This is a rough measure for Q2 and Q3.

3.1.

Continuous Benchmark Functions

In literature one can fin a manifold of continuous benchmark functions for the determination of the performance and comparison of different optimization methods in continuous solution space. In general one can specify the problem by the following definition D EFINITION 3..1 (C ONTINUOUS F UNCTION P ROBLEM ) (cp. [32]) Having a function f (x), which has several local and one or more global optima f (x⋆ ), and a search domain x ∈ S, we want to find the global optimum f (x⋆ ) ≤ f (x), ∀x ∈ S. Having a closer look at the various test functions from [32, 33], different versions and domain specification can be found. Therefore, we give a short overview of the function definition used here in Tab. 2 including their global minima and domains. Further, some of the functions are plotted in Fig. 5.

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Table 2. Selection of d-dimensional functions [32, 33]. Name

Function Definition

Domain and Minimum

Ackley Function

  q P N 1 2 f (x) = −20 exp − 51 N x i=1 i i h P N 1 cos(2πx ) + 20 + e − exp N i i=1

xi ∈ [−32.768, 32.768]

f (x) = (x1 − 1)2 +

Min. Pos.: x⋆i = 2(2 −2)/2 Min. Value: f (x⋆ ) = 0 xi ∈ [−1.28, 1.28]

Dixon & Price Function Function with Normal Dist. Density Griewangk Function

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Levy Function

f (x) =

d  P

1 4000

d−1 P

x2i −

i=1

N Q



cos

i=1

Rastrigin Function

f (x) = 10N +

Rosenbrock Function

f (x) =

d P

sin(xi ) · sin20

i=1

N P



ix2 i π



xi √ i



x2i − 10 cos(2πxi )

100 x2i − xi+1

i=1


2

√

f (x) = −

d P

xi · sin

i=1

f (x) =

Sphere Function

f (x) =

Step Function

f (x) =

d P

i=1

N P

i P

j=1

xj

p

 |xi |

!2

x2i

i=1

N P

⌊xi ⌋ + 6N

i=1

f (x) =

d P

(xi − 1)2 −

i=1

d−1 P i=1




+ (1 − xi )2

P sin2 ( y ) f (x) = − 1+10−3 y2 with y = di=1 x2i

Schwefel’s Double Sum Function

Trid Function

⋆2 √5 e−0.5xi i 2π P ⋆2 f (x⋆ ) = di=1 ix⋆2 i (xi



i=1

NP −1 h

i

Min. Pos.: x⋆2 i =

i=1

f (x) = −

Schwefel Function

i

i=2

  (yi − 1)2 1 + 10 sin2 (πyi + 1)   + sin2 (πy1 ) + (yd − 1)2 1 + 10 sin2 (2πyd ) with yi = 1 + 41 (xi − 1) f (x) =

Michalewicz Function

Schaffer Function

N P

xi ∈ [−10, 10]

i(2x2i − xi−1 )2

2 √20 e−0.5xi 2π

ix4i +

i=1

f (x) = 1 +

d P

Min. Pos.: x⋆i = 0 Min. Value: f (x⋆ ) = 0

xi · xi+1

i

Min. Value: xi ∈ [−600, 600] Min. Pos.: x⋆i = 0 Min. Value: f (x⋆ ) = 0 xi ∈ [−10, 10] Min. Pos.: x⋆i = 0 Min. Value: f (x⋆ ) = 0

+ 4)

xi ∈ [0, π] Min. Pos.: x⋆i = (2.21, 1.57, 1.29, 1.92, ..) Min. Value: f (x⋆ ) = −9.6602 (d = 10) xi ∈ [−5.12, 5.12] Min. Pos.: x⋆i = 0 Min. Value: f (x⋆ ) = 0 xi ∈ [−2.048, 2.048] Min. Pos.: x⋆i = 1 Min. Value: f (x⋆ ) = 0 xi ∈ [−100, 100] P Min. Pos.: di=1 x⋆2 i = 2.4436009 Min. Value: f (x⋆ ) = −0.9940049 xi ∈ [−500, 500] Min. Pos.: x⋆i = 420.9687463 Min. Value: f (x⋆ ) = −418.9828872 · d xi ∈ [−65536, 65536] P Min. Pos.: di=1 x⋆2 i =0 Min. Value: f (x⋆ ) = 0 xi ∈ [−5.12, 5.12] Min. Pos.: x⋆i = 0 Min. Value: f (x⋆ ) = 0 xi ∈ [−5.12, 5.12] Min. Pos.: x⋆i ∈ [−5.12, −5.0] Min. Value: f (x⋆ ) = 0 xi ∈ [−100, 100] Min. Pos.: x⋆i = i(d + 1 − i) Min. Value: f (x⋆ ) = −d(d + 4)(d − 1)/6

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Stefanie Thiem and J¨org L¨assig

(a) Levy

(b) Michalewicz

(c) Schwefel

(d) Step

Figure 5. Three-dimensional plots of a selection of continuous benchmark functions.

3.2.

Parameter Setup

To apply the different variants of the PSO algorithm to a selection of test functions introduced in the previous section, we firs give an overview over the parameter setup used for the experiments. The values are oriented at good setups proposed in literature and for all four variants summarized in Tab. 3. Table 3. Parameter setup for PSO variants.

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Parameter weight w cognitive parameter c1 social parameter c2 additional parameters references

PSO1 0.8 2.0 2.0 – [12]

PSO2 0.732 2.8 1.3 – [17]

PSO3 0.6 2.0 2.0 ϕ = 3.0 [16]

PSO4 0.732 2.8 1.3 p = 0.5 [22]

If not given different elsewhere, we assume a population size of 30 and a problem dimension of 10. Additionally, we used for all variants randomly distributed initial positions ri in the solution space, zero initial velocities vi and invisible boundary conditions, which are reported to show good results [29]. The different PSO algorithms are applied to the selection of test functions, summarized in Tab. 2. For PSO2 the parameter setup is oriented at [17] with c1 = 2.8, c2 = 1.3, which is reported to outperform the standard setup with c1 = c2 = 2.05.

3.3.

Performance Comparison for Continuous Test Functions

All results for the selected test functions are summarized in Tab. 4, where the best solution and the average solution for 20 runs are given. An optimization run terminates when the global optimum is approximated by 10−6 or at most 20,000 function evaluations are done. The results are summarized for the different functions in Tab. 4, where we give the best found solution from all twenty runs, the average best solution of these runs and how often the global optimum was found. Additionally, the average number of objective function evaluations is given for all runs. 1

The value of the weight factor w follows directly from Equ. (2.3).

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The results in Tab. 4 show that PSO converges very quickly to the global optimum for several test functions, especially the Ackley, Levy and Sphere function. For other functions we do not obtain the global optimum and the results also show significan differences between the algorithms.

Table 4. Performance comparison of different PSO variants in 10 dimensions. # Successful Runs

# of Function Evaluations

PSO1

Best Solution

Ackley Dixon & Price FWNDD Griewangk Levy Function Michalewicz Rastrigin Rosenbrock Schaffer Schwefel Double Sum Sphere Step Trid

5.517798 1.874654 71.494089 1.052350 2.364631 -7.032929 17.225120 7.931421 -0.025961 -2484.4861 3363016.3 0.004895 -31.900000 -173.2207

2.583130 0.262122 71.475196 0.127008 0.257143 -8.225467 10.810529 3.711308 -0.065927 -3221.4607 4812.62 0.000028 -43.000000 -209.8003

0 0 0 0 0 0 0 0 0 0 0 0 0 0

20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000

PSO2

Average Solution

Ackley Dixon & Price FWNDD Griewangk Levy Function Michalewicz Rastrigin Rosenbrock Schaffer Schwefel Double Sum Sphere Step Trid

0.000000 0.450000 71.474933 0.055521 0.000000 -9.221643 4.377886 3.275460 -0.791423 -3684.7215 0.000000 0.000000 -59.550000 -209.9853

0.000000 0.000000 71.474933 0.009865 0.000000 -9.613478 0.995682 0.795569 -0.994007 -4071.3905 0.000000 0.000000 -60.000000 -210.0000

20 2 20 0 20 0 0 0 7 0 19 20 12 12

8310 18690 4260 20000 4440 20000 20000 20000 15390 20000 18090 3720 14730 18060

PSO3

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Function

Ackley Dixon & Price FWNDD Griewangk Levy Function Michalewicz Rastrigin Rosenbrock Schaffer Schwefel Double Sum Sphere Step Trid

2.492030 0.450000 71.474933 0.282767 1.834876 -7.800517 15.824239 0.405592 -0.612497 -2600.3626 0.000000 0.000000 -45.700000 -210.000000

0.000000 0.000000 71.474933 0.054157 0.089528 -8.990948 6.764713 0.003044 -0.682340 -3163.3250 0.000000 0.000000 -53.000000 -210.000000

4 2 20 0 0 0 0 0 0 0 20 20 0 20

18030 19320 4470 20000 20000 20000 20000 20000 20000 20000 12660 4530 20000 8340

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144

Stefanie Thiem and J¨org L¨assig Table 4. Continued

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PSO4

Function Ackley Dixon & Price FWNDD Griewangk Levy Function Michalewicz Rastrigin Rosenbrock Schaffer Schwefel Double Sum Sphere Step Trid

Best Solution 0.000022 0.475000 71.474933 0.055168 0.000000 -9.222875 4.037273 4.061816 -0.853756 -3730.8773 0.000000 0.000000 -58.800000 -209.9419

Average Solution 0.000000 0.000000 71.474933 0.022151 0.000000 -9.581427 0.994959 1.532471 -0.994007 -4071.3905 0.000000 0.000000 -60.000000 -210.0000

# Successful Runs

# of Function Evaluations 16 1 20 0 20 0 0 0 11 0 19 19 9 2

15090 19860 13380 20000 12090 20000 20000 20000 13170 20000 18450 13290 18600 19980

Comparing the different PSO variants we clearly see the shortcomings of variant PSO1 to fin an optimal solution, which is caused by the collapse of the movement to line search as described in Sec. 2.2.1.. The variants PSO2 and PSO4 instead show a good behavior and seem to be quite equivalent in their performance, which is not unusual since both represent the same algorithm in the limit when all groups are merged in PSO4. The variant PSO3 is also able to fin good solutions to some of the continuous test functions but shows big differences to the algorithm variants PSO2 and PSO4. For instance PSO3 has trouble to optimize the Ackley and the Levy function unlike to the other two algorithms and on the other hand shows very good results for the Rosenbrock and the Trid function, where the variants PSO2 and PSO4 do not perform so well. However, during the computation we also noticed the already mentioned additional computation of random numbers for PSO3 and the repulsive force for PSO4 leading to an increased computation time compared to the other methods and will be one important point for the comparison of the algorithms later on. Since we were not able to really fin a superior method of the four PSO variants (with respect to this set of test functions) we will do some more experiments in the next section.

3.4.

Investigation of the Convergence Behavior

In this section we investigate the influenc of the dimensionality and the population size on the algorithm performance. To realize this we will restrict our considerations to only three test functions which showed distinctive behaviors for the four algorithms in the test runs which are reported in Tab. 4. ◦ The Levy function is chosen since the algorithms PSO2 and PSO4 were easily able to fin the global optimum in about the same number of objective function evaluations, whereas for the algorithm PSO3 we obtained the global optimum in only one run. ◦ The second function for our tests is the Rastrigin function. None of the four algo-

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rithms was able to fin the global optimum of this function in ten dimensions but we can use the best and the average objective function values of the twenty runs for a comparison of the algorithms, where these values differed noticeable for the different variants. ◦ The Rosenbrock function is chosen because the algorithm PSO3 performs here significantl better compared to the other algorithms. Population size: A widely asked question deals with the population size. Clearly, a larger number of individuals allows to explore the solution space in more detail but on the cost of a higher computational effort. Thus, we have to carefully adjust the population size to get a good performance. Carlisle and Dozier reported the best results for 30 individuals for the algorithm PSO2 [17] and this is also a commonly used population size for the other algorithm variants [16]. We want to test this result for the above given selection of benchmark functions from Tab. 4, i.e. we restrict our consideration to the Levy, Rastrigin and Rosenrock function. The results regarding the population size are summarized in Tab. 5.

N

PSO1

Levy

5 10 20 30 40 50 60

3.401786 3.579758 2.038136 2.364631 1.808642 1.375513 1.498357

0 0 0 0 0 0 0

2.558342 0.395127 0.031669 0.000000 0.000000 0.000000 0.000000

0 8 17 20 20 20 20

5.435948 4.113506 2.293984 1.834876 1.486812 1.356367 1.411355

0 0 0 0 2 3 1

0.006889 0.090743 0.000102 0.000000 0.000000 0.004743 0.000452

6 18 19 20 20 16 16

Rastrigin

5 10 20 30 40 50 60

55.228434 37.455299 21.449210 17.225120 16.178818 22.317950 19.463926

0 0 0 0 0 0 0

9.949586 8.656167 6.765715 4.377886 3.384456 3.587451 3.708781

0 0 0 0 0 0 0

27.229960 23.232239 21.988554 15.824239 21.093009 17.212759 18.854436

0 0 0 0 0 0 0

10.288322 6.964771 4.881877 4.037273 4.602819 6.661133 6.983927

0 0 0 0 0 0 0

Rosenbrock

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Table 5. Average results and number of discovered global optima over 20 runs for the PSO variants depending on the population size. PSO2

PSO3

PSO4

5 10 20 30 40 50 60

231.303016 52.901487 9.687132 7.931421 7.302739 7.538420 7.416475

0 0 0 0 0 0 0

29.558289 6.444312 3.789316 3.275460 3.437803 3.660280 3.908871

0 0 0 0 0 0 0

7.213911 1.007168 0.606198 0.405592 0.618989 0.820444 0.869289

0 0 0 0 0 0 0

7.404082 6.283546 4.689887 4.061816 4.640575 5.041940 6.024519

0 0 0 0 0 0 0

For the variants PSO2 and PSO4 the optimal population size seems to be around 30 to 40 when averaging over all three test functions. This is in good agreement with the results of Carlizle and Dosier [17]. A similar result is also obtained for the variant PSO3 with about 40 particles. Regarding the algorithm performance we can see similar results as before, namely the good performance of the variants PSO2 and PSO4 for the Levy and Rastrigin

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function and the algorithm PSO3 performed very good for the Rosenbrock function. The greatest discrepancy regarding the population size is obtained for variant PSO1, where the best results were computed for large population sizes. Especially for the Levy and Rosenbrock function the best result is given for a population size of 50 or 60. This behavior can be explained by the collapse of the movement to line search in the case that no new local or global optimum is found. Therefore, the high number of random walkers can prevent this behavior to a certain degree and also can cover a larger amount of the search space. Nevertheless, the results for PSO1 are not as good as for the other algorithms even for a population size of 60. Dependence on Dimensionality: Another important point is the performance of the algorithms depending on the dimensionality of the problem. In order to allow the algorithms to fin adequate optimization results also in higher dimensions, we adjusted the maximum number of allowed objective function evaluations to be 2000·d in d dimensions. The results for the three chosen benchmark functions are summarized in Tab. 6. From this table we can see some interesting behaviors which can help us to identify superior behavior.

D

PSO1

PSO2

PSO3

Levy

2 5 10 15 20 25 30 40 50

0.000000 0.112268 2.364631 4.075384 6.295161 11.126197 11.601619 19.356817 27.892140

0.000000 0.000000 0.000000 0.143859 0.558245 1.386592 2.194235 6.807962 11.452513

0.000000 0.256649 1.834876 4.529678 6.771720 9.929187 10.481226 16.191035 22.006409

0.000015 0.001136 0.000000 0.000025 0.035542 0.148262 0.663576 2.702797 7.412729

Rastrigin

2 5 10 15 20 25 30 40 50

0.310168 5.778121 17.225120 34.287237 58.688028 92.760770 117.669800 193.617488 276.711109

0.000000 0.649773 4.377886 9.502961 22.088305 39.883554 48.454424 93.333555 119.127738

0.149244 6.050416 15.824239 28.107544 42.832892 58.005958 68.154538 108.052266 125.961506

0.179326 1.229930 4.037273 9.854636 16.866711 26.151316 24.648357 62.263781 97.578342

Rosenbrock

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Table 6. Average results over 20 runs for PSO variants depending on the dimension. PSO4

2 5 10 15 20 25 30 40 50

0.000004 1.178776 7.931421 19.244341 38.064974 67.278711 104.720292 180.182921 257.749798

0.000002 0.418176 3.275460 9.290394 14.848237 22.725736 40.952690 70.631538 92.944353

0.000000 0.786169 0.405592 1.329256 8.200706 16.414125 23.451070 35.163330 44.817162

0.000093 0.539374 4.061816 9.473437 14.368681 19.372121 31.103120 37.103120 57.187222

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(a) PSO2

(b) CPSO

(c) Deterministic CPSO

(d) PSO4 (Random CPSO)

147

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Figure 6. Convergence Behavior of different PSO variants: the red line indicates the best so far value and the green line the current average value over the whole swarm (see [22]).

We already ruled out the algorithm PSO1 due to its worse performance, which can be again seen from these results. For the previously similar behavior of the algorithms PSO2 and PSO4, we now fin that PSO4 performs better with increasing dimensionality. As expected for higher dimensions the algorithm solutions become worse but for the algorithm PSO4 this process is much slower than for PSO2 and thus the results in 50 dimensions already differ significantl . This behavior is also visualized in Fig. 6, where the best so far solution as well as the current average solutions of the whole swarms are shown for different methods. This includes PSO2 and PSO4, but also CPSO (i.e. without merging of groups) and Deterministic CPSO (i.e. the pairwise collapsing of groups). While PSO2 converges very fast to a local optimum and the average state of CPSO does not converge at all, for the CPSO approaches with merging the convergence is postponed and better results are obtained. In the latter case the results for the almost constant reduction of the charge sum in Random CPSO (PSO4) are better compared to Deterministic CPSO. Regarding the comparison of the algorithms PSO3 and PSO4 the result is not so obvious because PSO3 yields better results for the Rosenbrock function and worse results for the Levy and Rastrigin function. This gives the algorithm PSO4 a small advantage compared to PSO3 but the main criteria for the preference of PSO4 is its significantl faster computation. Tests showed that for PSO4 already in 30 dimensions the computation was more than f ve times faster compared to PSO3. The difference is caused by the computation of the rotational matrices, which requires among others O(d2 N ) additional random numbers in PSO3, which requires more time as the computation of the repulsive force in PSO4. As a result PSO2 converges very quickly and find good solutions to problems of small dimensionality and PSO4 returns better results in complex search spaces with an acceptable increase in running time.

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Visualization of the Convergence Process: It can be useful to visualize the movement of the particles in the solution space in real time to get an insight into the convergence behavior.2 The visualization shows that the swarm slowly concentrates more and more around the global optimum during an optimization run. Thus, the swarm is able to explore the region around the optimum very well and therefore find in most cases the best possible solution. An exemplary behavior of the individuals is shown in Fig. 7 for the Ackley function in ten dimensions, where all plots show the individuals in the plane (x0 , x1 ) of the tendimensional space. Red dots are the current positions, blue circles the local best solutions and the green circle is the global best solution.

(a) 2D Plot

(b) Initial configuratio

(c) After 10 iterations

(d) Final configuratio

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Figure 7. Visualization of the search behavior of PSO for the Ackley function. The visualization can also help us to fin reasons for the bad convergence behavior for some of the test functions as shown in Fig. 8 for the Rastrigin and Rosenbrock function. For instance most of the vector components of the fina solution of the Rastrigin function have reached their optimal value but there are still one or more vector components in a neighboring local minima. For the Rosenbrock function instead we can see that the swarm converges very slowly and is still distributed along the valley of the Rosenbrock function, where the individuals mostly search at the slopes of the valley and only slowly move to the global optimum. Thus, with a limited number of function evaluations the swarm has not the change to converge at all.

(a) Rosenbrock function

(b) Rosenbrock function

(c) Rastrigin function

(d) Rastrigin function

Figure 8. Visualization of the search behavior of PSO for different functions.

2

The update frequency can be specifie as simulation parameter to save computation time.

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3.5.

149

Influence of Heuristics and Boundary Conditions

In this section we determine the influenc of different heuristics described in Sec. 2.3. on the optimization results. We again restrict our considerations to the Levy, Rastrigin and Rosenbrock function and test the performance of all four algorithm variants for a selection of heuristics introduced in Sec. 2.3. and different boundary conditions from Sec. 2.4.. The results are shown in Tab. 7 and the basic insights are summarized below. ◦ The application of the initial velocity restriction includes the choice of random velocities with a size of 10% of the search space domain. This heuristic has only a minor influenc on the performance of the algorithm for the variants PSO2, PSO3 and PSO4, where we observed no improvement or even a decline of the performance in all cases. In the case of the PSO1 algorithm this heuristic is beneficia due to the introduction of an additional diversity into the algorithm in order to prevent the collapse to line search. Unfortunately, the influenc of this heuristic ranges only to the firs iterations since the system will steadily forget its initial state and thus the improvement of PSO1 is minor.

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◦ The minimum velocity restriction restricts the velocities to a minimal value which is in our case chosen to be equal to vmin = 0.01. This heuristic worsens the performance of the algorithms PSO2, PSO3 and PSO4. Even for PSO1 there was almost no improvement of the performance in all cases since the performance gain due to the introduction of additional diversity to this algorithm variant was not predominant to the disadvantages of this heuristic, namely the prevention of the swarm to fi ally contract at the so far global optimum. ◦ The maximum velocity restriction limits the velocity components to a vmax value which was chosen to be one eighth of the search space size. This size was suggested by Carlisle and Dosier [17]. However, we did not obtain any improvement of the results for the algorithms PSO2, PSO3 and PSO4. As expected, the algorithm PSO1 was improved significantl since also the maximum velocity prevents the collapse to line search. ◦ We applied a number of different boundary conditions. For all experiments before we used invisible boundary conditions because this is automatically given due to the domain specification of the test functions and by not evaluating them outside of the search domain. The previously obtained results are again shown in the table for a better overview of the results. ◦ The absorbing boundary conditions sometimes improve the performance and sometimes not. Basically, there is no improvement for the variant PSO2, the results for PSO3 are ambiguous and we get a slight improvement for PSO4 as well as a clear improvement for algorithm PSO1. These boundary conditions yield the best results for PSO1 and thus should be used in connection with its application. Regarding the variant PSO4 the improvements are minor and could also be due to statistical noise in the tests. ◦ Bad convergence is obtained for reflecting boundary conditions for all four algorithms. This is most likely caused by not resetting the velocity of the particle. ThereParticle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

150

Stefanie Thiem and J¨org L¨assig fore, the particle has a very high velocity and randomly wanders often in the outer regions rather then converging to a good solution.

All in all, the greatest influenc of boundary conditions has been obtained for the algorithm PSO1, where absorbing boundary conditions clearly give the best performance increase. However, it cannot compete with the results from the other variants. The influenc of the heuristics and boundary conditions to the other three methods are smaller. For PSO2 and PSO3 no general improvement is found and one should use the plain algorithms instead of spending additional computational efforts for calculating the heuristics. For PSO4 absorbing boundary conditions show slightly better results. However, further testing is required to ensure this behavior. From our results we propose that the best method for continuous problems presented by this set of test functions of low dimensionality is PSO2 in combination with invisible boundary conditions and a parameter setup as given in Sec. 3.2.. For more complex solution spaces it can be beneficia to postpone the fast convergence of PSO2 for example by an repulsive force as in PSO4, which shows the best performance for higher dimensions.

PSO2

PSO3

PSO4

Levy

initial velocities minimum velocity maximum velocity invisible boundaries absorbing boundaries reflectin boundaries

1.543018 1.417102 0.452399 2.364631 0.100688 9.835597

0.000000 0.022792 0.000000 0.000000 0.027194 0.351737

1.890229 1.372000 2.068814 1.834876 0.758856 3.277196

0.000000 0.022785 0.000000 0.000000 0.000000 0.048566

Rastrigin

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Heuristics

4.

PSO1

initial velocities minimum velocity maximum velocity invisible boundaries absorbing boundaries reflectin boundaries

16.921589 19.545045 13.209399 17.225120 6.120772 72.858582

4.477325 4.172876 4.678266 4.377886 4.576811 11.323256

18.456456 19.850581 18.236328 15.824239 17.411761 30.223059

3.581899 3.728923 3.683502 4.037273 3.284141 15.873847

Rosenbrock

Table 7. Average results over 20 runs for the PSO variants depending on several heuristics and boundary conditions.

initial velocities minimum velocity maximum velocity invisible boundaries absorbing boundaries reflectin boundaries

7.546446 5.319546 0.518785 7.931421 5.280166 114.336727

3.102712 5.368353 3.074799 3.275460 3.490276 4.562828

0.607351 0.563634 0.605660 0.405592 0.607636 14.195142

3.298437 5.448133 3.578654 4.061816 3.552367 46.223531

Comparative Analysis to Alternative Methods

While the focus of our studies in this chapter is to compare different approaches to particle swarm optimization and to identify design characteristics, which seem to be superior compared to others, it is also important to know how good or how bad the PSO approach performs compared to other standard optimization heuristics because this may be essential for the choice and decisions of a practitioner.

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Hence, in this section we take different other meta-heuristics into account and compare their performance with the performance of the PSO approach for exactly the same problems as investigated above.

4.1.

Global Optimization Heuristics

For the solution of these problems, i.e. benchmark functions in the continuous domain, one can choose between a large number of general purpose optimization heuristics [34] (often called metaheuristics). An obvious disadvantage of this algorithm class is that the performance can be arbitrary bad, which means we do not know, how far from the optimal solution we are still away. But the advantage of these methods is that they perform very good in practice and can fin “good” solutions in moderate time. These methods can be classifie in two major categories: 1. There are relatively simple iterative improvement approaches as Simulated Annealing (SA) [7], Threshold Accepting (TA) [8] or the Great Deluge Algorithm (GD) [35] which deal only with one current solution and one candidate solution at the same time but,

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2. there are also more involved techniques which deal with more than one candidate solution at the same time as Genetic Algorithms (GA) [1] and Evolution Strategies [11] or newer paradigms as Ant Colony Optimization [3] or, as introduced in the paragraphs before, Particle Swarm Optimization (PSO) [4]. Also ensemble-based techniques of the simple iterative approaches as Ensemble-based Simulated Annealing or Ensemble-based Threshold Accepting (EBTA) [9] are representatives of this second group of algorithms. A general definitio of global optimization heuristics is, that these are iterative methods which perform a random walk in a solution space S based on one state α or a vector of states α to fin a global minimum or minimizer. Further, these methods are understood to be of the A(0) type, utilizing only function values of the objective function but no derivations.

4.2.

Single State Methods

Algorithm 7 is a template for a class of methods, which deal only with one current solution and one candidate solution, also called Monte Carlo-type techniques. The degrees of freedom within this template are the initial state, the temperature schedule, the move class, the acceptance condition and the terminating condition. Comparing different methods, the initial state can be chosen at random identically for all methods and the terminating condition is a fi ed number of iterations. For different designs of Pβα , the algorithm shows a different behavior. This is essentially equivalent to the choice of the strategy to get new states from the old ones. Different methods are described by Equ. (4.1) to (4.5). Focusing on this design criterion for algorithms which operate only on one state, the simplest approach besides random search with (RA) Pβα =1

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(4.1)

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Stefanie Thiem and J¨org L¨assig

is to apply the greedy algorithm using transition probabilities  1 | if ∆f ≤ 0 (GR) = Pβα 0 | if ∆f > 0 .

(4.2)

Algorithm 7 Monte Carlo-type techniques Input: problem P with a solution space S Output: solution αfina ∈ S 1: α ← generate initial state(S) 2: for t ← 1 to ∞ do 3: T ←get new temperature(α, t, T ) 4: β ←get new state(α, t, S) 5: acceptance condition ← true with probability Pβα , f alse otherwise 6: if (acceptance condition) then 7: α←β 8: if (terminating condition) then 9: return α

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The parameter ∆f = f (β) − f (α) is the difference between the objective function values of the new and the current state. A general disadvantage of the greedy algorithm is that it can get stuck in local optima. It is well known that Monte Carlo-type optimization algorithms like SA or TA are able to overcome local optima. For SA one uses then  1 | if ∆f ≤ 0 (SA) (4.3) Pβα = e−∆f /(kB ·T ) | if ∆f > 0 ,

where a temperature parameter T is introduced, which gets in general reduced during the algorithm execution. For TA one uses the simpler step function for Pβα :  1 | if ∆f ≤ T (TA) (4.4) Pβα = 0 | if ∆f > T . Franz et al. [36] were able to prove that Equ. (4.4) is an optimal decision rule for this kind of algorithm (but this depends on the choice of other parameters, especially on the choice of the temperature schedule, for which the optimal choice is in practice unknown). A further simplificatio of TA results in the Great Deluge Algorithm with  1 | if E(β) ≤ T (GD) Pβα = (4.5) 0 | otherwise .

4.3.

Temperature Parameter

For Monte Carlo-type algorithms there have been several strategies for the temperature schedule investigated in literature. Especially for the ensemble based strategies there are adaptive schedules which show excellent performance, as e.g. constant thermodynamic speed scheduling [10]. For comparative studies it is more convenient to use a temperature schedule which realizes a temperature reduction to a target temperature in a f xed number I of available iterations. This is necessary for a meaningful comparison. A standard technique to do this is to use an exponential schedule Tnew = c · T

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(4.6)

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153

in each iteration step, where the parameter c is chosen as     Tfina  c = exp ln I . T0

(4.7)

In the setup we used an adaptively obtained start temperature using T0 = c · σE|T =∞ =

q hE 2 iT =∞ − hEi2T =∞ ,

(4.8)

where c is a small constant, except for GD, where higher values are necessary. The energies at T = ∞ are obtained by realizing an unbiased random walk with the acceptance rule as given by Equ. (4.1). As temperature schedule we used exponential cooling according to Equ. (4.6) and (4.7), where I is an in advance fi ed number of allowed steps of the method. The fina temperature has been chosen to be Tfina = 10−7 in general but to a different value if this improved the results for the given function.

4.4.

Move Classes for the Continuous Domain

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Applying different move classes means in the continuous domain basically to apply different probability distributions to choose the step width of the next step. One possibility is the variation of single dimensions of the current solution by a random value. There are several distribution possible and we will present here the three used by us. Uniform-Random-Move: Changes one coordinate xi of state α = x by a uniform random value in the range −max step to max step. This is the simplest move class within this context and only needs numbers from a standard uniform random number generator and a user define value for max step. Gauss-Random-Move: Changes one coordinate xi of state α = x by a Gaussian random number with standard derivation σ = max step and expectation value µ = xi . The Gaussian probability distribution is given by (x − µ)2 N (µ, σ) = √ exp − 2σ 2 2πσ 2 1





(4.9)

.

The random numbers are generated from uniform random numbers u using the Box-Muller method [37], which enables the generation of N (0, 1) distributed number and is given by p −2 ln(1 − u1 ) · cos(2πu2 ) p r2 ← −2 ln(1 − u1 ) · sin(2πu2 ) r1 ←

.

Due to the relation N (µ, σ) ← µ + σr, where r is Gaussian distributed with N (0, 1), we can easily compute the desired numbers. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Lorentz-Random-Move: Changes one coordinate xi of state α = x by a random number generated from the Cauchy-Lorentz distribution L(s, t) =

s 1 . 2 π s + (x − t)2

(4.11)

For the Cauchy-Lorentz distribution all moments are not define and thus this distribution has no expectation value, standard deviation and variance. However, the parameters s and t still have a meaning. The median of the distribution is given by xm = t and additionally for xm the function reaches its maximum value given by f (xm ) = 1/(πs). To generate random numbers from the Cauchy-Lorentz distribution, we can use the inversion method for which we need the cumulative distribution function F (x) =

1 1 x−t + arctan . 2 π s

(4.12)

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Since y = F (x), where y is a uniform random number in [0, 1) and x is a random number according to the Cauchy-Lorentz distribution, we can easily construct the desired random numbers by x = F −1 (y). For the application to the continuous function problem, the parameter t is given by the xi component and the parameter s is set to max step, because for larger values s also larger moves are possible. For a closer look at these three move classes, they are visualized in Fig. 9.

(a) Uniform distribution

(b) Gaussian distribution

(c) Lorentz distribution

Figure 9. Different probability distributions for move classes in the continuous domain. It is beneficia to choose the step width adaptive in a way that a fi ed percentage of the tried moves is accepted when applying some acceptance strategy as SA. We do this by multiplying it with 1−ε, ε ≪ 1, if the acceptance is too low, or with 1+ε, if the acceptance is too high. It seems to be convenient to defin a lower bound cmin and an upper bound cmax for the step width additionally. Basin Hopping: An approach which tries to improve the simple iterative improvement schemes as SA or TA is Basin Hopping (BH). BH tries to reduce the original state space S to a state space S ⋆ ⊂ S, consisting only of the local minima of the original state space. In each iteration a nearby local minimum to the current state is determined and with the rules as given in Equ. (4.3) or (4.4) it is accepted as a new state or rejected. In detail, to fin a nearby local minimum, one applies a move in a certain neighborhood of some initial size to

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155

a state α, getting a different state α′ and then one applies local search, e.g. using Equ. (4.2) to converge to a state β. If β 6= α then a criterion Pβα as e.g. Equ. (4.3) is applied and β is accepted as new current state or rejected. Otherwise, i.e. if β = α after the local search step, the neighborhood has to be increased to hopefully get out of the local optimum with larger jumps.

4.5.

Evolutionary Algorithms

Evolutionary Algorithms (EA) are methods which try to mimic the principles of evolution which are variation and selection in combination with a fitnes function to solve an optimization problem. In general, three classes of EA are distinguished: ◦ Evolution Strategies (ES) [11], ◦ Genetic Algorithms (GA) [38] and

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◦ Evolutionary Programming (EP) [39]. Today it is difficul to clearly allocate each known algorithm to only one of these classes. Major differences are classically the different application of operators and parameter adaption. For an introduction to GA and EP we refer to the literature [38, 39]. ES have been implemented for a comparison to PSO and hence are introduced briefl . In general, the two parameters µ and λ are fi ed in advance. The parameter µ is the population size, i.e. individuals are chosen from a population P (i) of µ individuals. Then the crossover or mutation operators are applied to produce new individuals, which are then added to an intermediate population. This procedure is repeated, until λ new individuals have been generated. Then for (µ + λ)-ES, the new population P (i+1) is chosen from all µ + λ individuals, choosing deterministically the µ best individuals. For (µ, λ)-ES the new population is only chosen from the λ new individuals. This is a good choice especially for dynamic problems, where the fitnes landscape changes over time. Very sophisticated self-adaption mechanisms have been developed, which are not introduced here, see e.g. [40]. For our studies here a simple (µ + λ)-ES has been applied, choosing µ = 30 and λ = 30. To optimize functions in the continuous d dimensional space, the move classes from Sec. 4.4. have been applied as mutation operators. Crossover has been realized by different strategies as taking the mid point between two chosen parent states, exchanging coordinates at a randomly chosen dimension between two states and also by classical one point crossover. Of course there are many more methods which could be compared to PSO, as e. g. Tabu Search which makes extensive use of memory, Model-based Search or Extremal Optimization. A good overview on different techniques is given in [41]. Here we restrict ourselves to the methods as introduced so far.

4.6.

Experimental Comparison

The algorithms are applied to a selection of benchmark functions. All results for the selected test functions are summarized in Tab. 8 and 9, where the average solutions for twenty runs are given. In general for at maximum 20,000 objective function evaluations PSO is the Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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superior method. Only for larger running times the other approaches show their benefit and certain algorithms even fin better solutions. The optimization run terminates when the global optimum is approximated by an additive error of 10−6 or at most 100,000 function evaluations are done. The algorithms as introduced above are compared to the PSO2 and PSO4 variant, where the results are also determined for 100,000 function evaluations. Again we average the solutions over twenty runs. Table 8. Comparison of the average results over 20 runs for different algorithms on continuous benchmark functions using different move classes in d = 10. Lorentz adaptive

BH

Ackley

Gauss adaptive

Greedy Simulated Annealing Threshold Accepting Great Deluge Algo. Ensemble Based SA Ensemble Based TA

19.688993 6.422984 15.392034 19.663482 14.906920 18.142196

19.620313 1.478764 1.366079 19.750241 3.478268 5.211392

3.845109 2.213738 2.142136 18.193347 2.352309 2.210025

19.499258 19.882887 19.857492 17.552375 0.000043 0.000042

15.149652 20.509993 20.437377 19.698334 0.000057 0.000038

0.000055 0.000053 0.000055 0.000078 0.000054 0.000059

Griewangk

Lorentz

Greedy Simulated Annealing Threshold Accepting Great Deluge Algo. Ensemble Based SA Ensemble Based TA

0.591554 0.673403 0.633474 0.871103 0.665590 0.654103

0.731671 0.737713 0.781158 0.975350 0.760320 0.667472

0.939084 0.959845 0.910945 1.217548 0.941715 0.924869

6.860117 0.305347 0.775209 278.61054 0.071385 0.107090

0.413350 0.146674 0.117154 200.78203 0.080507 0.097247

0.010174 0.008945 0.007473 0.017534 0.008686 0.007209

Rastrigin

Gauss

Greedy Simulated Annealing Threshold Accepting Great Deluge Algo. Ensemble Based SA Ensemble Based TA

81.398649 42.998451 40.136802 33.231085 37.018373 47.171130

94.143229 31.825698 33.395000 40.755985 37.530228 35.981680

9.425358 8.585823 8.266775 18.027384 10.938434 10.098198

77.606349 35.719338 25.665389 56.539263 31.739151 25.072942

46.265433 1.392988 1.282169 3.942567 8.457204 5.074334

9.452164 8.755690 9.352667 11.541578 10.347622 9.452178

Rosenbrock

uniform

Greedy Simulated Annealing Threshold Accepting Great Deluge Algo. Ensemble Based SA Ensemble Based TA

3.005618 0.962249 2.243364 3352.0658 2.803984 3.237021

2.906340 1.654565 1.913408 4008.6484 2.438594 3.232502

5.923406 4.276056 4.089847 6.192481 4.213036 3.580854

2.745259 1.205173 9.085765 2174.2755 1.227660 1.860593

0.411125 0.015108 0.007784 1840.8896 1.242303 1.084215

0.129283 0.102862 0.081947 0.116360 0.065023 0.020712

Step

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Algorithm

Greedy Simulated Annealing Threshold Accepting Great Deluge Algo. Ensemble Based SA Ensemble Based TA

-10.7 -60.0 -60.0 -28.8 -60.0 -60.0

-7.1 -60.0 -60.0 -50.6 -60.0 -60.0

-60.0 -60.0 -60.0 -60.0 -60.0 -60.0

-7.9 -59.6 -59.9 -53.6 -60.0 -60.0

-46.4 -60.0 -60.0 -60.0 -60.0 -60.0

-60.0 -60.0 -60.0 -60.0 -60.0 -60.0

Tables 8 and 9 show that the Step function is relatively simple to optimize for each method, at least if the acceptance rule is different from greedy and great deluge. For PSO and BH only the Rastrigin function is difficul to optimize, where PSO performs better for this function. ES perform much better for this function but has weaknesses applied to the Rosenbrock function. The Lorentz moves have several weaknesses but Ensemblebased TA performs very good on most functions except for the Rastrigin function, where

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Table 9. Performance of Evolution Strategies compared to other approaches. Algorithm

Evolution Strategy

Best results of Tab. 8

PSO2

PSO4

Ackley Griewangk Rastrigin Rosenbrock Step

0.004966 0.301860 0.004522 6.968686 -60.0

0.000038 0.007209 1.282169 0.007784 -60.0

0.000000 0.046014 3.631600 2.665125 -60.0

0.000000 0.051536 2.313972 2.793544 -60.0

interestingly simple SA and TA perform best. An important result is that for each of the applied test functions we found an algorithm which find the optimum or comes very close to it. This strongly suggests to use hybrid methods combining different approaches in one algorithm, which has been tested as well but is not reported here in detail. Because this chapter mainly focuses on PSO we should especially point out that at least the PSO variants PSO2 and PSO4 are very competitive. All other combinations of a specifi method and a specifi move class and also ES show at least for one benchmark function more significan weaknesses than PSO. This marks the end of the section and also the end of the analysis and comparison of different PSO methods by artificia benchmark functions. Our analysis showed that PSO is very competitive for these instances. In the last section of this chapter we focus on the application of PSO to real world problems for the exemplary case of optimizing a hub and spoke inventory system by simulation-based optimization.

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5.

Simulation-based Optimization of a Hub and Spoke Inventory System

Often real-world problems in economy and engineering can not be solved strong mathematically and often no specifi algorithms are known for these problems. Hence, one has to apply optimization heuristics to obtain good solutions. In this context often simulationbased optimization has to be used since the real-world problem itself cannot be described by an easy and exact model but rather has to be simulated. As one example, in the following the optimization of a hub and spoke inventory system is presented. Such systems are of special interest because in the age of globalization and increasing competition more and more companies are faced with the problem of how to control the fl w of goods from the production locations to the points of demand, i.e. definin optimal transshipment resources and using them efficientl to serve demands with a given number of transportation units [42, 43]. Hub and spoke systems, which consist of one central hub, which is connected to all spokes, arranged similarly to a bicycle wheel, have several benefit including the fact that for a network of d spokes, only d routes are necessary to connect all nodes and this generally leads to a more efficien use of scare transportation resources. Thus, the model is commonly used in industry, in particular in transport, telecommunications and freight, as well as in distributed computing. Additionally, complicated operations can be consolidated at the hub, rather than maintained separately at each

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node and therefore the spokes are simpler and new spokes can be connected rapidly [44]. Inventory systems with hub and spoke structure have been investigated before [42, 45]. In this section results are shown which in a given sense combine approaches from inventory theory and logistics to investigate the following problem [46, 47]: ◦ The system consists of one single warehouse, the hub, and a number of d retailers, the spokes. ◦ The hub owns an infinit amount of one specifi product, which can be ordered by the retailers, and a finit number of T transport units. Transportation orders from retailers, which cannot be served immediately due to a stock-out or no available transportation units, are queued in an order queue with given capacity. Ordering and storing of products as well as holding and leasing of transportation units and realizing transshipment generates costs. Thus, the following parameters should be concretized: number of products, storage capacities for all products, capacity of the order queue for waiting transshipment orders, number of own transportation units, and cost and gain structure with its parameters. ◦ The systems allows the realization of different classes of transportation units (trucks, pickups, ships, etc.). The parameters of each class are: the number of transportation units in this class, the loading capacity Q of a single transportation unit, the transportation time (velocity), which may be deterministic or random, and the cost structure with its parameters. ◦ Each retailer has a storage for the products with the current level x and there is a stochastic demand of these product units. Depending on the ordering policy each retailer can order new products from the hub. Arriving demand is handled according to an acceptance-rejection policy. Thus, the following parameters describe a retailer: distances from the warehouse, the number of products, the client classes with parameters of their demand process, the storage capacities for the products, and the cost and gain structure including holding costs for inventory on hand as well as waiting and rejection costs for backlogged or rejected demand. The whole system is realized in a simulation tool written by Hochmuth [48], which we will use as an integrated part in the optimization process. A screen shot of this system is shown in Fig. 10. Concretizing all the introduced parameters and choosing the variables and corresponding criteria to be optimized allows to model and optimize a large variety of real-world hub and spoke inventory systems. The optimization problem is to defin an allocation policy of free transportation resources to retailer orders for the warehouse, and for the retailers such ordering policies that minimize the long-run average cost in the system. We basically have the choice between two different options for the ordering policy: ◦ The (s, S)-ordering policy is the restocking to x = S product units when the inventory level of the retailer has fallen below the reorder point s. ◦ The (s, nQ)-ordering policy is the ordering of an integer multiple n of the capacity Q of product units when the inventory level of the retailer has fallen below the reorder point s.

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Figure 10. Screenshot of simulation for of hub and spoke inventory system. For our inventory system the solution αi of a swarm individual i encodes the variables for the ordering policy, i.e. the reorder levels sl for all retailers l and, if necessary, the number of transportation units tk for each class k as a real-valued vector. Since a feasible solution can only contain integer numbers, we simply round the real-valued entries of ri to obtain the input variables for the simulation. In our view this approach leads to the most natural movement of the swarm individuals in an integer solution space and, further, allows the unchanged use of the common PSO variants originally designed for continuous optimization problems. Another problem we faced for these systems is the stochastic noise in the objective function, which adds non-static local maxima and minima to the solution space. To reduce this effect one has to choose a sufficientl large time period for the simulation, and it is also advisable to average over several optimization runs. Further, we found by extensive tests that it is reasonable to initially start with relatively low reordering levels since the simulation of the system shows strong fluctuation of the total profi for higher reorder levels s due to fluctuation of the storage costs. In particular, for our test cases the initial values for the reorder level are chosen randomly in a region of the solution space with s < 20 and the initial velocities vi are set to zero. Since an evaluation of the objective function during the optimization in the PSOapproach requires the complete simulation of the whole system, we are dealing with a very time-consuming optimization problem. For instance a single simulation of the examples discussed in the following can take up to f ve seconds. Therefore, it is desirable to use optimization algorithms with powerful move classes instead of simple ones in order to minimize the number of such objective function evaluations. One example of such a move class is PSO2, which was identifie in Sec. 2.2.2. to converge very fast to an optimum.

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System Setup for Hub and Spoke System: In the following we show the optimization results for two test cases. The firs deals with a system of identical retailers and the second case with retailers varying either in their distances to the hub or in their demand structure. However, the general setup for all model elements is define at first For the warehouse we assume ◦ a FIFO service policy, and ◦ an allocation policy with priority sequence according to maximum velocity, maximum loading, minimum empty units, and minimum transport costs. The retailers are specifie by ◦ 200 km distance from the warehouse, ◦ a demand with exponentially distributed inter-arrival times with average value of 1 hour and demand of 1 product unit, ◦ impatient clients with zero waiting time (i.e. the lost-sales case), ◦ a storage capacity of 500 product units, ◦ cost parameters: inventory costs 50e per day and product unit, rejection costs 50e per rejected client and reward 200e per satisfie demand unit, and ◦ all retailers use a (s, nQ) ordering policy with reorder batch size Q, which is equal to the capacity of the used transportation units.

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For the transportation units we distinguish between two different classes: ◦ fast, small transportation units (delivery vans) and not so fast but larger transportation units (trucks), ◦ their average velocity is equal to 80 km/h and 50 km/h, respectively, ◦ the capacities are 1 product unit or 4 product units, respectively, and ◦ the corresponding costs for amortization 50e or 100e per day a transportation unit is used, 0.10e or 0.20e transportation costs per km and transported product unit, 10e or 30e loading/unloading costs, respectively. The optimization criterion is maximization of the total profi over the planning period, which is chosen to equal 3 years. For the evaluation of the objective function for each solution we average over 5 independent simulation runs with the same parameters in order to minimize fluctuations Further, the starting inventory is 10 product units for each retailer. By a few pre-considerations, we obtain that the average demand per day is about 120 product units. One fast transport unit can deliver at maximum 4.8 product units per day, whereas a slow transport unit can deliver 12 product units. This corresponds to either the usage of 25 fast or 10 slow transport units a day and also motivates the fi st example, where we will investigate the influenc of different numbers of available transportation units.

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Example – 5 identical retailers: In this example we use the setup for the model components as specifie above and the numbers of transportation units are chosen according to the following cases: a) 20 transportation units with velocity 80 km/h and capacity 1 product unit, b) 8 transportation units with velocity 50 km/h and capacity 4 product units, c) 10 fast transportation units and 4 slow transportation units with capacities according to a) and b), and d) optimization of the number of fast and slow transportation units. Additionally, we consider two setups for the reorder level s. Because we assume identical retailers, we firs choose identical s for all retailers. After this we optimize s individually for each retailer. The results are expected to be nearly identical for both cases and are shown in the format x⋆ = (s⋆ , t⋆ ) with the optimal reorder levels s⋆ = (s⋆1 , ..., s⋆5 ) and the number of transportation units t⋆ = (t⋆fast , t⋆slow ) (cp. Table 10). Table 10. Results for a hub and spoke inventory system with 5 identical retailers.

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Case

Ordering Strategy

identical reorder levels s for each retailer

a b c d

s⋆ s⋆ s⋆ s⋆

individual reorder levels s for each retailer

a b c d

⋆

s s⋆ s⋆ s⋆

=6 =7 = 18 =7 = (5, 5, 6, 5, 6) = (3, 11, 11, 11, 3) = (18, 18, 18, 18, 18) = (8, 9, 7, 7, 6)

Total profit in e

t⋆ = (33, 0)

14.425.646 13.487.500 13.058.093 19.014.137

t⋆ = (32, 0)

14.407.362 13.770.108 13.043.158 18.917.484

The search finishe for identical s in less than 10 iterations for PSO with 10 concurrent individuals. The results are shown in Tab. 10 and are identical or very close for this case, surely caused by the very low number of degrees of freedom, which are either only 1 or 3. For the case in which the reorder levels sl are optimized for each retailer l individually, we expect to obtain similar results. This is more or less satisfied Further, in case b), i.e. 8 large transportation units, there are noticeable fluctuation in the reorder level, which seems to be caused by only small differences between different reordering strategies in this area of the solution space. The solutions are also very close to that of identical s, which shows that PSO is able to adequately optimize this inventory systems with more degrees of freedom. When optimizing also the number of transportation units, we can increase the solution quality considerably. The results show that the operation of fast transportation units yields in this example much better results, and hence, the optimized solution only uses fast ones, where the optimal number of 32 or 32 transportation units is greater than necessary for the satisfaction of the average daily demand, given by 25 fast TUs. The reason might be that it is optimal for the system to have enough products in the storage in order to avoid rejection costs.

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Example – 5 different retailers, 2 types of TUs: In this example we use the setup for the model components as specifie above and optimize the reorder levels of the retailers as well as the number of fast and slow TUs for the following cases: a) Different demand in the retailers: The retailers demand is given by an exponential distribution with average inter-arrival times ET1 = 1.5h, ET2 = 1.2h, ET3 = 1h, ET4 = 6/7h and ET5 = 3/4h. This results in an average daily demand of 16, 20, 24, 28 and 32 product units respectively, which also sums up to the former 120 product units.

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b) Different distances from the warehouse: Again this is based on Example 1, but with different retailer-warehouse distances. We consider two variants, where variant 2 has the same average distances as Example 1: Variant 1: d1 = 50km, d2 = 100km, d3 = 200km, d4 = 300km, d5 = 400km Variant 2: d1 = 100km, d2 = 150km, d3 = 200km, d4 = 250km, d5 = 300km For the setup in a) we expect a general tendency towards an increase of the reorder levels from retailer 1 to 5 since the average daily demand also increases in this direction. The results are summarized in Tab. 11. For case a) we clearly can see the expected increase in the reorder levels. Further, the results are almost identical to the total profi of about 19 Mio. e obtained in Example 1. However, in the PSO solution the number of TUs is significantl increased compared to Example 1 and for both algorithms also the usage of a few slow TUs is found to be optimal. The reason for the different numbers of TUs is probably again the relatively fla shape of the solution space. In case b) we again obtain for both variants the expected increase in the reorder levels. This has been expected because the retailers with larger distance have to hold more products in the storage to compensate the longer delivery times. Since variant 2 has a shorter average distance than variant 1, the total profi of variant 2 is slightly better. Further, comparing the results of variant 2 with Example 1, which has the same average setup, we obtain very similar results for the total profi as expected. Table 11. Results for a hub and spoke inventory system with 5 different retailers. Case different demand different distances variant 1 different distances variant 2

6.

Ordering Strategy s⋆ = (5, 6, 7, 8, 8) s⋆ = (6, 6, 8, 9, 10) s⋆ = (5, 8, 7, 8, 9)

Total profit in e

t⋆ = (41, 5) t⋆ = (34, 0) t⋆ = (31, 0)

19.032.411 18.847.544 18.895.547

Conclusion

In this chapter we compared different approaches to particle swarm optimization. First four different variants have been introduced: the relatively simple classical approach by Kennedy and Eberhart which needs only two random numbers per iteration and individual in the swarm, a more involved version using matrices of random numbers and an optimized

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parameter setup by Clerc and Kennedy, a computational expensive approach applying matrix rotations and charged PSO, a version which comes in different fl vors, where a repulsive force term gets introduced. Besides the different equations of motion, which is the major ingredient of a PSO algorithm, many other characteristics can be modifie or constrained, i.e. the initial particle velocities, the feasible velocities during the algorithm execution, the weight factor and the neighborhood of the particles. Also different boundary conditions for the handling of the boundaries of the search space are applicable. A firs broadly based comparison of the four PSO variants for fourteen standard benchmark functions in ten dimensions for 20,000 function evaluations showed, that the PSO1 variant is not competitive compared with the other three approaches, but the results were inconclusive to fin a superior method within this group. But a disadvantage of PSO3 was already here a much higher computation time compared to PSO2 and PSO4.

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A closer look at the convergence behavior of the different methods for a subset of three hard test functions was the next step. First, the swarm size has been varied from 5 up to 60 individuals. Where the performance gain by using larger swarms is high for the simple PSO1 approach, the optimum for the other three variants is given with about 30 to 40 individuals in the swarm. Also the dimension of the problems has been varied between 2 and 50. This showed again that the PSO1 variant may not be the best choice. Comparing the other three methods applied to higher-dimensional problems, PSO4 performs better than the other two approaches, except PSO3 for the Rosenbrock function. But the advantage is moderate and PSO4 achieved its results in a much lower running time. In further investigations, some of the heuristics and boundary conditions turned out to be contra-productive as e.g. initial velocities for PSO2 to PSO4 or minimum velocities. Also maximum velocities had only for PSO1 a positive influence The same holds basically for absorbing boundary conditions (few improvements have been obtained for PSO4). Reflectin boundary conditions were contra-productive for all variants. In further comparative studies, other algorithms as Simulated Annealing, Threshold Accepting, Basin Hopping for different move classes, and Evolution Strategies were compared to PSO. The results show that PSO is competitive, especially for relatively low numbers of function evaluations. For higher numbers of function evaluations, other methods perform for some functions equally well or better. Again, the results are inconclusive - there is no method, which dominates all other methods. Finally, a study of the application of PSO to the simulation-based optimization of hub and spoke inventory systems showed that PSO can not only be applied successfully to benchmark functions but also to quite complex practical problems.

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[48] C. A. Hochmuth. Modellierung, Simulation und Visualisierung eines Hub-and-SpokeSystems. Term Subject, Chemnitz University of Technology, 2007.

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Chapter 8

PARTICLE S WARM O PTIMIZATION FOR C OMPUTER G RAPHICS AND G EOMETRIC M ODELING : R ECENT T RENDS Akemi G´alvez∗ and Andr´es Iglesias† Department of Applied Mathematics and Computational Sciences, University of Cantabria, E.T.S.I. Caminos, Canales y Puertos, Avda. de los Castros, s/n, E-39005, Santander, SPAIN

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Abstract Particle Swarm Optimization (PSO) is an evolutionary computation scheme originally described in 1995 by James Kennedy and Russell Eberhart. This technique has received increasing attention from the scientifi community during the last few years because of its ability to fin a solution to an optimization problem in a search space, and model and predict social behavior in the presence of objectives. Amazingly enough, PSO has barely been applied so far in the field of computer graphics and geometric modeling, with only a few references found in the literature regarding these topics. The present chapter is aimed at fillin this gap in two different ways: on one hand, by providing the interested reader with a gentle and updated overview on some applications of the PSO methodology to relevant problems in the field of computer graphics and geometric modeling; on the other hand, by identifying new lines of research in which PSO has the potential to become a fundamental tool in solving relevant problems. From our discussion it becomes clear that the original PSO technique along with its modification have a great potential as a powerful computational tool for many problems in those fields

Key Words: Particle swarm optimization, swarm intelligence, computer graphics, geometric modeling, artificia life, virtual crowds, human body pose estimation, geometric constraint solving, curve fitting surface fitting AMS Subject Classification: 65D10, 65D17, 65D18, 68T20, 68T32, 68T35, 68U05. ∗ †

E-mail address: [email protected] E-mail address: [email protected] ; Web site: http://personales.unican.es/iglesias

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1.

Akemi G´alvez and Andr´es Iglesias

Introduction

Swarm Intelligence is a concept and a methodology used in artificia intelligence, possibly firs proposed by Beni and Wang [4] in 1989. Swarm intelligence has been define as the property of a system whereby the collective behaviors of (unsophisticated) agents or boids interacting locally with one another and with their environment cause coherent functional global patterns to emerge [18]. A distinctive feature of swarm intelligence is the development of a collective behavior arising from decentralized systems comprised of (generally mobile) agents which communicate with each other (either directly or indirectly). Instead of a central behavior determining the evolution of the population, are these local interactions between agents which lead to the emergence of a global behavior for the swarm. In other words, there is no centralized control to manage the agents, but all agents or some of the agents, depending on the adopted neighborhood structure or other equivalent algorithmic design, exchange their information to cooperate and emerge group behaviors. Agents in a swarm intelligence system obey very simple rules, have limited perception or intelligence and cannot individually carry out the task it intends to. A typical example of swarm intelligence is the behavior of a floc of birds when moving all together following a common tendency in their displacements. Other examples from nature include ant colonies, animal herding, fis schooling and many others. Nowadays, swarm intelligence is attracting increasing attention from researchers and practitioners because of its potential applications in several fields For instance, selforganizing swarm robots can potentially accomplish complex tasks and thus replace sophisticated and expensive robots by simple inexpensive drones [61, 62], a research subfiel usually referred to as swarm robotics. Military and civil applications of swarm intelligence have also been reported in the literature with regards to the control of unmanned vehicles [63, 64, 65]. Applications of swarm intelligence range from crowd simulation in computer movies and video games to ant-based routing in telecommunication networks, and new exciting areas of research are constantly arising. The objective of swarm intelligence schemes is to model the simple behaviors of individual agents as well as their interactions with both the environment and their own neighbors in order to obtain more sophisticated behaviors that can be applied to solving complex problems, for instance optimization problems. In fact, a very popular swarm intelligence technique called Particle Swarm Optimization (PSO) has been proposed to solve optimization problems and fin solutions in a given problem space [40, 41]. Basically, PSO is a global stochastic optimization algorithm for dealing with problems where potential solutions can be represented as vectors in a N -dimensional space (the search space). In PSO, particles representing potential solutions are distributed over such space and provided with an initial velocity and the capacity to communicate with other neighbor particles, even the entire swarm. Particles ‘fl w” through the solution space and are evaluated according to some fitnes function after each instance. Particles evolution is regulated by two memory factors: their memory of their own best position and knowledge of the global or their neighborhood’s best. Particles of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. As the swarm iterates, the fitnes of the global best solution improves so the swarm eventually reaches the best solution. Original PSO algorithm was firs reported in 1995 by Kennedy and Eberhart in

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[15, 40] (see also [16]). The reader is referred to the excellent book in [41]. See also [18] for a gentle analysis of PSO from a computational point of view. In this context, the present chapter explores some recent trends regarding current and potential applications of PSO in order to solve relevant problems in the computer graphics and geometric modeling domains. Although the idea of applying PSO to simulation and optimization problems in computer graphics and geometric modeling has not been explored very well so far, some preliminary works have shown that PSO can be successfully applied to a number of problems, such as the realistic simulation of the behavior of synthetic actors evolving in digital 3D worlds, the evolution of virtual crowds in computer movies, virtual reality and videogames, the estimation of human body pose from pictures for computergenerated scenes, the geometric constraint solving problem, or the fittin of curves and surfaces from clouds of data points. The aim of this chapter is twofold: on one hand, it describes the main developments achieved so far regarding the application of PSO schemes to the above-mentioned problems; as such, the paper can be useful for beginners and practitioners. On the other hand, it identifie new lines of research in which PSO has the potential to become a key tool in solving relevant problems, so it can be of interest for experienced researchers in those fields We should remark however that the chapter has been influence by both the huge magnitude of this task and the limitations of space. Some references have had to be omitted and many explanations have been reduced to the minimum. Due to these reasons, neither the description nor the bibliographic entries in this chapter are intended to comprise a totally exhaustive survey on the subject. However, we hope that we have included enough comments and references to make the chapter useful for our readers.

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2.

Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a stochastic algorithm based on the evolution of populations for problem solving. PSO is commonly regarded as an efficient simple, and effective global optimization algorithm that can solve discontinuous, multimodal, and nonconvex problems. In PSO the particle swarm simulates the social optimization commonly found in communities with a high degree of organization. For a given problem, some fitnes function is needed to evaluate the proposed solution. In order to get a good one, PSO methods incorporate both a global tendency for the movement of the set of individuals and local influence from neighbors [15, 40]. PSO procedures start by choosing a population (swarm) of random candidate solutions in a multidimensional space, called particles. Then they are displaced throughout their domain looking for an optimum taking into account global and local influences the latest coming form the neighborhood of each particle. To this purpose, all particles have a position and a velocity and evolve all through the hyperspace according to two essential reasoning capabilities: a memory of their own best position and knowledge of the global or their neighborhood’s best. The meaning of the ‘best” must be understood in the context of the problem to be solved. For example, in a minimization problem that means the position with the smallest value for the target function. The dynamics of the particle swarm is considered along successive iterations, like time instances. Each particle modifie its position Pi along the iterations, keeping track of its best position in the variables domain implied in the problem. This is made by storing for

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each particle the coordinates Pib associated with the best solution (fitness it has achieved so far along with the corresponding fitnes value, fib . These values account for the memory of the best particle position. In addition, members of a swarm can communicate good positions to each other, so they can adjust their own position and velocity according to this information. To this purpose, we also collect the best fitnes value among all the particles in the population, fgb , and its position Pgb from the initial iteration. This is a global information for modifying the position of each particle. Finally, the evolution for each particle i is given by: Vi (k + 1) = w Vi (k) + γ1 R1 (k)[Pgb (k) − Pi (k)] + γ2 R2 (k)[Pib (k) − Pi (k)]

(2.1)

Pi (k + 1) = Pi (k) + Vi (k)

(2.2)

where Pi (k) and Vi (k) are respectively the position and the velocity of particle i at time k, w is called inertia weight and decide how much the old velocity will affect the new one, R1 (k), R2 (k) ∼ U (0, 1) are random values in the range [0, 1] sampled from a uniform distribution to introduce a stochastic element to the algorithm (and hence enrich the searching space) and coefficient γ1 and γ2 are constant values called learning factors, which decide the degree of affection of Pgb and Pib . In particular, γ1 is a weight that accounts for the “social” component, while γ2 represents the “cognitive” component, accounting for the memory of an individual particle along the time. The effect of the social component is that each particle is drawn towards the best position found by the particle’s neighborhood, while the effect of the cognitive component is that particles are drawn back to their own positions, resembling the tendency of individuals to return to situations or places that most satisfie them in the past. Finally, a fitnes function must be given to evaluate the quality of a position. Such a fitnes function is usually specialized for each given problem. Inertia weight plays an important role in directing the exploratory behaviour of the particles. Higher inertia values push the particles to explore more of the search space and emphasize their individual velocity. This kind of behaviour is useful when trying to coarsely explore the entire search space to fin a good starting point for a multimodal optimisation. On the other hand, lower inertia values force particles to focus on a smaller search area and move towards the best solution found so far. This approach makes sense when the global optimum region has been successfully identifie and all that remains is findin the exact optimum location in the search space. Similar to the choice of a fitnes function, the optimal value for the inertia weight is problem-dependent. Initial implementations of PSO used a static value for inertia weight while later implementations changed it dynamically. In the later case, it was usual to start with large inertia values, which decreases over time to smaller values. The rationale behind is that at the beginning particles are allowed to explore the search space, while exploitation is favored as time increases. The topology of the social network used by a PSO is at least as important in determining its behaviour and performance as are the details of the velocity update equation. Many variants of the canonical PSO and many topologies (such as star, ring, wheel, pyramid, four clusters, von Neumann and others) have been developed and tested in the last decade. We cannot review them here. The interested reader is referred to [18, 54] for a more detailed discussion on this issue. This procedure is repeated several times (thus yielding successive generations) until a termination condition is reached. Common terminating criteria are that a solution is found

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Table 1. General structure of the particle swarm optimization algorithm Create a swarm of population Pop begin k=0 initialization of individual positions Pi and velocities Vi in Pop(k) fitnes evaluation of Pop(k) while (not termination condition) do Calculate best fitnes particle Pgb for each particle i in Pop(k) do Calculate particle position Pib with best fitnes Calculate velocity Vi for particle i according to (2.1) while not feasible Pi + Vi do Apply scale factor to Vi end Update position Pi according to (2.2) end k =k+1 end end

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that satisfie a lower threshold value, or that a fi ed number of generations has been reached, or that successive iterations no longer produce better results. Global PSO procedure is briefl sketched in Table 1.

3. 3.1.

Applications of PSO to Computer Graphics Artificial Life

The simulation and animation of virtual worlds is one of the most exciting topics in computer graphics [1, 2, 5]. It is indeed a common task in the entertainment industry, ranging from virtual and augmented reality in movies to videogames and computer animation. But while it is unanimously recognized that the quality of the current computer generated scenes is very high, the simulation of the behavior of virtual agents (or avatars) still needs further development, specially for real-time applications. The challenge here is to provide the virtual agents with a high degree of autonomy, so that they can evolve freely with a minimal input from the animator [19, 38, 39]. In addition, this evolution is expected to be accurate, in the sense that the virtual agents must behave in a realistic way from the point of view of a human observer. This is the goal of a very exciting fiel known as Artificial Life, which is mostly based on the observation of natural systems. The core is to create a virtual environment inhabited by virtual agents that evolve in a life-like way. To this aim, several techniques emulating biological organisms and designed to manage different types of natural phenomena have been proposed (cellular automata, L-systems, neural networks, genetic

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algorithms, etc.). A pioneering work on self-animating virtual creatures is due to Reynolds1 . In his seminal paper [60] he discussed the problem of simulating the motions of a floc in computer animation. He realized that scripting each individual object path is both very tedious and error prone. Further, it would be almost impossible to avoid collisions. To overcome these limitations he proposed a different approach, based on the idea that the fl ck is the result of interactions between the behaviors of different boids and hence we must simulate the behavior of an individual boid. This represents a generalization of the particle systems, as boid particles interact with each other whereas simple particles do not. In addition, each boid has an internal state and a set of behaviors. The corresponding behaviors were expressed in rules of descending precedence for collision avoidance, velocity matching and floc centering, in order to avoid collisions with neighbors or obstacles, keep the same velocity that the neighbors and stay close to neighbors, respectively. With respect to the perception, it was assumed that each boid only takes into account the closest neighbors, so the sensitivity was restricted to a sphere of a certain radius centered on the boid itself. The corresponding function for the degree of sensitivity is an inverse exponential of distance, so it is dependent upon the value of the exponent. In addition, a certain degree of attraction-repulsion (similar to that of blobs or water droplets, for example) given inverse square relationship was also considered. From the point of view of swarm intelligence, this work is clearly a milestone towards the development of social-based heuristic techniques. It is even considered one of the main sources of many fundamental concepts in swarm intelligence, and more particularly in PSO. Inspired by Reynolds’ work, Terzopoulos [21, 68] proposed a framework for animating natural ecosystems with minimal input from the animator. In their seminal paper “Artificia fishes [68] presented at Siggraph’94, the authors described an approach to emulate the natural complexity of a virtual marine world inhabited by artificia fishe that can swin hydroynamically in simulated water through a motor control of internal muscles that motivate fins The core of that paper was to create realistic fishes in the sense that not only they look realistic, but also they move in a realistic fashion, and even behave realistically. From this point of view, the work was an example of both physically based modeling and behavioral simulation. His physically based modeling system was inspired by previous Miller’s work on motion dynamics of snakes and worms [49]. This author considered that each worn or snake body consists of segments, each modeled as a cube of masses with springs along each edge and across the face diagonals. At each time interval the forces exerted on the masses at the end of each spring can be computed by the formula: f = k(E − L) − D

dL dt

(3.1)

where f is the force along the spring direction, and k, D, L and E are the spring force constant, the damping force, the current length of the spring and the minimum energy spring length, respectively. With this previous idea in mind, the authors in [68] used a spring-mass scheme integrated through an implicit Euler simulation. Roughly, it consists of a set of 23 1

We recommend to visit his Web page on boids: http://www.red3d.com/cwr/boids/ with a lot of information links, source codes, software and others. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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point masses and 91 springs conceived in such a way that the fis is allowed to fl x while maintaining its structural stability. Twelve of the springs are used as muscles of a given length and with a contraction factor between 0 and 1, thus allowing swimming. The motor controllers are parameterized procedures with each one assigned to a specifi task (swim, left-turn and right-turn) for efficien y purposes and with swimming speed and turn angles proportional to the contraction amplitudes and frequencies of the muscle springs. The sophisticated motion system also includes other more complex motions, such as pitching (up and down motion) and yawing (side to side motion). The perception subsystem works in two different stages: firs the basic behaviors such as obstacle avoidance, and then more complex ones. There are two sensors: one for temperature located at the center of the body and another one for vision. The visual sensor has access to geometries, materials, and lighting. The vision also allows the fishe to identify nearby objects, as well as to obtain information about the velocity of objects and others. The range of vision covers a 300 degree spherical angle and the criterion is that an object is “visible” if any part of it is in this view volume and not occluded by another object. The behavioral system generates different intentions based on the fis ’s habits, mental state, and incoming sensory information. Habits are determined by the animator by choosing the set of preferences, (male or female, likes brightness or darkness, etc). There are also three mental states: hunger, libido and fear, valued between 0.0 and 1.0 according to some mathematical expressions depending on parameters such as the digestion rate, the appetite of the fish time since last mating, distance to predators, etc. Finally, each fis has eight behavior routines: avoiding-static-obstacle, avoiding-fish eating-food, mating, leaving, escaping, wandering-about, and schooling (see [68] for details on each routine). The combination of perception, behavior, and motor systems create fis that are autonomous while unprogrammed behaviors may be eventually created. These advanced features allowed the authors to obtain a very realistic simulation of a marine world. However, the number of different routines that fishe can exhibit is not certainly quite large and the system seems to replicate the behavior phenomena over the time. Undoubtely, human behavior is much more complex than that of fishe and, consequently, the simulation of human behavior is, by large, a much more varied and difficul task. It has been accomplished by, among others, Prof. Thalmann and collaborators at the Swiss Institute of Technology [7, 50, 67]. In their papers, emphasis was put on giving the virtual actors a higher degree of autonomy without losing control. For instance, in [7] the virtual agents were handled by means of an extended version of the BDI (beliefs, desires and intentions) architecture described in [58]. The extension is given by a categorization of beliefs in order to include the short-term and long-term memories, the inclusion of internal states for the agents to simulate basic emotions and mental states, realiability on trust so that agents trust to each other and the inclusion of emotions in the evolution plans. This scheme was later improved in [50] to integrate into the same framework a set of different techniques for simulating behaviors of virtual humans. In particular, the terminology of low and high-levels were used to split up the physical elements (for the virtual human, it includes the body, the basic motions and objects interactions), to be included in the low-level structures, and the behavioral elements that simulate the human brain, which are collected in the high-level modules. Finally, the human-objects interaction was accomplished through the so-called smart objects [38], i.e., objects that have not only a graphi-

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cal representation but also a description of their functionalities and how they can be used. With this combined architecture the authors obtain convincing simulations of many human behaviors evolving in a carefully chosen scenario (a journal office that exhibits a lot of potential inter-agents cooperations and interactions. More recent developments include the application of neural networks and other Artificia Intelligence techniques to the animation and control of physics-based models [26], the analysis of the behavior of crowds [59] (see also next section), the interaction with objects in an intelligent way [23] or the control of virtual agents with PDAs [25]. From previous paragraphs, it turns out that there is still a long way ahead towards the realistic simulation of the behavior of virtual avatars. In our opinion, PSO might play a very important role in the design of this much-needed behavioral engine for our synthetic actors. However, up to now no serious attempts have been described in the literature. Some preliminary trials were partially done in papers [33, 34, 35, 36, 45, 46, 47, 48] although PSO was just a part of an artificia intelligence module comprised of several components that cannot be decomposed in its fundamental building blocks, making it hard to determine the potential benefit of PSO at its own. It is our belief however that PSO may become a promising tool for some specifi optimization-oriented tasks.

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3.2.

Realistic Simulation of Virtual Crowds

Once behavioral animation of a single avatar is carried out, next task is to extend it to a (possibly massive) population of virtual avatars. Nowadays, computer simulation of virtual crowds is becoming a very important issue in several entertainment-related domains such as videogames, movies, commercials and the like. This problem also arises in other field such as site planning, education, training, and human factors analysis for building evacuation. Other applications include simulations of scenarios where masses of people gather, fl w, and disperse, such as transportation centers, sporting events, and concerts. Generating computer models of virtual crowds is a technique with increasing importance in computer animations and theater movies (Mulan, Lord of the Rings, Star Wars, Jurassic Park, Shrek, Bug’s Life). It is also a very challenging one, with many issues to be taken into account, including the extremely high computational cost required by virtual crowds simulation systems. For instance, to produce AntZ, two systems were built in order to provide tools to be used for controlling the background characters (over 60.000) with almost the same fl xibility as they had with main characters. A few works in the literature have shown that evolutionary algorithms can assist programmers and computer scientists in many tasks involved in virtual crowds simulation. In [44] some experiments of dynamic swarm behavior patterns using an interactive evolutionary algorithm were discussed, while [42] incorporated several specificall designed mechanisms into the conventional particle swarm optimization methodology for simulating decentralized swarm agents. Previous papers did not consider the PSO technique as a primary tool for virtual crowds simulation. But in a very recent paper [10] authors used PSO for controlling the movement of crowds in computer graphics. According to their proposal, the ability of PSO to reach the position of the optimum creates the possibility to automatically generate non-deterministic paths of virtual human beings from one specifie position to another. The key is to create

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an analogy between a particle swarm and a human crowd. A person can be considered as a particle, which would like to fin a way to reach the best solution. However, the authors realized that although PSO has certain characteristics of the crowd behavior, further work is still needed for controlling the crowd movement in computer graphics. Firstly, particles in PSO are absolutely free to fl everywhere in the given multidimensional space, i.e., the search space. However, the given environment for a crowd may have obstacles, and the pedestrians in the crowd must avoid collisions, including those with given obstacles and their fellow pedestrians, where other pedestrians can be considered as dynamic obstacles. These dynamic obstacles are not predictable and may appear and disappear in the environment at any moment. Moreover, it is important to make a virtual pedestrian to walk smoothly and naturally, instead of just oscillate uncertainly and strangely. The walking path must be reasonable and appropriate. To make a truly reliable and realistic model for the movement of virtual crowds, some additional considerations are to be made. For instance, PSO only concerns the fina positions of particles but cares nothing about the particle paths at each step. It is very different from crowds in the real world, where not everywhere can be stepped on or gone through. In real world scenarios there are also obstacles, such as holes and walls. Moreover, a person normally also does not step on another person. To deal with these situations, the authors considered a modificatio of PSO in which velocity is decomposed into two components, the direction part and the speed part. The speed part is used to model and match the various paces of different people and has a maximum limit. Each particle holds the information about itself, including a position, a direction, a speed, and an objective value. The direction part is computed according to PSO rules, and so is the position vector, which depends on the speed part, which takes values between zero and a maximum speed set adopted by the user. The speed is updated proportional to the reverse of the particles objective value so that it can fi the different pace of each person and make the environment more dynamic. For example, if a particle approaches an obstacle, the speed will be slower, due to greater objective values for the particle to avoid the obstacle. Authors also considered a cost function used during the optimization process. If a particle approaches the target, it should get a better objective value, modulated by this cost function. All the particles in the crowd move toward the region with lower objective values. On the other hand, the objective value rises as a penalty if the particle comes close to or even possibly touches obstacles. This general strategy can eliminate the collision issues and reduce the oscillatory situation that occurs in PSO. Therefore, generated paths may look more natural and realistic. The proposed model can be used in several different scenarios, including static obstacles, moving targets, and multiple crowds. This work clearly exemplifie the power of the PSO methodology in solving problems for virtual crowds simulation.

3.3.

Human Body Pose Estimation with PSO

Human body pose estimation is currently a very active research area with remarkable applications in many domains including surveillance, motion capture, behaviour analysis, medical analysis, human-computer interaction and animation. Basically, this fiel concerns the study of algorithms and systems that recover the pose of an articulated body, which consists of joints and rigid parts using image-based observations. As such, it is a classical problem

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in computer vision, and also a very challenging one because of the complexity of the models that relate observation with pose, and because of the variety of situations it might be applied to. One of the major difficultie in recovering pose from images is the high number of degrees-of-freedom (DOF) involved in the body model to be recovered. Any rigid object requires six DOF to fully describe its pose, and each additional rigid object connected to it adds at least one DOF. A human body contains at least 10 large body parts, meaning that we have more than 20 DOF. Besides, we should face the problem of self-occlusion, where body parts occlude each other depending on the configuratio of the parts. Other challenges are related to lighting conditions, which affect appearance, different human attire or body type, camera configuratio settings and demanding computation time. In a recent paper [37] PSO has been successfully applied to address the problem of upper-body pose estimation for application in immersive videoconferencing. In the target application the person is seated at a conference table with only the upper body visible, so only modeling of the human body from the waist up is actually concerned. In that work, the human body is based on a layered subdivision surface model comprised of two layers, the skeleton and the skin. The skeleton layer is represented as a set of transformation matrices which encode the information about the position and orientation of every joint with respect to its parent joint in the hierarchy. The skin layer represents the second layer in the model and is connected to the skeleton through the joints’ local coordinate systems. Each of the joints controls a certain area of the skin. Starting from a coarse model, subdivision is performed by successive refinemen according to Catmull-Clark subdivision rules to produce the fina smooth body model. In that work, search space is the space of all plausible skeleton configur tions. Body pose is represented with 20 degrees of freedom, 3 translations and 17 rotations. Individual particle’s position vector in the search space is specifie by the position of the root joint with respect to the world coordinate system (3 parameters) and rotations around the coordinate axes of the root joint coordinate system. In the reported experiments, the values of the learning factors γ1 and γ2 in Eq. (2.1) were both set to integer 2, which on average made the weights for social and cognition components of the swarm equal to 1. The evaluation function compares the silhouettes of the original images acquired by the cameras and the silhouettes generated by the model in its current pose. Original images were acquired from four different viewpoints, namely left, centre, right and top. Execution of PSO showed that the method as described in Section 2 fails as the optimisation regularly returned local minima, i.e., the estimated pose offered a plausible interpretation of that of the person in the sequence, however, the silhouette overlap of the estimated pose and the real pose was not the best possible. To tackle this issue, authors noticed that there is an inherent hierarchy in the problem: hierarchy of the upper body’s kinematic structure begins at the root joint, the movement of which influence the movement of all other joints. Similarly, the movement of the clavicle joint influence the movement of the shoulder and the rest of the arm. Taking advantage of this hierarchical structure, they could formulate the problem as a sequence of optimisation problems for which PSO is able to fin the correct solution. In particular, combination of original PSO algorithm applied hierarchically with the full overall optimisation to decrease the effects of error propagation achieved pretty reasonable results. Next goal of that work is to deal with video sequences, where temporal consistency is extremely important. To this purpose, the authors propose the use of PSO for tracking in order to

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produce a smoothly varying pose estimation for the entire video sequence.

4.

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4.1.

Applications of PSO to Geometric Modeling Geometric Constraint Solving

Geometric constraint solving refers to the problem of solving systems of geometric constraints. This is a problem that has been considered in many different fiel s (such as computer-aided design, mechanical engineering or manufacturing) and using qualitatively different approaches. For instance, the geometric modeling community has traditionally addressed this problem for the purpose of developing sketching systems in which a rough sketch, annotated with dimension and constraints, is instantiated to satisfy all constraints. Typically, two-dimensional sketches are comprised of points, lines, circles, segments and arcs, while constraints are given in terms of distances and angles, although other geometric constraints (such as parallelism, incidence, perpendicularity, tangency, collinearity, and others) are often considered as well. To make the problem even harder, over-constrained and under-constrained configuration may occur, deliberately or erroneously. Several schemes have been proposed to overcome previous limitations. They are generally based on the idea of translating the geometric constraints into a system of algebraic equations and solve them by using iterative methods [20]. Other approaches use rewrite rules for the discovery and execution of the construction steps or are based on an analysis of the constraint graph. However, PSO seems to be a very promising approach to deal with this problem. In [53] the standard PSO was applied to reconstruct 3D models from line drawings in sketching-based geometric modelers. To this purpose, learning factors have both been set to 1.5, while the number of particles was chosen in the range 3m ∼ 5m where m is the number of vertices in the drawing. The termination criteria is a combination of a maximum number of iterations (set to 1000) and that of no improvement after 20 consecutive iterations. The scheme was tested on some examples described in the literature. Obtained results were acceptable and they did not improve when increasing the number of particles. From the reported experiments, authors concluded that PSO is slower than other methods such as descent or gradient-based algorithms for those problems where all these algorithms work well, while PSO performs better under noisy conditions where descent algorithms are usually trapped in local minima or even in trivial optima. The main disadvantages of PSO are the high computational time required to achieve a good solution and that PSO parameters have to be adjusted manually for each example, meaning that no automatic procedure can actually be achieved for this process. A more recent approach for geometric constraint solving uses an improved chaos-PSO algorithm (labelled as MPSO) to transform the problem into an optimization problem [66]. The basic idea of this approach is to combine PSO and a chaos searching strategy in order to get away from local optima. In fact, a major drawback of PSO schemes is their premature convergence, especially while handling problems with several local optima. A possible solution is to consider sequences generated from chaotic systems as random numbers for the PSO parameters where it is necessary to make a random-based choice. By this way, it is possible to improve the global convergence and to prevent to stick on a local solution. On the other hand, chaotic sequences have been proven easy and fast to generate and store, as

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there is no need for storage of long sequences. Merely a few functions (chaotic maps) and few parameters (initial conditions) are needed even for very long sequences. In addition, an enormous number of different sequences can be generated simply by changing its initial condition. Moreover these sequences are deterministic and reproducible. In [66], authors used a one-dimensional quadratic chaotic mapping to generate different chaotic values randomly. Such values are then mapped onto the optimization variable area. By this procedure the optimization problem becomes a non-linear and multi-variable optimization function. A potential advantage of this approach is that it can solve not only under-constrainted but also over-constrained problems. Then MPSO is used to search the minimum of the transformed function; gbest is searched by PSO, when PSO gets into local extremum, particles are activated by improved chaos search strategy, which guides particle swarm toward the gbest by reducing search area of variables around the current best solution. This scheme was applied to some geometric constraint solving instances. Computer experiments showed that MSPO outperformed standard PSO in all trial cases and can solve geometric constraint solving problems efficientl .

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4.2.

Curve and Surface Fitting

Fitting curves and surfaces to measurement data plays an important role in real problems such as manufacturing of car bodies, ship hulls, airplane fuselage, and other free-form objects [11, 51]. It is a major issue in regression analysis in statistics [14], usually fittin an explicit polynomial function in one or several variables, plus a random noise term. Best approximation of points or functions is one of the main topics of numerical analysis, and is the source of most of the methods for performing practical fittin processes by computer [12, 13]. One typical geometric problem in Reverse Engineering is the process of converting dense data points captured from the surface of an object into a boundary representation CAD model [55, 56, 57, 69, 70]. Most of the usual models for fittin in Computer Aided Geometric Design (CAGD) are free-form parametric curves and surfaces, such as B´ezier, B-spline and NURBS [29, 52]. Curve/surface fittin methods are mainly based on the least-squares approximation scheme, a classical optimization technique that (given a series of measured data) attempts to fin a function which closely approximates the data (a “best fit”) Suppose that we are given a set of np data points {(xi , yi )}i=1,...,np , and we want to fin a function f such that f (xi ) ≈ yi , ∀i = 1, . . . , np . The typical approach is to assume that f has a particular functional structure which depends on some parameters that need to be calculated. The procedure is to fin the parameter values minimizing the sum S of squares of the ordinate differences (called residuals) between the points generated by the function and corresponding points in the data: np X (yi − f (xi ))2 . (4.1) S= i=1

It is well-known that if the function f is linear, the problem simplifie considerably as it essentially reduces to a system of linear equations. By contrast, the problem becomes more difficul if such function is not linear, since we then need to solve a general (unconstrained) optimization problem.

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In the following, we consider the case of f being a free-form parametric curve or surface. In the former case, we have a curve C(t) given by: C(t) =

M X

Pj Bj (t)

(4.2)

j=0

where Pj = (Pjx , Pjy , Pjz ) are the vector coefficient (usually called control points), {Bj (t)}j=0,...,M are the basis functions (or blending functions) of the parametric curve C(t) and t is the parameter, usually define on a finit interval [α, β]. Note that in this section vectors are denoted in bold. Now we can compute, for each of the cartesian components (x, y, z), the minimization of the sum of squared errors referred to the data points according to (4.1), but we need a parameter value ti to be associated with each data point (xi , yi , zi ), i = 1, . . . , np :  2 np M X X µi − Errµ = Pjµ Bj (ti ) ; µ = x, y, z (4.3)

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i=1

j=0

Coefficient Pj , j = 0, . . . , M , in (4.2) have to be determined from the information given by the data points (xi , yi , zi ), i = 1, . . . , np . Note that performing the componentwise minimization of these errors is equivalent to minimizing the sum, over the set of data, of the Euclidean distances between data points and corresponding points given by the model in 3D space. This problem is far from being trivial: because our curves and surfaces are parametric, we are confronted with the problem of obtaining a suitable parameterization of the data points. As remarked in [3] the selection of an appropriate parameterization is essential for topology reconstruction and curve/surface fitness For instance, many current methods have topological problems leading to undesired surface fittin results, such as noisy self-intersecting surfaces. In general, algorithms for automated surface fittin [6, 28] require knowledge of the connectivity between sampled points prior to parametric surface fitting This task becomes increasingly difficul if the capture of the coordinate data is unorganized or scattered. Most of the techniques used to compute connectivity require a dense data set to prevent gaps and holes, which can significantl change the topology of the generated surface. Therefore, in addition to the coefficient of the basis functions, Pj , the parameter values, ti , i = 1, . . . , np , corresponding to the data points also appear as unknowns in our formulation. Due to the fact that the blending functions Bj (t) are nonlinear in t, the least-squares minimization of the errors becomes a strongly nonlinear problem [70], with a high number of unknowns for large sets of data points, a case that happens very often in practice. Some recent papers have shown that the application of Artificia Intelligence (AI) techniques can achieve remarkable results regarding this problem [17, 24, 27, 30, 32, 43]. Most of these methods rely on some kind of neural networks, either standard neural networks [24], Kohonen’s SOM (Self-Organizing Maps) nets [3, 27], or the Bernstein Basis Function (BBF) network [43]. In some cases, the network is used exclusively to order the data and create a grid of control vertices with quadrilateral topology. After this preprocessing step, any standard surface reconstruction method (such as those referenced above) has to be applied. In other cases, the neural network approach is combined with partial differential

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equations [3] or other approaches. The generalization to functional networks, an extension of neural networks based on functional equations [8, 9], is also analyzed in [17, 31, 32] while [22] discusses the application of genetic algorithms and functional networks yielding pretty good results for both curves and surfaces. 4.2.1.

Best Least-Squares Approximation

Let us consider a set of 3D data points Di = (xi , yi , zi ), i = 1, . . . , np . We describe the procedure in more detail for the x’s coordinates (the extension to y’s and z’s is immediate). The goal is to calculate the coefficient cxj , j = 0, . . . , nb of (4.3) which give the best fi in the discrete least-squares sense to the column vector X = (x1 , . . . , xnp )T where (.)T means transposition, by using the model x(t) =

nb X

cxj Bj (t),

(4.4)

j=0

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supposing that ti (i = 1, . . . , np ) are parameter values assigned to the data points and the Bj (t) are the known blending functions of the model. Considering the column vectors Bj = (Bj (t1 ), . . . , Bj (tnp ))T , j = 0, . . . , nb and solving the following system gives the coefficient cxj :  T  x    B0 .B0 . . . BTnb .B0 c0 XT .B0    ..    .. .. .. .. (4.5)   .  =   . . . . x T T T cnb B0 .Bnb . . . Bnb .Bnb X .Bnb The elements of the coefficien matrix and the independent terms are calculated by performing a standard Euclidean scalar product between finite-dimensio al vectors. This system (4.5) results from minimizing the sum of squared errors referred to the xi coordinates of the data points. Considering all the xi , yi , zi coordinates, the solution of the three linear systems with the same coefficien matrix provides the best least-squares approximanb X tion for the curve C(t) = Pj Bj (t), where the coefficient Pj = (cxj , cyj , czj ) represent j=0

3D vectors. For the case of surfaces given in parametric form, one uses for Eq. (4.4) the strucnv nu X X Pi,j Bi (u)Bj (v), which is a tensor-product surface, a very common ture S(u, v) = i=0 j=0

model in CAGD. The coefficient Pi,j are the control points in 3D, arranged in a quadrilateral topology, and functions Bi (u) and Bj (v) are the same basis functions used for representing curves, for example Bernstein polynomials in the examples described below. The parameters u and v are valued on a rectangular domain [um , uM ] × [vm , vM ], a Cartesian product of the respective domains for u and v. If Bi (u) and Bj (v) are B´ezier basis functions, the (nu +1).(nv +1) bivariate polynomials Bi,j (u, v) = Bi (u).Bj (v), i = 0, . . . , nu , j = 0, . . . , nv constitute a vector basis for a linear vector space of polynomials in u and v on the square domain [0, 1] × [0, 1]. Given a cloud of points (xl,k , yl,k , zl,k ), in 3D, with a quadrilateral structure, l = 1, . . . , npu , k = 1, . . . , npv , and a set of parameter values

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(ul , vk ) associated one-to-one with the data points in the cloud such that these points form a cartesian set in the parameter domain, a discrete formulation similar to that for f tting points to a curve can be made. The best least-squares tensor-product surface fittin the points can be obtained using the system (4.5), in which the role of the B’s is now assumed by the bivariate basis functions Bi,j (u, v) described earlier. Next sections will discuss the application of these ideas to the cases of a B´ezier curve and a B´ezier surface. Fitting a B´ezier Curve

As a firs example we consider a B´ezier curve of degree d whose parametric representation is  given by Eq. (4.2) where the basis functions of degree d are define as: d Bid (t) = ti (1 − t)d−i , i = 0, . . . , d and t ∈ [0, 1]. In this example, we have choi sen a set of eigth 3D points to be fitte to a B´ezier curve of degree d = 4. The unknowns are 23 scalar values: a vector of 8 parameter values associated with the 8 data points, plus 15 coefficient (3 for the coordinates of each of the 5 control points of the curve of degree 4). The parameter values for the PSO algorithm are: population size: 100 individuals or particles, where each one is a vector with 8 components initially taken as an increasing sequence of random uniform numbers on [0,1]; inertia coefficien w = 1; coefficien for the global influenc γ1 = 0.2; coefficien for the neighbors local influenc γ2 = 0.8; number of neighbors locally influencin each particle: 5; and limit for not improving error iterations: 10. With these parameters, we obtain a curve with: best fittin error: 1.8104; mean error: 1.8106; number of iterations: 192; computation time (Pentium IV, 3 GHz, 1 GB. RAM, running Matlab v7.0): 2.6562 seconds. For these values, the optimum parameter vector is: [0.0028,0.0420, 0.0841, 0.2756, 0.4866, 0.7155, 0.8056, 0.9978]. The resulting curve is displayed in Fig. 1(left) while Figure 1(right) shows the evolution of the mean and best Euclidean errors along the successive iterations.

15

10

z

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4.2.2.

5

0 40 40

20 20

0 0 −20 y

−20 −40

−40

x

Figure 1. B´ezier curve fittin through particle swarm optimization: (left) B´ezier curve, its control points (stars) and the data points (spheres); (right) evolution of the mean (solid line) and the best (dotted line) Euclidean errors along the generations.

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Table 2 shows 20 executions of the PSO algorithm for this example and the same parameters used in the previous paragraphs. Columns of this table show the example number, number of iterations, the best and mean errors and the computation time (in seconds), respectively. For the sake of clarity, executions with the best results have been boldfaced. Table entries show that PSO is able to yield a very good approximation of the optimal solution. Such good results are obtained for a larger number of iterations in the PSO executions thus leading to larger computation times as well. One observation is that PSO introduces a great variation in the values of the variables, and that increases the time required for the stabilization of the fittin error value.

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Table 2. 20 executions of a B´ezier curve fitting through particle swarm optimization. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.2.3.

# iter. 53 198 20 42 172 46 78 140 187 65 104 200 146 137 105 34 45 76 152 89

Best error 1.823 1.810 1.830 1.815 1.810 1.813 1.811 1.810 1.810 1.813 1.810 1.811 1.810 1.810 1.813 1.820 1.814 1.810 1.811 1.811

Mean error 1.833 1.810 1.877 1.838 1.810 1.826 1.812 1.811 1.810 1.820 1.810 1.813 1.810 1.810 1.817 1.853 1.829 1.811 1.811 1.813

CPU time (secs) 0.9687 2.8281 0.5156 0.8125 2.1718 0.7968 1.2187 2.0468 2.4687 1.0156 1.5312 2.6875 2.0937 1.8437 1.5122 0.6718 0.8125 1.1406 2.2031 1.3906

Fitting a B´ezier Surface

We consider now a parametric B´ezier surface of degree nu in u and nv in v whose representation is given by: S(u, v) =

nv nu X X

Pi,j Binu (u)Bjnv (v)

(4.6)

i=0 j=0

where the basis functions (the Bernstein polynomials) are define as above and the coefficient Pi,j are the surface control points. For this example we consider an input of 256 Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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data points generated from a B´ezier surface as follows: for the u’s and v’s of data points, we choose two groups of 8 equidistant parameter values in the intervals [0, 0.2] and [0.8, 1]. Our goal is to reconstruct the surface which the given points come from. To this purpose, we consider a bicubic B´ezier surface, so the unknowns are 3 × 16 = 48 scalar coefficient (3 coordinates for each of 16 control points) and two parameter vectors for u and v (each of size 16) associated with the 256 data points. That makes a total of 80 scalar unknowns. The input parameter values for the PSO algorithm are: population size: 200 individuals or particles, where each particle is represented by two vectors, U and V , each with 16 components initialized with random uniform values on [0, 1] sorted in increasing order; inertia coefficien w = 1; coefficien for the global influence γ1 = 0.2; coefficien for the neighbors local influenc γ2 = 0.8; number of neighbours influencin locally to each particle: 20 and threshold limit for not improving iterations: 10. 14

12

10

8

6

4

2

0

0

100

200 300 Number of Generations

400

500

1 0.9 0.8 0.7

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v

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

u

Figure 2. B´ezier surface fittin through particle swarm optimization: (left) bicubic B´ezier surface and data points; (right-top): evolution of the mean (solid line) and the best (dotted line) Euclidean errors along the generations; (right-bottom): optimum parameter values for u and v on the parametric domain. Figure 2(left) shows the obtained surface and the data points. In Fig. 2(right-top) we display the evolution of mean error (solid line) and best (dotted line) distance error for each generation along the iterations, while the optimum parameter values for u and v are shown in Fig. 2(right-bottom). The convergence in this example is attained at iteration 473 at which the best and mean errors are 0.287 and 0.289 respectively. The computation time for this example is 48.7 seconds. Figure 2(left) shows that the PSO algorithm yields a good fittin to the data points. Points at the corners are very well fitte and the surface is close to the cloud of points thus

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Akemi G´alvez and Andr´es Iglesias

reconstructing the shape of the surface very well. From other trials not reported here because of limitations of space, we can conclude that the surfaces obtained with PSO actually reflec the real geometry of the data while preserving well the smoothness at the boundaries. Another remarkable issue concerns the distance between the best and the mean error for each iteration. In our PSO example, they converge to roughly the same value. Finally, the distribution of points, (Figs. 2(right-bottom)) is quite uniform for this example and fit well into the intervals [0, 0.2] and [0.8, 1] for both u and v (although the obtained intervals are slightly larger). In short, a good quality is achieved at a reasonable computation time.

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Table 3. 20 executions of a B´ezier surface fitting through particle swarm optimization # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

# iter. 185 488 368 441 372 242 264 261 143 242 382 135 136 109 106 143 289 344 289 413

Best error 0.448 0.274 0.350 0.259 0.889 0.390 0.420 0.685 0.633 0.505 0.544 0.633 0.552 0.634 0.378 0.845 0.368 0.207 0.421 0.319

Mean error 0.466 0.275 0.351 0.260 0.890 0.397 0.420 0.692 0.659 0.510 0.550 0.646 0.559 0.654 0.434 0.853 0.369 0.209 0.422 0.322

CPU time (secs) 19.2031 50.6406 37.7968 45.2812 35.1875 24.2343 26.1562 26.5625 14.0625 24.9843 38.2812 15.4843 13.1718 10.8281 11.5000 15.6562 26.9687 35.0156 28.9375 39.6562

Table 3 reports the results of 20 executions on the surface example for a population size of 200 candidate solutions, inertia coefficien w = 1; coefficien for the global influence γ1 = 0.2; coefficien for the neighbors local influenc γ2 = 0.8; number of neighbours influencin locally to each particle: 10 and threshold limit for not improving iterations: 10. In general, these results are in good agreement with our previous results.

5.

Conclusion

This chapter book focuses on describing some actual and potential applications of the original particle swarm optimization scheme and its later modification to some relevant probParticle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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lems in computer graphics and geometric modeling. In particular, we discussed the interesting problems of the realistic simulation of the behavior of synthetic actors in digital 3D worlds, the simulation of virtual crowds, the estimation of human body pose from pictures, the geometric constraint solving problem and the fittin of curves and surfaces from clouds of data points. We showed that the PSO methodology can provide reliable solutions to many of these problems and open the door to a series of new research developments and fruitful ideas. Although the application of PSO to these and other problems in computer graphics and geometric modeling is still in its infancy, the works discussed here clearly show that PSO-based schemes have the potential to become a very powerful computational paradigm in order to solve many problems in those domains. It is our hope that this chapter can encourage our readers to follow this approach.

Acknowledgments The authors would like to thank Prof. Frank Columbus, editor of this book, for his kind invitation to write this chapter and to all staff of Nova Publishers for their diligent work towards the publication of this contribution. This research has been supported by the Computer Science National Program of the Spanish Ministry of Education and Science, Project Ref. #TIN2006-13615 and the University of Cantabria.

References

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[23] Goncalves, L. M., Kallmann, M., & Thalmann, D. (2001). Programming Behaviors with Local Perception and Smart Objects: An Approach to Solve Autonomous Agent Tasks. Proc. SIGGRAPI’2001. Florianopolis, Brazil: ACM. [24] Gu, P., & Yan, X. (1995). Neural network approach to the reconstruction of free-form surfaces for reverse engineering. Computer Aided Design, 27(1) 59-64. [25] Gutierrez, M., Vexo, F., & Thalmann, D. (2003). Controlling Virtual Humans Using PDAs. Proceedings of 9th International Conference on Multi-Media ModelingMMM’03 (pp. 150-166). Taiwan. [26] Grzeszczuk, R., Terzopoulos, D., & Hinton, G. (1998). NeuroAnimator: fast neural network emulation and control of physics-based models. Proceedings of SIGGRAPH’99 (pp. 9-20). New York: ACM. [27] Hoffmann, M., & Varady, L. (1998). Free-form surfaces for scattered data by neural networks. J. Geometry and Graphics, 2 1-6. [28] Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., & Stuetzle, W. (1992). Surface reconstruction from unorganized points. Proceedings of SIGGRAPH’92 (pp. 71-78). New York: ACM. [29] Hoschek, J., & Lasser, D. (1993). Fundamentals of Computer Aided Geometric Design. Natick, MA: A.K. Peters.

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[30] Iglesias, A., & G´alvez, A. (2001). A New Artificia Intelligence Paradigm for Computer-Aided Geometric Design. Lectures Notes in Artificial Intelligence, 1930 200-213. [31] Iglesias, A., & G´alvez, A. (2001). Applying functional networks to fi data points from B-spline surfaces. In: Ip, H.H.S., Magnenat-Thalmann, N., Lau, R.W.H., Chua, T.S. (eds.) Proceedings of the Computer Graphics International, CGI’2001. (pp. 329-332) Hong-Kong, China: IEEE Computer Society Press. [32] Iglesias, A., Echevarr´ıa, G., & G´alvez, A. (2004). Functional networks for B-spline surface reconstruction. Future Generation Computer Systems, 20(8) 1337-1353. [33] Iglesias A., & Luengo, F. (2004). A New Based-on-Artificial-Int lligence Framework for Behavioral Animation of Virtual Actors. In: Computer Graphics, Imaging and Visualization, CGIV’2004 (pp. 245-250) Penang, Malaysia: IEEE Computer Society Press. [34] Iglesias A., & Luengo, F. (2004). Intelligent Agents for Virtual Worlds. Computer Graphics, Imaging and Visualization, CyberWorlds, CW’2004 (pp. 62-69). Tokyo, Japan: IEEE Computer Society Press. [35] Iglesias A., & Luengo, F. (2005). New goal selection scheme for behavioral animation of intelligent virtual agents. IEICE Trans. on Inf. and Systems, E88-D(5) 865-871. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[36] Iglesias A., & Luengo, F. (2007). AI Framework for Decision Modeling in Behavioral Animation of Virtual Avatars. Lectures Notes in Computer Science, 4488 89-96. [37] Ivekovic, S., Trucco, E., & Petillot, Y.R. (2008). Human Body Pose Estimation with Particle Swarm Optimisation. Evolutionary Computation, 16(4) 509-528. [38] Kallmann, M., & Thalmann, D. (1999). A behavioral interface to simulate agentobject interactions in real-time, Proceedings of Computer Animation’99 (pp. 138146). Menlo Park: IEEE Computer Society Press. [39] Kallmann, M., de Sevin, E., & Thalmann, D. (2001). Constructing Virtual Human Life Simulations. In: Deformable Avatars (pp. 240-247). Dordrecht, The Netherlands: Kluwer Publications. [40] Kennedy, J., & Eberhart, R.C. (1995). Particle swarm optimization. IEEE International Conference on Neural Networks (pp. 1942-1948). Perth, Australia: IEEE Computer Society Press. [41] Kennedy, J., Eberhart, R.C., & Shi, Y. (2001). Swarm Intelligence, San Francisco: Morgan Kaufmann Publishers. [42] Kim, D.H., & Shin, S. (2006). Self-organization of decentralized swarm agents based on modifie particle swarm algorithm. Journal of Intelligent and Robotic Systems, 46(2) 129149.

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[43] Knopf, G.K. & Kofman,J. (1999). Free-form surface reconstruction using Bernstein basis function networks. In: C.H. Dagli et al. (Eds.): Intelligent Engineering Systems Through Artificial Neural Networks. Vol. 9. (pp. 797-802). New York: ASME Press. [44] Kwong, H., & Jacob, C. (2003). Evolutionary exploration of dynamic swarm behaviour, in: Proceedings of IEEE Congress on Evolutionary Computation, CEC’2003 (pp. 367374). Sacramento, California: IEEE Computer Society Press. [45] Luengo, F., & Iglesias A. (2003). A new architecture for simulating the behavior of virtual agents. Lectures Notes in Computer Science, 2657 935-944. [46] Luengo, F., & Iglesias A. (2003). Animating the behavior of virtual Agents: the virtual park. Lectures Notes in Computer Science, 2669 660-669. [47] Luengo, F., & Iglesias A. (2004). Framework for simulating the human behavior for intelligent virtual agents. Part I: Framework architecture. Lectures Notes in Computer Science, 3039 229-236. [48] Luengo, F., & Iglesias A. (2004). Framework for simulating the human behavior for intelligent virtual agents. Part II: Behavioral system. Lectures Notes in Computer Science, 3039 237-244. [49] Miller, G.S.P. (1988).The Motion Dynamics of Snakes and Worms. Proceedings of SIGGRAPH’88, Computer Graphics 22(4) 169-173. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[50] Monzani, J.S., Caicedo, A., & Thalmann, D. (2001). Integrating behavioral animation techniques, Proceedings of EUROGRAPHICS’2001, Computer Graphics Forum, 20(3) 309-318. [51] Patrikalakis, N.M., & Maekawa, T. (2002) Shape Interrogation for Computer Aided Design and Manufacturing. Berlin Heidelberg New York: Springer Verlag. [52] Piegl, L., & Tiller, W. (1997). The NURBS Book. Berlin Heidelberg: Springer Verlag. [53] Piquer, A., Company, P., & Contero, M.: Particle swarm optimization based 3D reconstruction of sketched line-drawings. In: Recent Advances in Artificial Intelligence Research and Development (pp. 367-374). Valencia, Spain: IOS Press. [54] Poli, R., Kennedy, J., & Blackwell, T. (2007). Particle swarm optimisation: an overview. Swarm Intelligence, 1(1) 33-57. [55] Pottmann, H., Leopoldseder, S., & Hofer, M. (2002). Approximation with active Bspline curves and surfaces. In Proceedings of Pacific Graphics. (pp. 8-25). Beijing, China: IEEE Computer Society Press. [56] Pottmann, H., & Hofer, M. (2003). Geometry of the squared distance function to curves and surfaces. In Hege, H., Polthier, K. (Eds.) Visualization and Mathematics III (pp. 223-244). Berlin Heidelberg: Springer Verlag.

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[57] Pottmann, H., Leopoldseder, S. Hofer, M., Steiner, T., & Wang, W. (2005). Industrial geometry: recent advances and applications in CAD. Computer-Aided Design, 37 751766. [58] Rao, A.S., & Georgeff, M.P. (1991). Modeling rational agents within a bdiarchitecture. In: Allen, J., Fikes, R., Sandewall, E. (eds.) Proceedings of the Third International Conference on Principles of Knowledge Representation and Reasoning (pp. 317-328). San Mateo: Morgan Kaufmann. [59] Raupp, S., & Thalmann, D. (2001). Hierarchical Model for Real Time Simulation of Virtual Human Crowds. IEEE Transactions on Visualization and Computer Graphics 7(2) 152-164. [60] Reynolds, C.W. (1987). Flocks, herds and schools: a distributed behavioral model. Computer Graphics, 21(4) 25-34. [61] Rimon, E., & Kodischek, D. E. (1992). Exact Robot Navigation Using Artificia Potential Functions. IEEE Transactions on Robotics and Automation, 8(5) 501-518. [62] Sauter, J. A., Matthews, R., Parunak, H. V. D., & Brueckner, S. A. (2002). Evolving Adaptive Pheromone Path Planning Mechanisms. In: Proceedings of First International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS 2002). Bologna, Italy: ACM Proceedings.

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[63] Sauter, J. A., Matthews, R., Parunak, H. V. D., & Brueckner, S. A. (2005). Performance of Digital Pheromones for Swarming Vehicle Control. In: Proceedings of Fourth International Joint Conference on Autonomous Agents and Multi-Agent Systems. Utrecht, Netherlands: ACM Proceedings. [64] Sauter, J. A., Matthews, R., Parunak, H. V. D., & Brueckner, S. A. (2007). Effectiveness of Digital Pheromones Controlling Swarming Vehicles in Military Scenarios. Journal of Aerospace Computing, Information, and Communication, 4(5) 753-769. [65] SMDC-BL-AS. (2001). Swarming Unmanned Aerial Vehicle (UAV) Limited Objective Experiment (LOE). Huntsville, AL: U.S. Army Space and Missile Defense Battlelab, Studies and Analysis Division. [66] Sun, L.Q., & Gao, X.Y. (2008). Improved chaos-particle swarm optimization algorithm for geometric constraint solving. In: 2008 International Conference on Computer Science and Software Engineering (pp. 992-995). Los Alamitos, California: IEEE Computer Society Press. [67] Thalmann, D., & Noser, H. (1999). Towards autonomous, perceptive and intelligent virtual actors. Lecture Notes in Artificial Intelligence, 1600 457-472. [68] Tu, X., &Terzopoulos, D. (1994). Artificia fishes physics, locomotion, perception, behavior, Proceedings of SIGGRAPH’94 (pp. 309-318). New York: ACM.

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[69] Varady, T., & Martin, R. (2002). Reverse Engineering. In: Farin, G., Hoschek, J., Kim, M. (eds.): Handbook of Computer Aided Geometric Design. Amsterdam, Netherlands: Elsevier Science. [70] Weiss, V., Andor, L., Renner, G., & Varady, T. (2002). Advanced surface fittin techniques. Computer Aided Geometric Design, 19 19-42.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 193-225

ISBN 978-1-61668-527-0 c 2011 Nova Science Publishers, Inc.

Chapter 9

PARTICLE S WARM O PTIMIZATION U SED FOR M ECHANISM D ESIGN AND G UIDANCE OF S WARM M OBILE ROBOTS Peter Eberhard∗ and Qirong Tang† Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany

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Abstract This chapter presents particle swarm optimization (PSO) based algorithms. After an overview of PSO’s development and application history also two application examples are given in the following. PSO’s robustness and its simple applicability without the need for cumbersome derivative calculations make it an attractive optimization method. Such features also allow this algorithm to be adjusted for engineering optimization tasks which often contain problem immanent equality and inequality constraints. Constrained engineering problems are usually treated by sometimes inadequate penalty functions when using stochastic algorithms. In this work, an algorithm is presented which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrangian particle swarm optimization (ALPSO). It is used for the stiffness optimization of an industrial machine tool with parallel kinematics. Based on ALPSO, we can go a further step. Utilizing the ALPSO algorithm together with strategies of special velocity limits, virtual detectors and others, the algorithm is improved to augmented Lagrangian particle swarm optimization with special velocity limits (VLALPSO). Then the work uses this algorithm to solve problems of motion planning for swarm mobile robots. All the strategies together with basic PSO are corresponding to real situations of swarm mobile robots in coordinated movements. We build a swarm motion model based on Euler forward time integration that involves some mechanical properties such as masses, inertias or external forces to the swarm robotic system. The results show that the stiffness of the machine can be optimized by ALPSO and simulation results show that the swarm robots moving in the environment mimic the ∗ E-mail

† E-mail

address: [email protected] address: [email protected]

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Peter Eberhard and Qirong Tang real robots quite well. In the simulation, each robot has the ability to master target searching, obstacle avoidance, random wonder, acceleration/deceleration and escape entrapment. So, from this two application examples one can claim, that after some engineering adaptation, PSO based algorithms can suit well for engineering problems.

1.

Introduction

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Particle swarm optimization (PSO) originated in different areas, among them Complex Adaptive Systems (CAS). The theory of CAS was formally proposed in 1994 by Holland and later collected in his book in (1995). PSO was inspired by the main ideas of CAS and from Heppner and Grenander’s biological population model (Heppner and Grenander, 1990). In a complex adaptive system, like a swarm of birds, see Figure 1, each member is taken as an adaptive agent and members of CAS can communicate with the environment and other agents. During the process of communication, they will ‘learn’ or ‘accumulate experience’. Base on this, the agent can change its structure and behavior. Such processes, e.g., in the floc of birds system which is the basis of PSO, include the production of new generations (have young birds), the emergence of differentiation and diversity (the floc of birds is divided into many small groups), and findin new themes (discover new food). Particle swarm optimization comes from the research on a CAS system - a social system of birds. The computational algorithm was firs introduced by Kennedy and Eberhart in (1995). The framework of basic PSO can be described in a very simple form, see Figure 2. After PSO’s emergence, it has evolved greatly. Many research groups focus on

Figure 1. Swarm of birds. this algorithm due to its robustness and simple applicability without the need for cumbersome derivative calculations. Basically, there are four classes of PSO which focus on the algorithm itself, i.e., 1. variants of the standard particle swarm optimization algorithm, 2. hybrid particle swarm optimization algorithms,

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start initialization

recursive update equation find best particle as the best best,k solution: x swarm no

terminate? yes best,k get x swarm

end

Figure 2. Framework of basic PSO. 3. binary particle swarm optimization algorithms, and

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4. cooperative particle swarm optimization algorithms. Besides the algorithm itself, there are many research works for engineering applications. This chapter also mainly describes researches of using PSO in engineering problems. Since 1995, PSO has been modifie and improved into many variants. Here we list some main developments of PSO, see Table 1. Around the year 2000, researchers changed their methods and made further improvements. We can then fin hundreds of articles in many application fields Also, then researchers focused on how to use PSO for practical problems, such as power systems, IC design, system identification state estimation, lot sizing, composing music, or modeling markets and organizations. Some classical applications are listed in Table 2. Although the application of basic PSO for practical problems in engineering is often successful, most engineering problems contain problem immanent equality and inequality constraints. The basic PSO or so called standard PSO, can not be applied to constrained problems. Thus, often constrained engineering problems are treated by inadequate penalty functions when using stochastic algorithms. This chapter will present an algorithm which utilizes the simple structure of the basic PSO technique and combines it with an augmented Lagrangian multiplier approach described in the following section.

2.

Algorithm for Constrained Engineering Problems

PSO is traditionally used for unconstrained optimization problems. However the general optimization problem is define by the objective function Ψ, which is to be minimized with respect to the design variables x and the linear or nonlinear equality and inequality Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

196

Peter Eberhard and Qirong Tang Table 1. Some main developments of PSO

year

contributors

main contributions

1995 1997

Kennedy, Eberhart Kennedy, Eberhart

1998

Shi, Eberhart

1999

Clerc

1999 1999

Angeline Suganthan

2000

Carlisle, Dozier

2001

Lovbjerg, Rasmussen, and Krink van den Bergh

provided the algorithm of PSO firs modifie to binary PSO algorithm which is used for structural optimization of neural networks, provided a helpful way to compare the performance of genetic algorithms improved the performance of convergence by putting inertia weight w into the velocity item, dynamically adjust w during the iterative process so as to counterpoise the global superiority and the rate of convergence (standard PSO) introduced a contraction factor to ensure the convergence of PSO, relaxed the limitation of velocity used the idea of evolution selection to improve the convergence introduced the idea of neighborhood operators into standard PSO so as to maintain the diversity of particles proposed a PSO model which can be adapted to dynamic environment automatically introduced the concept of subpopulations from Genetic Algorithm into PSO

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2001

proved that the standard PSO algorithm can not guarantee that it will converge to the global optimum and even can not guarantee the convergence to a local optimum

constraints. This can be formulated by minimize Ψ(xx) x ∈X

 with X = x ∈ RΘ | g(xx) = 0, h(xx) ≤ 0, xl ≤ x ≤ xu ,

(1)

where g (xx) = 0 and h (xx) ≤ 0 are the me equality and mi inequality constraints, respectively. The lower limit is x l while x u is the upper limit.

2.1.

General Methods for the Constrained Optimization Problem

To solve optimization problems with equality and inequality constraints, different methods and algorithms are used. Generally there are two kinds of methods, deterministic and stochastic ones. The deterministic algorithms usually use gradient information and often locate a minimum within a few iteration steps, but they sometimes get struck in local minima and require a smooth performance function. Stochastic methods have the advantage that they may fin a global minimum without posing severe restrictions on, e.g., the differentiability, convexity, or separability. Unfortunately the number of iterations is also increasing requiring many criteria evaluations.

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Table 2. Some classical applications of PSO year

contributors

1998 1999 2001 2000 2001 2001 2002 2002 2002 2003 2003 2003 2004 2004

He, Wei, Yang, Gao, Yao, Eberhart, and Shi Fukuyama, Takayama, Nakanishi, Yoshida, and Kawata Peng, Chen, and Eberhart Cockshott, Hartman Tandon Blackwell, Bentley Coelho, Oliveira, and Cunha El-Gallad, El-Hawary, Sallam, and Kalas Emara, Ammar, Bahgat, and Dorrah Lu, Fan, and Lo Srinivasan, Loo, and Cheu Onwubolu, Clerc Teo, Foo, Chien, Low, and You

2006 2007 2008 2009

Sedlaczek, Eberhard Yu, Zhu, Yu, and Li Han, Zhao, Xu, and Qian Holden, Freitas

application fiel fuzzy neural network electric power systems battery biochemistry CAD/CAM, CNC music greenhouse air temperature control economics induction motors environmental protection traffi drilling placement of wavelength converters in an arbitrary mesh network robot stiffness optimization bellow optimum design multivariable PID controller design hierarchical classificatio of protein function

In recent years, stochastic methods such as evolutionary algorithms, simulated annealing, and particle swarm optimization are applied frequently. However, basic PSO can’t handle constrained optimization problems. Hence, combining it with other strategies from deterministic optimization will greatly widen its applicability.

2.2.

Extending the Basic PSO to ALPSO for Efficient Constraint Handling

The recursive update equations of basic PSO can be formulated by k k k k xbest,k (xxbest,k ∆xxk+1 = ω ∆xxki + c1 ri,1 swarm − x i ). i i,sel f − x i ) + c2 ri,2 (x

(2)

This is the so called ‘velocity’ update equation and the position update is done in the traditional PSO algorithm by x k+1 = x ki + ∆xxk+1 . (3) i i Here ω is the inertia weight, usually c1 , c2 ∈ (0, 2) are referred to as cognitive scaling k ∼ U(0, 1), r k ∼ U(0, 1) are two independent random and social scaling factors, and ri,1 i,2 functions. One saves x best,k as the best position of particle i reached till now, i.e., the indii,sel f vidual best value, and x best,k swarm is the best previously obtained position of all particles in the

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Peter Eberhard and Qirong Tang

entire swarm, that is to say, the current global best position. In Eq. (3), the PSO technical term ‘velocity’ really corresponds to a time step ∆t times the mechanical velocity to give , but usually in PSO literature = ∆t x˙ k+1 the equation a physically correct format, i.e., ∆xxk+1 i i the time step ∆t is omitted or set to be one. So the basic PSO algorithm is summarized for all n particles by #  k+1   k  " x˙ k+1 x x = + , (4) k k x best,k − x k ) x˙ k+1 ω x˙ k c1 r k1 (xxbest,k swarm sel f − x ) + c2 r 2 (ˆ where 

  r k1 =  

0 r1,1 I 3 ··· 0 r2,1 I 3 · · · .. .. .. . . . 0 0 ···

0 0 .. .





   k   , r2 =   

rn,1 I 3

0 r1,2 I 3 ··· 0 r2,2 I 3 · · · .. .. .. . . . 0 0 ···

0 0 .. . rn,2 I 3



  , 

(5)

and n is the number of particles. Considering the spatial case, I 3 is a 3 × 3 unit matrix. Similarly the matrices 

k x1,i





best,k x1,i,sel f





best,k x1,swarm



 best,k  best,k   k  best,k  best,k    x ki =  x2,i  , x i,sel f =   x2,i,sel f  , x swarm =  x2,swarm  , i = 1(1)n, k best,k best,k x3,i x3,swarm x3,i,sel f

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

    xk =     

x k1 xk2 .. . x ki .. . x kn





x best,k 1,sel f

 best,k  x   2,sel f    ..   .  ∈ R3n×1 , xbest,k =  sel f  best,k   x i,sel f    ..   .  best,k x n,sel f



  best,k  x swarm   best,k   x swarm    ∈ R3n×1 , xˆ best,k = swarm   ..   .  best,k  x swarm 

(6)



   ∈ R3n×1 (7)  

are defined Engineering problems usually have constraints as in the two examples shown in the following sections, i.e., optimizing the stiffness of a hexapod machine and the guidance of swarm mobile robots. For such constrained optimization problems, the augmented Lagrangian multiplier method can be used where each constraint violation is penalized separately by using a finit penalty factor r p,i . Thus, the minimization problem with constraints in Eq. (1) can be transformed into an unconstrained minimization problem minimize x

LA (xx, λ , r p )

with LA (xx, λ , r p ) = Ψ(xx) +

(8) me +mi

∑

i=1

λi Pi (xx) +

me +mi

∑

r p,i Pi2 (xx),

i=1

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

(9)

PSO Used for Mechanism Design and Guidance of Swarm Mobile Robots ( gi (xx),   for i = 1(1)me , and Pi (xx) = −λi max hi−me (xx), 2r p,i , for i = (me + 1)(1)(me + mi ).

199 (10)

Here λi are Lagrangian multipliers and r p,i are finit penalty factors. Note that λi and r p,i are unknown in advance and are adaptively adjusted during the simulation. According to Sedlaczek and Eberhard (2006) this problem can be solved by dividing it into a sequence of smaller unconstrained subproblems with subsequent updates of λi and r p,i . Then, λi and r p,i are changed based on the iteration equations

λis+1 = λis + 2rsp,i Pi (xx),

rs+1 p,i

rs+1 p, j+me

2rsp,i 0.5rsp,i =  rsp,i  

2rsp, j+me 0.5rsp, j+me =  rsp, j+me  

i = 1(1)(me + mi ),

(11)

if |gi (xxs )| > |gi (xxs−1 )| ∧ |gi (xxs )| > εequality , if |gi (xxs )| < εequality , else,

i = 1(1)me ,

if h j (xxs ) > h j (xxs−1 ) ∧ h j (xxs ) > εinequality , if h j (xxs ) < εinequality , j = 1(1)mi , else,

(12)

(13)

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where εequality and εinequality are user define tolerances for constraint violations which are still acceptable. The initial values are λi0 = 0 and r0p,i = 1. This work uses the update equations (11) which come from partial differentiation of the Lagrangian function (9) yielding the subproblem "

∂L ∂x

#

x =xxs

"

∂ Ψ(xx) me +mi s ∂ Pi (xx) me +mi s ∂ Pi (xx) + ∑ λi + ∑ 2r p,i Pi (xx) = ∂x ∂x ∂x i=1 i=1

#

≈ 0.

(14)

x =xxs

We apply a heuristic update scheme for the penalty factors r p , see Eqs. (12) and (13). If the intermediate solution x s is not closer to the feasible region define by the i-th constraint than the previous solution x s−1 , the penalty factor r p,i is increased. On the other hand, we reduce r p,i if the i-th constraint is satisfie with respect to user define tolerances. During the algorithm tests we also experienced that for active constraints a lower bound on the penalty factors yields improved convergence characteristics for the Lagrangian multiplier estimates. This work maintains the magnitude of the penalty factors such that an effective change in Lagrangian multipliers is possible. This lower bound is 1 r p,i ≥ 2

s

|λi |

εequality, inequality

.

(15)

So far, the basic PSO algorithm is extended to augmented Lagrangian particle swarm optimization (ALPSO) which is well suited for handling constrained problems. In the following sections 3. and 4., two application examples will be shown which are based on ALPSO with several extensions.

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3. 3.1.

Peter Eberhard and Qirong Tang

PSO Based Algorithm Used for Mechanism Design Mechanism Design and Optimization

Optimization is a very important aspect in mechanism design and production especially for some complex mechanical systems. As a relatively new subject optimal design is the result of applying optimization techniques and computing technologies in the product design area. The basic idea is to chose the design parameters in a systematic way and to try to achieve the most satisfactory particular mechanical property under the condition of meeting requirements from various design constraints. Following, an example of optimizing the stiffness behavior for a hexapod robot based on ALPSO is presented. Machines with parallel kinematics feature low-inertia forces due to low moving masses of the structure. In contradiction to this are the desired high accuracy and stiffness required in various fields e.g., like robotics or measurement systems. We investigated and optimized the stiffness behavior of the hexapod robot HEXACT using the ALPSO method. HEXACT is a research machine tool with parallel kinematics, developed by Prof. Heisel and his coworkers at the Institute of Machine Tools, University of Stuttgart, Germany (Heisel et al. 1998), see Figure 3 left.

ζ

de

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le

Figure 3. Spatial mechanical model of the hexapod robot HEXACT (see also http://www.ifw.uni-stuttgart.de) A frequent drawback of parallel kinematics robots is the appearance of kinematically singular configuration that have to be avoided during operation. For the design of the investigated hexapod machine, such singularities are located along the tool axis in the central position of the machine, where the rotational stiffness around the tool axis decreases to zero, see Henninger et al. 2004.

3.2.

Optimization Design of the HEXACT

For parallel kinematics machines, the stiffness of the end-effector depends on its translational and rotational position in the workspace, the so-called pose. The relation between the applied forces and torques F on the one hand and the evasive displacements ∆yye = [∆rr e ∆φ e ] of the end-effector on the other hand is highly nonlinear. For the evalua-

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tion of the stiffness and fl xibility behavior, it is sufficien to regard the linearized relations F = K t · dyye dF

and

F, dyye = C t · dF

(16)

where the tangential stiffness matrix K t describes the change in forces caused by a given rotational and translational displacement change and the tangential fl xibility (or compliance) matrix C t gives the relationship between a given force and torque load change and the resulting end-effector’s displacement change. In robotics, the elastic behavior is usually described in a global coordinate system using the vector of infinitesima rotation dsse = ω e dt to describe the deflectio of orientation of the end-effector. A simplifie description of the compliance behavior can be achieved by assuming all driven joints behavior according to linear elasticity, which is sufficien to show the basic behavior of the machine. In our case, the stiffness of the inner and outer joints of each of the six driving struts can be summarized to a resulting strut elasticity k. The F and the displacement of the end-effector relation between the infinitesima applied load dF dqqe = [drr e dsse ] is then given as 1 F dqqe = − J · J T · dF | k {z } =: C tx

(17)

with the Jacobian matrix J also describing the relation between the actuator velocities θ˙ = [θ˙1 · · · θ˙6 ], and the end-effector velocity

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q˙ e = J (yy) · θ˙ ,

(18)

see Henninger et al. 2004. It should be mentioned that different definitio s of the Jacobian matrix exist in literature and one has to defin clearly which one is used. To describe the deflectio of orientation by the change of elementary rotation angles like Cardan angles which are applied here with d φ e = [d αe d βe d γe ], we can write       I 0 drr e I 0 y Ctx · dF F dy e = · = ·C (19) 0 HR dsse 0 HR | {z } =: C t

with the identity matrix I and the Jacobian matrix of rotation H R , see, e.g., Schiehlen and Eberhard 2004. The inverse relation is then given by (20)

F = K t · dyye dF with the tangential stiffness matrix K t = −kJJ −T · J −1 ·



I 0 0 H −1 R



.

(21)

In the general case, both matrices K t and C t are symmetric and fully occupied. This means that a load in one direction causes rotational and translational evasive motions in Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Peter Eberhard and Qirong Tang

every direction. Furthermore, they consist of four blocks having different units. For the following considerations, it is useful to normalize the coefficient using characteristic forces, torques, lengths, and angles. These values should be chosen with care as they influenc the results of the following optimization process. For our example, we have chosen the normalization as described by Henninger et al. (2004), where more detailed information about the stiffness analysis of the HEXACT machine tool including information about its parameter values can be found. One way to reduce the fl xibility and to eliminate the singular configuration is, to alter the angles ζ between the telescope struts and the end-effector, which are perpendicular in the initial design, see Figure 3. In addition, the diameter de and the length le of the endeffector and thus the mounting points of the struts can be varied to improve the stiffness behavior of the hexapod robot, yielding the optimization parameter vector x = [ζ de le ].

(22)

To gain more insight into the global fl xibility behavior of the machine, it is necessary to evaluate the stiffness matrix at several poses in the workspace. In this study, N = 65 sample poses on a regular grid are regarded. As a global fl xibility criterion is to be minimized, we use the negative average of the minimum principal stiffness of the N sample poses in the entire workspace

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1 minimize fk (xx) with fk (xx) = − x N

N

∑ min(ki∗ ) j ,

(23)

j=1

where min(ki∗ ) j is the minimum eigenvalue of the tangential stiffness matrix K t at pose j. The optimization problem described by Eq. (23) is solved using particle swarm optimization which enables a gradient free and global search without any diff cult or expensive gradient calculations or without any restrictions to a local solution. Two different sets of design variables were considered. The firs variant considers only ζ as a design variable, whereas variant 2 takes into account both ζ and the dimension of the end-effector described by de and le . Regarding design variant 1, the average of the minimum principal stiffness could be improved by a factor of 35 with respect to the initial design, see Henninger et al. (2004). The optimization of design variant 2 improved the average stiffness by a factor of approximately 200. As expected, the geometry of the end-effector has a great inf uence on the stiffness behavior of the hexapod robot. However, the maximization of the minimum stiffness as formulated by Eq. (23) impairs the stiffness distribution of the hexapod machine in the workspace. The standard deviation of the minimum principal stiffness in the workspace increased by a factor of 11 for design variant 1 and by a factor of 82 for design variant 2. The resulting more non-uniformly distributed stiffness behavior is undesirable for manufacturing processes. Therefore, design variant 3 is define by solving the following modifie nonlinearly constrained optimization problem v u u 1 N minimize fs (xx) with fs (xx) = t ∑ (min(ki∗ )q + fk (xx))2 , x N − 1 q=1 (24) subject to h(xx) = fk (xx) − 0.8 fk∗ ≤ 0,

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where fs (xx) describes the standard deviation of the stiffness distribution. The inequality constraint restricts the decrease in the optimized average principal stiffness fk∗ from design variant 2. This nonlinear constrained problem is solved using the augmented Lagrangian particle swarm optimization algorithm with respect to the design variables ζ , de , and le . For this optimization process as well as for design variants 1 and 2, we used n = 20 particles. As a result, we could improve the standard deviation of the minimum principal stiffness by 25% with an acceptable worsening in the average stiffness of 20% compared to variant 2. The experimental setup and results are summarized in Table 3 and Table 4, respectively. For this engineering problem, these results are useful to analyze the design modification and to quantify the potentially achievable improvements. Further information about the hexapod robot and its stiffness behavior can be found in (Henninger et al., 2004) and (Henninger, 2009).

4. 4.1.

PSO Based Algorithm Used for Guidance of Swarm Mobile Robots Background of Robot Navigation

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Robot motion planning or so called path planning is a very important topic in robot development especially for mobile robots, robotic arms or walking robots. This research area belongs to robot navigation which includes location, path planning, and obstacle avoidance. These are three different topics. In fact, in engineering applications, they are often considered at the same time and are summarized in the term ‘motion planning’. In Table 5 some methods/strategies used in robot navigation are mentioned. Table 3. Setup for optimizing the stiffness behavior of the hexapod robot design variant

design variables

objective function

constraints

variant 1 variant 2 variant 3

x = [ζ ] x = [ζ de le ] x = [ζ de le ]

fk fk fs

fk − 0.8 fk (xx∗2 ) ≤ 0

Table 4. Results of optimizing the stiffness behavior of the hexapod robot design variant

solutions

objective function value

initial design variant 1 variant 2 variant 3

x 0 = [0◦ 0.4m 0.44m] x ∗1 = [76.72◦ ] x ∗2 = [65.77◦ 0.6m 1.0m] x∗3 = [79.57◦ 0.509m 1.0m]

fk (xx0 ) = −0.025, fk (xx∗1 ) = −0.876, fk (xx∗2 ) = −5.083, fk (xx∗3 ) = −4.066,

fs (xx0 ) = 0.039 fs (xx∗1 ) = 0.435 fs (xx∗2 ) = 3.167 fs (xx∗2 ) = 2.349

Local path planning strategies include APF (Artificia Potential Field) firs proposed by Khatib in 1968, genetic algorithms and fuzzy logic algorithms. Meanwhile global environment modeling can be realized by the methods of graph, free-space, grid and global path Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

204

Peter Eberhard and Qirong Tang Table 5. Robot navigation strategies location robot navigation (motion planning)

path planning

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obstacle avoidance

relative absolute local global

dead reckoning imaging, laser, GPS APF, genetic algorithms, fuzzy logic environment graph methods, free-space modeling methods, grid methods A* algorithms, path search D* optimal algorithms VFH, APF, VFH+, VFH*

search usually achieved by the optimal algorithms of A* (Hart, Nilsson and Raphael, 1968) or achieved by D* (Stentz, 1994). The third part of robot navigation is obstacle avoidance which has many solutions like VFH (Vector Field Histogram) proposed by Borenstein and Koren in 1991, the updated versions VFH+ (Ulrich and Borenstein, 1998) and VFH* (Ulrich and Borenstein, 2000). Actually, each of the algorithms has its limitations, e.g., in some situations it can work reliably, but with conditions changing, the algorithm may loose its effectiveness. Thus, many researchers enhance old algorithms or develop new algorithms with innovations. In spite of this, for swarm robots there still is no solution which can be used in general and works robustly. So, besides the above mentioned conventional methods, some researchers dedicated their attention to biology inspired algorithms. Because nature can motivate unusual approaches, it inspired countless scientifi innovations, and helps humans to solve practical problems. Some of the most innovative and useful discoveries have arisen from a fusion of two or more seemingly unrelated field of study just like some algorithms invented from models of biology. In the robotics area, there are several biology inspired algorithms for robot motion planning which are briefl presented in the following. Genetic algorithms (GA) are famous biology inspired algorithms and are used in many fields However, in the view of swarm mobile robots they look not so suitable. The main reason is that GAs show sudden large changes during the iterations. Such jumps are not feasible for real robots. Bacterial colony growth algorithms (BCGA) are new biology inspired approaches which were proposed in 2008 by Gasparri and Prosperi. They use the idea of bacterium growth and bunching to the nutrient areas to be a colony. Each robot is seen as a bacterium in a biological environment which reproduces asexually. Unfortunately, these algorithms are very complex. If used in swarm robots, the calculation cost can be prohibitively high. Recently, a reactive immune network (RIN) was proposed and employed for mobile robot navigation (Guan and Wei, 2008). This is an immunological approach to mobile robot reactive navigation. The same shortcoming as in the aforementioned BCGA is its complexity. Additionally, it is difficul to be scaled for swarm robots. The heuristic ant algorithms (AA) use a group of modelled ants to navigate the multirobot, but the definition are not simple, and the iterative process is sophisticated (Zhu, 2006).

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In contrast to the above mentioned biology inspired algorithms, the PSO algorithm is more appealing due to its clear ideas, simple iteration equations, and also the ease to be mapped onto robots or even swarm robots. In the following section, the mechanical ‘PSO’ motion model of swarm mobile robots is presented.

4.2.

Mechanical PSO Model of Warm Mobile Robots

Usually the PSO algorithm is used as a mathematical optimization tool without physical meaning. In the following example, a PSO based algorithm is presented which is used for the motion planning of swarm mobile robots and so their physical background must be considered. Additionally, we want to interpret the PSO algorithm as providing the required forces in the view of multibody system dynamics. Each particle (robot) is considered as one body in a multibody system which is influence by forces and torques from other bodies in the system but without direct mechanical connection to them. The forces are artificiall created by corresponding drive controllers. First, one starts from the Newton-Euler equations. These equations are presented in a general form. The motion of the particle i in the ‘multi-particle’ system is governed by the two matrix equations mi a i = f i , (25)

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J iα i + ω i × J iω i = l i,

i = 1(1)n

(26)

which correspond to the balance of linear and angular momentum. Here mi is the mass of particle i, a i its linear acceleration and f i are the forces acting on particle i. In Eq. (26), the matrix J i is the moment of inertia, α i the angular acceleration, while ω i is the angular velocity. At the right side of Eq. (26), l i contains the moments or torques acting on body i. Equation (25), the so called Newton equation, is consisting of three scalar equations that relate the forces and the accelerations of the particle in the three Cartesian dimensions. Equation (26), on the other hand, relates the rotational acceleration to a given set of moments or torques. This matrix equation consists also of three scalar equations and is called Euler equation. Further more, Eqs. (25) and (26) can be combined in the Newton-Euler equations for one particle i      mi I 3 0 ai fi = . (27) 0 Ji αi l i − ω i × J iω i If no constraints or joints exist, this equation contains no reaction forces. Otherwise, a projection using a global Jacobian matrix can be performed eliminating them. In both cases, the general form of the equations of motion for swarm robots can be formulated as M x¨ + k = q

or

x¨ = M −1 (qq − k ) = M −1 F .

(28)

For a free system without joints, M = d i a g (m1 I 3 , m2 I 3 , · · · , mn I 3 , J 1 , J 2 , · · · , J n ) = M T ≥ 0 is the mass matrix collecting the masses and inertias of the particles (robots), x¨ is the general acceleration   a x¨ = , (29) α

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k is a term which comes from the Euler equation, and q contains forces and moments acting on the robots. One can also write Eq. (28) as a state equation with the state vector   x , (30) y= x˙ where x and x˙ are the translational and rotational positions and velocities of the robots. Thus, the firs order differential equation follows     x˙ x˙ y˙ = = . (31) x¨ M −1 F Together with the initial conditions, the motion of the swarm robots over time can be computed, e.g., by the simple Euler forward integration formula y k+1 = y k + ∆t y˙ k ,

(32)

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where ∆t is the chosen time step, and the superscript k, k + 1 means the k-th and (k + 1)-th point in time. Rewriting Eq. (32) yields  k+1   k    x˙ k x x = + ∆t . (33) x˙ k+1 x˙ k M −1 F k Next, the connection between the mechanical motion of a particle or robot and the PSO algorithm should be made. For this reason we firs assume that the robot should at the moment only be driven by forces such that no torques appear, i.e. l i = 0. Also, the motion is described for a mean-axis system in the center of gravity so that the second term in Eq. (26) vanishes, too. So, where do the forces f i come from? The force f k is determined by three parts, f k1 , f k2 and f k3 , which are define as   f k1 = −hhkf1 x k − x best,k , sel f
(34) f k2 = −hhkf2 x k − xˆ best,k swarm , f k3 = −hhkf3 x˙ k

with the matrices h kf1 = d i a g (hk1, f1 I 3 , hk2, f1 I 3 , · · · , hkn, f1 I 3 , 0 3n ), h kf2 = d i a g (hk1, f2 I 3 , hk2, f2 I 3 , · · · , hkn, f2 I 3 , 0 3n ), h kf3

=

(35)

d i a g (hk1, f3 I 3 , hk2, f3 I 3 , · · · , hkn, f3 I 3 , 0 3n ).

Here I 3 is a 3 × 3 unit matrix, 0 3n is a 3n × 3n zero matrix. The forces f k1 and f k2 are attraction forces to the last self best robot position and the last swarm best robot position. They are proportional to their distances. The vector f k3 represents the force which is proportional to the last velocity and is a kind of inertia which counteracts a rapid change in direction. This work maps the swarm mobile robots’ ‘PSO’ model to the original PSO algorithm

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so as to fin the theoretical support for motion planning of swarm mobile robots. Inserting Eq. (34) to Eq. (33) yields 

x k+1 x˙ k+1



=

"

xk   M −1 h kf3 x˙ k I 6n − ∆tM

+ ∆t

"

# k  x˙


best,k k + M −1 h k x k M −1 hkf1 xbest,k f2 ˆ swarm − x sel f − x

#

(36) .

Comparing the mechanical ‘PSO’ model of swarm mobile robots in Eq. (36) to the original PSO model in Eq. (4), one can see that they are quite similar and the corresponding relationships are M −1 h kf1 ←→ c1 r k1 , ∆tM M −1 h kf2 ←→ c2 r k2 , ∆tM

(37)

M −1 h kf3 ←→ ω . I 6n − ∆tM Of course, similar relations can be derived if also torques are entered and rotations of the robots occur since neither the system description (31) nor the integration formula (32) change. The random effects are included in h and all forces must be created by local drive controllers in the robots.

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4.3.

Modification of the Neighborhood

From Eqs. (4) and (36), one can see that for using a PSO based algorithm for swarm mobile robots, each robot needs the information of the current self best value and the current swarm (global) best value. It is trivial to get the self best value since it can be stored in the local on-board position memory. When the size of the swarm becomes larger, it’s sometimes infeasible or even undesired to distribute and store the swarm best value to all swarm membest,k bers. Therefore, it is proposed to replace xˆ best,k swarm with x nhood , that is, using the best value in the neighborhood to replace the global one during the iterative processes. This replacement has great practical significanc since it adds a lot of robustness to the robots, especially for the motion planning. It reduces the communication expense of the algorithm since they only need to communicate with other robots which are close to them. So the original PSO in Eq. (4) and the mechanical ‘PSO’ for swarm mobile robots in Eq. (36) are modifie by replacing the global best to the best in the neighborhood. The strategy to defin the neighborhood depends on the size of the swarm and the communication ability of practical robots. Basically, there are two kinds of neighborhood models, i.e., indexed neighborhoods and spatial neighborhoods. In this work, the latter are used. The radius of the neighborhood in this work is fl xible as the size and density of swarm are changing. Here, the algorithm selects the nearest one-third particles from the complete swarm to the current particle as the neighborhood field Then, the neighborhood best particle/robot will be determined after comparing their performance values.

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4.4.

Peter Eberhard and Qirong Tang

Extension of the Basic PSO Algorithm to VL-ALPSO for Coordinated Movements of Swarm Mobile Robots

It is proposed to use a PSO based algorithm for the motion planning of swarm mobile robots with the goal to search a target in the environment. Usually, the target in the environment can be described by an objective function. Several kinds of information are available to a robot, i.e., 1. local information like the evaluation of a function value or a gradient at the current position of a robot, 2. information about the surrounding, e.g., obtained from distance sensors, and 3. information communicated by other robots. The firs kind of information is treated in the optimization problem and its usage is basically described in Section 2.. The second kind of information will be described later in Subsection 4.4.1. and is only considered by the robot itself. Finally the third kind of information described in Subsection 4.4.2. is the core of the VL-ALPSO algorithm where several particles/robots search in parallel and exchange information.

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4.4.1.

Swarm Particle Robots

If PSO should be applied to a real mobile robot, the algorithm must be modifie and adjusted, e.g., to include the mechanically feasible motion and to take care for obstacle avoidance. In classical PSO or ALPSO, the particles can move in any direction and with any velocity. Nevertheless, the robots have to move with limited velocity and acceleration since only finit forces can be generated by the drives yielding only finit accelerations and so any change in velocity needs time and energy. Also, there are obstacles in the environment. In the process of adapting the algorithm, there are some aspects that need to be considered. 1. The basic structure of ALPSO should be used for guiding the motion including its ability to deal with constraints. 2. The algorithm should not yield too much computational effort since no strong processor is available on a robot. 3. The robots should take care locally for obstacle avoidance. 4. The algorithm should be applied for physical real robots. On the basis of the before-mentioned requirements, this study proposes the structure of a sub-algorithm for obstacle avoidance, especially for static obstacles in an environment. Of course, obstacles from an internal map could be considered simply as constraints which are treated in the optimization code. However, in robotics it is desirable that the collision detection and avoidance are done locally in each robot based on its obtained own sensor signals. This is also important, e.g., for handling moving obstacles. Constraints treated in the optimization algorithm are, e.g., considering energy, minimal distances between the

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robots or avoiding prohibited areas. In the algorithm, we restrict the maximal displacement of a particle in a single step to Z which is define as Z = max(z1 , z2 ). (38) Here z1 and z2 come from two different aspects. The distance z1 is due to the computer performance. In the scenario of a real mobile robot, it depends on the embedded MicroCPU and the volume of robots. The value z2 depends on the braking distance. This braking distance must guarantee that the robot won’t collide with obstacles before it can stop or turn away. Because the stopping force or steering force is finit and strongly depends on the maximal velocity, this maximal braking distance must be chosen. In order to make sure that particles (robots) in the simulation do not pass through (fl over or jump over) obstacles under any circumstances, even for a very small or very thin obstacle, one can set dangerous regions surrounding the static obstacles in the environment. If any particle (robot) enters into such a dangerous region, the algorithm will give alarm and focus on collision avoidance instead of concentrating on optimization. In fact, this principle of obstacle avoidance is achieved in the actual mobile robot because the robot can make the collision avoidance based on the data from distance sensors. In Figure 4, a length of Z + Tol aside each edge of the obstacle is used to defin the dangerous regions. dangerous region obstacle particle/robot Z + Tol

Z + Tol

step k+1

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step k

Figure 4. Setting the dangerous region for each obstacle. In the situation shown in Figure 4, during the iterative process of ALPSO for updating the velocity and position it is identifie that the next step will enter a dangerous region. Of course the step will not go so far as to collide with the obstacle, because each single iteration step for the particle or the robot is controlled with a maximal velocity. The maximal step is related to Z and the width of the dangerous region is at least Z + Tol . If the dangerous region is entered, the algorithm takes care for obstacle avoidance first Here Tol is a small positive tolerance value. The fl w chart is shown in Figure 5. If the robot is in danger to collide with an obstacle, this study assumes an additional force to Eq. (36) with the effect of generating an acceleration and then the velocity is changed which yields a steering rotation angle θi or braking of the robot. The sub-algorithm makes a decision for the direction of steering. Due to comparison of θi > 0 or θi < 0, it controls the rotation direction to left or right, respectively. If the robot is not in danger to enter into the dangerous regions, then the movement to the new position guided by ALPSO will be performed. However, we need a little further consideration. How to determine the angle and the steering direction when the robot meets the obstacle? Based on the concept of minimal energy to be used in the system, and also since we assume at firs that the obstacles in the

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Peter Eberhard and Qirong Tang start calculate the position of the next step

will enter dangerous region?

no

yes rotate once calculate the new increment

still enter dangerous region?

yes

no move to new position end

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Figure 5. Flow chart of the firs sub-algorithm of obstacle avoidance. environment are static and known locally from the sensor signals, the above problem can be treated by boundary scanning, see Figure 6. dangerous region

Z + Tol

obstacle particle/robot A

α

O′

β

B

Z + Tol

O

Figure 6. Boundary scanning. The robot looks for a point O′ which is the closest point on the obstacle. The extremal scanned points on the obstacle are A and B. This yields the the angles α and β , respectively. If α > β , then θi should be set to rotate in the right direction. If α < β , the robot should turn left. In this way, the robot can save energy bypassing the obstacle since a big rotation angle will require a lot of steering and cost much energy for the robot. However, scanning the obstacles for the practical robots will take some time and not every robot has the ability of scanning. Usually, it needs cameras and sophisticated image processing algorithms.

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PSO Used for Mechanism Design and Guidance of Swarm Mobile Robots 4.4.2.

211

Volume Constrained Swarm Robots

So far, the question of obstacle avoidance for non volume restricted swarm particle robot is treated. Several references use PSO based algorithms to navigate a group of mobile robots but only very few of them take into account the size of the robot itself when they map the approach from particles to real robots. Here one must consider motion planning for swarm mobile robots, where the volume of the robots itself is not ignored. For this, one needs to work on the following issues. 1. Collision avoidance between the volume robot and static obstacles,

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2. mutual avoidance during the motion of all volume robots. In the following descriptions the word ‘robot’ means a volume robot, and a ‘particle robot’ is one with no volume. We enclose each robot in its enclosing circle with radius Ri . Hence, the firs question becomes trivial because simply the value of Tol has to be adjusted considering the volume. This strategy enables the robot to bypass all obstacles including thin barriers. To the second question of mutual avoidance during the movement of all volume robots, the simulation in this work uses ‘virtual detectors’ on each robot. Take robot A as an example, see Figure 7, around its forward velocity v A , the virtual detector has, e.g., a 120 degrees fiel of view. This view area can be taken as a dynamic detection area, and can also be referred to as a dynamic dangerous region. Of course, the angle can be adjusted to a suitable value based on the available sensors. The radius of the detector area is set slightly larger than 2Z since the situation of two robots moving in opposite directions at full speed must be considered. So actually the detector radius is set to be 2(Z + Tol ). In this way, during each step of trial calculation in the iterative process, if robot A detects that its next step will lead to other peers entering its dynamic dangerous region, an immediate breaking will be started by this robot. Then it turns to the obstacle avoidance sub-algorithm. However, the question is not so simple since the rotations can fall into a endless loop which is of course not reasonable. Such a case is depicted in Figure 8 where the considered robot W is surrounded by its peers. B

2( Z + Tol ) 60 o

A

vA

60 o

C

Figure 7. Virtual detector. In this situation the sub-algorithm stops this robot and, at the same time, increases the scope of its virtual detector to 360 degrees. Robot W in Figure 8 waits until the other robots move away, and then it takes the next step guided by ALPSO. Such a robot surrounded by

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Peter Eberhard and Qirong Tang E D A W C B

2( Z + Tol )

Figure 8. Robot surrounded by peers. other robots occurs frequently especially at the end of the search since all the individual robots then concentrate in a small area near the target. The fl w chart of this sub-algorithm is shown in Figure 9 and the fl w chart of VL-ALPSO can be seen in Figure 10.

4.5.

Simulation and Results

Some assumptions for the simulation experiments must be made: 1. The static obstacles are represented by polygons, 2. all simulations treat only the planar case, and 3. during the avoidance of obstacles, the robots can rotate full 360 degrees.

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4.5.1.

Objective Function and Constraints

As an objective function, a group of robots should search a target in a certain area. The target could be a light source, odor source, etc., with its potential described, e.g., as f (xx) =

1 (x1 − xm1 )2 + (x2 − xm2 )2 + ε

.

(39)

In Eq. (39), xm1 , xm2 is the location of the maximum, ε is a very small positive value used only for avoidance of an infinit value of f if x = xm . Robot i has the information of its position x i = (x1,i , x2,i ) and velocity x˙ i = (x˙1,i , x˙2,i ). The task of the group robots is to search for the source in the environment, and of course they do not know x m in advance. The robots themselves are equipped with sensors to measure the local strength of the source f (xx), and they can exchange information to their surrounding neighbors. Interferences only exist in certain local districts, they will reduce the individual robot’s ability of exchanging information, but they can not go so far as to undermine this robot since it always has its local information. If the center of the source x m is infeasible due to constraints or obstacles, then the robots should at least get as close as possible. For the following simulations we maximize the performance function from Eq. (39) and add three inequality constraints h1 , h2 and h3 . Later also some obstacles will be added. The problem can then be formulated by minimize

Ψ(xx)

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(40)

PSO Used for Mechanism Design and Guidance of Swarm Mobile Robots

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start calculate the position of the next step

will enter dangerous region?

no

yes rotate once (count=count+1) calculate the new increment

no

yes count>=N ? yes

do a short stop for the current step

still enter dangerous region? no move to new position end

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Figure 9. Flow chart of second sub-algorithm of obstacle avoidance. with

Ψ(xx) =

subject to

1 − ε = (x1 − xm1 )2 + (x2 − xm2 )2 f (xx)

  h1 (xx) = 3 − x1 ≤ 0, h2 (xx) = 2 − x2 ≤ 0,  h3 (xx) = 1 + x12 − x22 ≤ 0,

xl ≤ x ≤ xu.

Of course this is quite trivial as an optimization problem but it allows us to test the described √ algorithms. Here, we choose x m = (3, 10) and the constrained minimum is x opt = x m with Ψ(xxopt ) = 0. 4.5.2.

Experimental Setup

The main purpose of firs using simulation instead of a real robotic system is convenience. A simulation environment enables the systems to be developed rapidly and transferred to the real system with hopefully minimal change in behavior. In addition, simulation offers a programmer access to data which is not easily attainable with real robots. This study classifie the simulation experiments as shown in Table 6. In the experiments, the basic PSO parameters are set to c1 =0.1, c2 =0.8 and ω =0.6 which are equivalent to the force factors acting on h f1 , h f2 and h f3 in mechanical PSO. The population of the swarm mobile robots is n = 4 or 30. The symbol dot ‘.’ is used for a particle robot, and a

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Peter Eberhard and Qirong Tang start GUI, set initial parameters search target under PSO, iterate one step

no

constraints?

iterate to next step

yes

yes

only math. optimization constraints?

use ALPSO subprogram

meet the termination criteria?

no

use VL-ALPSO subprogram

no

yes return the necessary results (end)

yes

static obstacles?

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velocity limit + first subalgorithm of obstacle avoidance

no

virtual detector + second subalgorithm of obstacle avoidance

Figure 10. Flow chart of VL-ALPSO.

circle ‘ f’for a volume robot. Bounds are x l = [−10 − 10] and x u = [10 10] in this planar case. All algorithms are programmed using MATLAB. This work utilizes the idea of modular programming. The main function calls different sub-modules (sub-functions) depending on different situations. The benefit by doing so are not only to be able to focus on a module to solve a specifie problem and to improve the overall efficien y of the codes, but also to prepare the adaptation of the MATLAB code onto other platforms, especially to the robot itself. At the stages of exploration and fina application, in order to get a balance between convergence and accuracy, it is required to modify the basic parameters of VL-ALPSO frequently. So the development of a convenient user interface becomes necessary. Based on the MATLAB GUI functions, a user interface is developed which is shown in Figure 11. All such interfaces allow testing the simulation conveniently. Furthermore, the modules are encapsulated behind the user interface.

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Table 6. Type of experiment no. particles experiment description 1

n=4

2

n=4

3

n=30

4

n=30

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4.5.3.

particle robots coordinated movement under constraints without obstacles based on ALPSO as experiment one, but with static obstacles and also using the firs subalgorithm of obstacle avoidance volume robots under constraints with static obstacles, considering the mutual collision of robots, the target locates outside of obstacle, based on VL-ALPSO, firs and second sub-algorithms of obstacle avoidance, plus virtual detector as experiment three, but the target is within an obstacle

Results and Discussions

For experiment one and two, we want to show the aggregate behavior of the particles and their trajectories. In order to make their motion easier to be visualized, only 4 particles are used there. The objective function and constraints used are described in Eq. (40). The experiments one and two are performed for 50 times. Almost every run √ and all particles converge to the exact optimal value of Ψ(xx) = 0 at position x opt = (3, 10). The trajectories of experiment one are shown in Figure 12 while Figure 13 shows the results of experiment two. In Figure 13, one can see that particles/robots not only search the target, but also take care for collision avoidance with obstacles in the environment. Experiment three and four use 30 volume robots to search the target under constraints and also with obstacles in the environment. The purpose of this two groups of experiments is to check the algorithm of VL-ALPSO for its performance in mutual avoidance, i.e., that the robots neither during the process nor at the end overlap with each other. Thus, at most, finall only one particle/robot can occupy the exact minimal position, and the other robots should distribute themselves at other positions in the close surrounding. Here comes the important question, how to place the other robots? As VL-ALPSO is designed, other robots should at least get as close as possible to the target. So, the distribution situation becomes an important evaluation indicator to the results of the experiments. In experiment three, a Box&Whisker diagram (Tukey, 1977) is used to evaluate the distribution results. Box&Whisker methods are commonly used in the visualization of statistical analysis in many applications including the robotic area. In this method, a box and whisker graph is used to display a set of data so that one can easily see where most of the members are. Experiment three runs 50 times and all of them siege the target. One of the runs at four searching stages is shown in Figure 14. One can see that all volume robots fin and siege the target but do not overlap with each other. Furthermore, a statistics from the 50 runs is made which evaluates their distribution status. The distribution status of 50 runs of experiment three can be seen from Figure 15. Also from this figur one can see that the 25th and 34th run have several robots that are far away from the minimum and have big objective function values and are even infeasible. Nevertheless, all other 48 runs get the desired results well, even in those two worse runs the main group of robots also find and surrounds the target. Most runs get the firs quartile and the third quartile below objective function value 2 which means that about 75% of the members of the swarm robots locate close to the target with

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Peter Eberhard and Qirong Tang

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Figure 11. GUI interface for VL-ALPSO.

10 5 0 −5 −10 −10

−5

0

5

10

Figure 12. Experiment one.

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10 5 0 −5 −10 −10

−5

0

5

10

Figure 13. Experiment two.

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10

10

(a)

5

5

0

0

−5

−5

−10 −10 −5 10 (c)

0

5

10

−10 −10 −5 10 (d)

5

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−5

−5

−10 −10

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(b)

−10 −10

−5

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Figure 14. Experiment three, (a) t=0 s, (b) t=46.73 s, (c) t=115.76 s, (d) t=223.04 s. high density. In summary, all 50 runs in this experiment can be accepted as obtaining the correct results from VL-ALPSO. In experiment four, the difference is that the triangular obstacle is moved with its cen√ troid at (3, 10) so that the target is infeasible.

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15

80

18

14 13 12 11 objective function value

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10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 runs

Figure 15. Statistics of distribution for experiment three. The horizontal lines correspond to the best, the worst and the four quartiles.

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One of the 50 runs in experiment four can be seen in Figure 16. The volume robots try to get close to x m , and they do not overlap with each other during the whole motion and also the obstacles are avoided. In this case, one of the most important aspects is how the volume robots distribute around the target. Several processes of the 50 runs are chosen randomly and the mean distance between robots is computed during the motion over time, see Figure 17. The distances calculated in Figure 17 are normalized. From this figur one can see that

10

10

(a)

5

5

0

0

−5

−5

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−10 −10 −5 10 (c)

0

5

10

−10 −10 −5 10 (d)

5

5

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0

−5

−5

−10 −10

−5

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−10 −10

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Figure 16. Experiment four, (a) t=0 s, (b) t=59.30 s, (c) t=138.06 s, (d) t=226.22 s.

the mean distance between robots declines over time. The results from experiment four are as desired for real applications of swarm mobile robots. Usually, for the swarm robotic system, there are two critical factors, one is the density of the swarm robots, and another is collision detection and avoidance. In the proposed algorithm design, both of them are satisfied From the experiment results, there are some aspects which should be emphasized. First, the algorithm has a robust convergence. During the process of searching for the target in the experiments, the robots converge to the correct location. Second, although this study makes many adjustments and improvements to the basic

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normalized mean distance

normalized mean distance

1.2 1.1 (a) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 30 60 90 120 150 180 210 time (s) 1.2 1.1 (c) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 30 60 90 120 150 180 210 time (s)

normalized mean distance

Peter Eberhard and Qirong Tang

normalized mean distance

220

1.2 1.1 (b) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 30 60 90 120 150 180 210 time (s) 1.2 1.1 (d) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 30 60 90 120 150 180 210 time (s)

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Figure 17. Normalized mean distance of experiment four, (a) 8th run, (b) 15th run, (c) 30th run, (d) 45th run. PSO algorithm, the complexity in VL-ALPSO is still low. Moreover, since the robots are taking care of collision avoidance, VL-ALPSO it is not difficul to be modifie for avoidance of dynamic obstacles in an unknown environment. This is because during the mutual avoidance of robots, they already take other moving robots as dynamic obstacles, so that the robots won’t collide with each other on the way of moving toward the convergence position which originates from a real engineering problem. Third, the algorithm uses the information of each single robots neighborhood of certain radius, rather than the swarm best. This reduces the requirements for the robots detection ability to its surrounding environment. So, it can narrow down the exploration scope. Based on this, each single robot will be implemented with its own VL-ALPSO algorithm to realize a distributed computing environment. Thus, it is not necessary to have a central processor. Fortunately, not all of the robots are needed for findin an optimal solution. So the system can be scalable to a large number of robots. In the simulation, even more than 10000 particle robots have been used.

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PSO Used for Mechanism Design and Guidance of Swarm Mobile Robots

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5.

221

Conclusion

This chapter has presented several extensions to the stochastic particle swarm optimization algorithm and also two application examples. Through the augmented Lagrangian multiplier method we extend PSO to ALPSO which can easily be adjusted for solving optimization tasks with problem immanent equality and inequality constraints. The behavior of the basic PSO method is comparable to other stochastic algorithms, especially to evolutionary strategies. Our extension resulting in ALPSO shows convincing results without the need for infinit penalty factors which are yielding poor conditioning. The applicability to complex engineering problems was demonstrated by optimizing the stiffness behavior of a hexapod machine tool. To increase the stiffness of the hexapod robot, we successfully applied ALPSO to the optimization problem formulated by the use of the tangential stiffness matrix. From ALPSO to VL-ALPSO, some extension work for practical use of this algorithm for the motion planning of large scale mobile robots is given. This requires many simulation experiments. The results show that the algorithm is simple, reliable, and transferable to swarm robots. In the simulation, the robots move corresponding to their mechanical properties such as masses, inertias or external forces. This work investigates the concept of collective coordinated movement under constraints and obstacles in the environment. For the constraints, it also uses the augmented Lagrangian multiplier method. For avoiding obstacles and other robots in the environment, it uses strategies of velocity limiting, virtual detectors, and others. The proposed algorithm has no need of a central processor, so it can be implemented decentralized on a large scale swarm mobile robotic system and makes the robots move well coordinated. The natural next step after the simulations are finishe will be to distribute the simulation to many different processors, to identify clearly the mechanical properties of a single robot and then to run everything on a small swarm of real robots.

Acknowledgments The authors greatly appreciate the help of Dr.-Ing. K. Sedlaczek, Dr.-Ing. Ch. Henninger and Dipl.-Ing. T. Kurz, as well as of Prof. M. Iwamura from Fukuoka University, Japan. The example of investigating the stiffness behavior of the hexapod machine tool is part of the work finance by the German Research Council (DFG) within the framework of the SPP-1156 ‘Adaptronics for Machine Tools’. The work of motion planning for swarm mobile robots is partly done in the framework of SimTech, the Stuttgart Research Center for Simulation Technology. The authors also want to thank the China Scholarship Council for providing a Ph.D. fellowship to Qirong Tang to study in Germany. All this support is highly appreciated.

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Coelho, J. P., de Moura Oliveira, P. B., and Boaventura Cunha, J. (2002). Greenhouse air temperature control using the particle swarm optimisation algorithm. In Proceedings of 15th Triennial World Congress of the International Federation of Automatic Control, page 65, Barcelona, Spain. El-Gallad, A. I., El-Hawary, M. E., Sallam, A. A., and Kalas, A. (2002). Particle swarm optimizer for constrained economic dispatch with prohibited operating zones. In Canadian Conference on Electrical and Computer Engineering, volume 1, pages 78–81, Winnipeg, Canada. Emara, H. M., Ammar, M. E., Bahgat, A., and Dorrah, H. T. (2003). Stator fault estimation in induction motors using particle swarm optimization. In Proceedings of the IEEE International on Electric Machines and Drives Conference, volume 3, pages 1469–1475, Madison, USA. Fukuyama, Y., Takayama, S., Nakanishi, Y., and Yoshida, H. (2001). A particle swarm optimization for reactive power and voltage control in electric power systems. In Proceedings of the Congress on Evolutionary Computation, volume 1, pages 87–93, Seoul, South Korea. Gasparri, A. and Prosperi, M. (2008). A bacterial colony growth algorithm for mobile robot localization. Autonomous Robots, 24:349–364. Guan, C. L. and Wei, W. L. (2008). An immunological approach to mobile robot reactive navigation. Applied Soft Computing, 8:30–45. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Han, K., Zhao, J., Xu, Z. H., and Qian, J. X. (2008). A closed-loop particle swarm optimizer for multivariable process controller design. Journal of Zhejiang University SCIENCE A, 9(8):1050–1060. Hart, P. E., Nilsson, N. J., and Raphael, B. (1968). A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics, 4(2):100–107. He, Z. Y., Wei, C. J., Yang, L. X., Gao, X. Q., Yao, S. S., Eberhart, R. C., and Shi, Y. H. (1998). Extracting rules from fuzzy neural network by particle swarm optimization. In Proceedings of IEEE Congress on Evolutionary Computation, pages 74–77, Anchorage, USA. Heisel, U., Maier, V., and Lunz, E. (1998). Auslegung von Maschinenkonstruktionen mit Gelenkstab-Kinematik-Grundaufbau, Tools, Komponentenauswahl, Methoden und Erfahrungen. wt – Werkstatttechnik, 88(4):75–78. Henninger, C. (2009). Methoden zur simulationsbasierten Analyse der dynamischen Stabilit¨at von Fr¨asprozessen (in German). Schriften aus dem Institut f¨ur Technische und Numerische Mechanik, volume 15. Shaker Verlag, Aachen. (in print). Henninger, C., K¨ubler, L., and Eberhard, P. (2004). Flexibility optimization of a hexapod machine tool. GAMM-Mitteilungen, 27:46–65.

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Heppner, F. and Grenander, U. (1990). A stochastic nonlinear model for coordinated bird flocks In Krasner, E., editor, The ubiquity of chaos, pages 233–238. AAAS Publications, Washington D.C. Holden, N. and Freitas, A. (2009). Hierarchical classificatio of protein function with ensembles of rules and particle swarm optimisation. Soft Computing, 13(3):259–272. Holland, J. H. (1995). Hidden Order: How Adaptation Builds Complexity. Perseus Books, New York. Kennedy, J. and Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings of the international conference on neural networks, volume 4, pages 1942–1948, Perth, Australia. Kennedy, J. and Eberhart, R. C. (1997). A discrete binary version of the particle swarm algorithm. In Proceedings of the World Multiconference on Systemics, Cybernetics and Informatics, volume 5, pages 4104–4109, Piscataway, USA. Khatib, O. (1968). Real time obstacle avoidance for manipulators and mobile robots. International Journal of Robotics Research, 5(1):90–98. Lovbjerg, M., Rasmussen, T. K., and Krink, T. (2001). Hybrid particle swarm optimiser with breeding and subpopulations. In Proceedings of the Genetic and Evolutionary Computation Conference, pages 469–476, San Francisco, USA. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Lu, W. Z., Fan, H. Y., and Lo, S. M. (2003). Application of evolutionary neural network method in predicting pollutant levels in downtown area of Hong Kong. Neurocomputing, 51:387–400. Onwubolu, G. C. and Clerc, M. (2004). Optimal path for automated drilling operations by a new heuristic approach using particle swarm optimization. International Journal of Production Research, 4:473–491. Peng, J. C., Chen, Y. B., and Eberhart, R. C. (2000). Battery pack state of charge estimator design using computational intelligence approaches. In The Fifteenth Annual Battery Conference on Applications and Advances, pages 173–177, Long Beach, USA. Schiehlen, W. and Eberhard, P. (2004). Technische Dynamik (in German). Teubner, Wiesbaden. Sedlaczek, K. and Eberhard, P. (2006). Using augmented Lagrangian particle swarm optimization for constrained problems in engineering. Structural and Multidisciplinary Optimization, 32(4):277–286. Shi, Y. H. and Eberhart, R. C. (1998). A modifie particle swarm optimizer. In Proceedings of the IEEE International Conference on Evolutionary Computation, pages 69–73. Srinivasan, D., Loo, W. H., and Cheu, R. L. (2003). Traffi incident detection using particle swarm optimization. In Proceedings of the IEEE Swarm Intelligence Symposium, pages 144–151, Indianapolis, USA.

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Stentz, A. (1994). Optimal and efficien path planning for partially-known environments. In Proceedings of IEEE International Conference on Robotics and Automation, volume 4, pages 3310–3317, San Diego, USA. Suganthan, P. N. (1999). Particle swarm optimizer with neighbourhood operator. In Proceedings of the Congress on Evolutionary Computation, volume 3, pages 1958–1962, Washington, D.C., USA. Tandon, V. (2001). Closing the gap between CAD/CAM and optimized CNC end milling. Master’s thesis, Purdue School of Engineering and Technology, Purdue University Indianapolis. Tukey, J. (1977). Exploratory data analysis. Addison-Wesley, Boston. Ulrich, I. and Borenstein, J. (1998). VFH+: reliable obstacle avoidance for fast mobile robots. In Proceedings of IEEE International Conference on Robotics and Automation, volume 2, pages 1572–1577, Leuven, Belgium. Ulrich, I. and Borenstein, J. (2000). VFH*: local obstacle avoidance with look-ahead verification In Proceedings of IEEE International Conference on Robotics and Automation, volume 3, pages 2505–2511, San Francisco, USA. van den Bergh, F. (2001). An Analysis of Particle Swarm Optimizers. PhD thesis, University of Pretoria, Pretoria, South Africa. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Yoshida, H., Fukuyama, Y., Takayama, S., and Nakanishi, Y. (1999). A particle swarm optimization for reactive power and voltage control in electric power systems considering voltage security assessment. In Proceedings of IEEE International Conference on Systems, Man, and Cybernetics, volume 6, pages 497–502, Tokyo, Japan. Yu, Y., Zhu, Q. N., Yu, X. C., and Li, Y. S. (2007). Application of particle swarm optimization algorithm in bellow optimum design. Journal of Communication and Computer, 4(7):50–56.

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Zhu, Q. B. (2006). Ant algorithm for navigation of multi-robot movement in unknow environment. Journal of Software, 17(9):1890–1898.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 227-247

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Chapter 10

A N EW N EIGHBORHOOD T OPOLOGY FOR THE PARTICLE S WARM O PTIMIZATION A LGORITHM 1,∗ Angel Eduardo Munoz-Zavala , Arturo Hern´andez-Aguirre2,† ˜ and Enrique Raul ´ Villa-Diharce3,‡ 1 Departamento de Estad´ıstica, Centro de Ciencias B´asicas, Universidad Aut´onoma de Aguascalientes Universidad #940, Ciudad Universitaria, Aguacalientes, AGS, M´exico 2 Computer Science Department, Center for Research in Mathematics Jalisco S/N, Valenciana, Guanajuato, GTO, M´exico 3 Statistical Department, Center for Research in Mathematics Jalisco S/N, Valenciana, Guanajuato, GTO, M´exico

Abstract Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

This chapter introduces a new neighborhood structure for the Particle Swarm Optimization (PSO) algorithm, called Singly-Linked Ring. In the PSO algorithm, a neighborhood enables different communication paths among its members, and therefore, the way the swarm searches the landscape. Since the neighborhood topology changes the flyin pattern of the swarm, convergence and diversity differ from topology to topology. The approach proposes a neighborhood whose members share the information at a different rate. Its main property is a single link between two particles. The objective is to avoid the premature convergence of the floc and stagnation into local optimal. The approach is compared against two neighborhoods which are the state-of-theart: ring structure and von Neumann structure. These structures possess a mutual attraction between two particles. A set of controled experiments is developed to observe the transmission behavior (convergency) of every structure. Besides, a well-known set of global optimization problems is used to compare the 3 structures with the same PSO parameters. A statistical test was performed for every experiment to compare the mean values of the 3 structures. Our approach is easy to implement, and its results and its convergence performance are better than the other 2 structures. E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected]

∗ †

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1.

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce

Introduction

The particle swarm optimization (PSO) algorithm is a population-based optimization technique inspired by the motion of a bird floc [5]. Such groups are social organizations whose overall behavior relies on some sort of communication between members. Any member of the floc is called a “particle”. Most models for flock s motion are based on the interaction between the particles, and the motion of the particle as an independent entity. In PSO, the particles fl over a real valued n-dimensional search space, where each particle has three attributes: position x, velocity v, and best position visited after the firs fl PBest . The best of all PBest values is called global best GBest . Its position is communicated to all floc members such that, at the time of the next fl , all the particles are aware of the best position visited. By “flying a particle we mean to apply the effect of the local and global attractors to the current motion vector. As a result, every particle gets a new position. The floc must keep flyin and looking for better positions even when the current one seems good. Thus, findin the next best position is the main task of the floc for which exploration and therefore population diversity is crucial. In PSO, the source of diversity, called variation, comes from two sources. One is the difference between the particle’s current position and the global best, and the other is the difference between the particle’s current position and its best historical value. vt+1 = w ∗ vt + c1 ∗ U (0, 1) ∗ (xt − PBest ) + c2 ∗ U (0, 1) ∗ (xt − GBest )

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xt+1 = xt + vt+1

(1.1) (1.2)

The equation above reflect the three main features of the PSO paradigm: distributed control, collective behavior, and local interaction with the environment [3]. The firs term is the previous velocity (inertia path), the second term is called the cognitive component (particle’s memory), and the last term is called the social component (neighborhood’s influ ence). w is the inertia weight, and c1 and c2 are called acceleration coefficients The whole floc moves following the leader, but the leadership can be passed from member to member. At every PSO iteration the floc is inspected to fin the best member. Whenever a member is found to improve the function value of the current leader, that member takes the leadership (see Figure 1). A leader can be global to all the flock or local to a flock s neighborhood. In the latter case there are as many local leaders as neighborhoods. Having more than one leader in the floc translates into more attractors, possibly scattered over the search space. Therefore, the use of neighborhoods is a natural approach to figh premature convergence [9]. To update the particle position the equation for local best PSO is similar to that of the global best PSO. One would simply substitute GBest by LBest in Equation 1.1. In the PSO algorithm, a neighborhood enables different communication paths among its members, and therefore, the way the swarm searches the landscape. Since the neighborhood topology changes the flyin pattern of the swarm, convergence and diversity differ from topology to topology. The present chapter propose a new neighborhood structure that improves the performance of the original ring structure. In Section 2. a brief analysis of structures commonly used is presented. Our proposal is described in Section 3.. A comparison of our approach

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 229

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Figure 1. Neighborhood structures for PSO. against the state-of-the-art is discussed in Section 4.. Finally, in Section 5. the conclusions are presented.

2.

Neighborhood Structures

Flock neighborhoods have a structure which defin the way the information fl ws among members. Virtually, the information of the best particle, or leader, is concentrated and then distributed among its members. The most common floc organizations are shown in Figure 1. The organization of the floc affects convergence and search capacity. For instance, the fully connected structure (Figure 1-a) has reported the fastest convergence speed [3, 6]. In a fully connected structure, the communication between particles is expeditious; thereby, the floc also moves quickly toward the best solution found. On multimodal functions, the population will fail to explore outside of locally optimal regions. In a ring structure (see Figure 1-b), the information is slowly distributed among the floc members. This behavior does not contribute necessarily to improve the performance because it may be very inefficien during the refinin phase of the solutions. The ring structure has been reported to traverse larger areas of the search space [6]. Flock members organized in a ring structure communicate with n immediate neighbors, n/2 on each side (usually n = 2). Every particle is initialized with a permanent label which is independent

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A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce

of its geographic location in space. Finding the local best LBest neighbor of particle k is done by inspecting the particles in the neighborhood: k + 1, k + 2, . . . , k + n/2 and k − 1, k − 2, . . . , k − n/2. Other neighborhood structures have been developed to improve the PSO performance. Kennedy and Mendes analyzed the effects of various neighborhoods structures on the PSO algorithm [7]. They recommended the von Neumann structure because performed more consistently in their experiments than the other neighborhoods tested. Figure 1-c shows the von Neumann structure. The population is arranged in a rectangular matrix and every particle is connected to the individuals above, below, and to each side of it, wrapping the edges. It is important to mention that the LBest in a neighborhood could be the particle itself LBest = k, thus the neighbors are worst than particle k. Nonetheless, there is a variation called selfless model [4, 7, 8]. In a selfles model, the LBest of particle k can only be chosen from the neighbors, even when they can be worst than particle k. Albeit there are several neighborhood structures for PSO, their importance have been only addressed by a few researchers. It is worth to mention [5], [8], [10], [11]. This chapter studies a common characteristic in all the neighborhoods: the Double Link. In the next section, an analysis about the double link in the neighborhoods and a new neighborhood with single link are presented.

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3.

The Singly-Linked Ring

The ring neighborhood structure is used by most PSO implementations. In its simplest version, every particle k has two neighbors, particles k − 1 and k + 1. Likewise, particles k − 1 and k + 1 have particle k as a neighbor. Therefore, there is a mutual attraction between consecutive particles because they are shared by two neighborhoods. This can be represented as a doubly-linked list, as shown in Figure 2.

Figure 2. Ring neighborhood structure for PSO. In the same way, the von Neumann structure possesses a doubly-linked list. A particle k has four neighbors, particles k − 1 and k + 1 at the sides, and particles k − δ and k + δ at the top and bottom, where δ is a distance determined by the floc size. Likewise, particles

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 231 k −1, k +1, k −δ and k +δ have particle k as a neighbor. Again, there is a mutual attraction between particles, as shown in Figure 3.

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Figure 3. von Neumann neighborhood structure for PSO. The new approach introduced in this chapter organizes the floc in a modifi d ring fashion called singly-linked ring, (SLR). This structure improves the success of experimental results by a very important factor. The advantages of the new structure can be explained as follows. In the SLR topology, a particle k has particles k − 2 and k + 1 as neighbors (not k − 1 and k + 1 as in the original ring structure). In turn, particle k + 1 has particles k − 1 and k + 2 as neighbors, and k − 1 has particles k − 3 and k as neighbors. Then, k attracts k − 1, but k − 1 only attracts k through particle k + 1. Therefore, the particle in between cancels the mutual attraction (see Figure 4). Additionally, the information of the leader is transmitted to the particles at a lower speed. In other words, how many iterations does the whole swarm need to know the location of the leader?. The single linked ring takes more iterations than the ring topology. The findin reported by this chapter is that by reducing the transfer information speed, the singly-linked ring keeps the exploration of the search space, and increases the exploitation of the best solutions. Algorithm 1 shown the procedure to fin the neighbors for particle k in a singly-linked ring structure; where Nk is the neighborhood of particle k, and P(k+m) is the particle located m positions beyond particle k. The singly-linked ring structure allows neighborhoods of size N = f lock/2 − 1 without mutual attraction. In neighborhoods of size N = f lock/2 appears a double link between particles k and k + f lock/2. In the next section, we perform a comparison between singly-linked ring against von Neumann and ring structures. Neighborhoods of size n = 4 were used for every structure.

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A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce

Figure 4. Singly-Linked Ring neighborhood structure for PSO. Algorithm 1 Singly-linked ring neighborhood Nk = ∅ Step = 1 Switch = 1 repeat Nk = Nk ∪ P(k+Switch∗Step) Step = Step + 1 Switch = −1 ∗ Switch until Nk = N

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The ring structure with 4 neighbors is presented in Figure 5.

Figure 5. Ring neighborhood structure of size n = 4. A detailed explanation of the singly-linked ring structure with 4 neighbors is given in Table 1. Table 1 shows the neighbors for every particle k and the information fl w on the communications channels (neighbors) spawn by the proposed topology.

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 233

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Table 1. Singly-Linked Ring Structure with n = 4 neighbors.

4.

N

Neighbor

1

k+1

2

k−2

3

k+3

4

k−4

Structure

Experiments

The objective is to perform a fair comparison between the 3 structures using the same PSO model with the same parameters. A controled test was developed to show the convergence of each algorithm. Also, a set of well-known global optimization problems was chosen to test the performance of the 3 structures: Ring, von Neumann and Singly-Linked Ring.

4.1.

Controled Tests

Two functions was designed to show the convergence of PSO applying the 3 neighborhoods. A swarm size of 20 particles with a neighborhood size of 4 particules. The neighbors for every particle k are listed in Table 2. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

234

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 2. Neighbors for every particle Particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.1.1.

Ring 2,3,19,20 1,3,4,20 1,2,4,5 2,3,5,6 3,4,6,7 4,5,7,8 5,6,8,9 6,7,9,10 7,8,10,11 8,9,11,12 9,10,12,13 10,11,13,14 11,12,14,15 12,13,15,16 13,14,16,17 14,15,17,18 15,16,18,19 16,17,19,20 1,17,18,20 1,2,18,19

Von N. 2,4,5,17 1,3,6,18 2,4,7,19 1,3,8,20 1,6,8,9 2,5,7,10 3,6,8,11 4,5,7,12 5,10,12,13 6,9,11,14 7,10,12,15 8,9,11,16 9,14,16,17 10,13,15,18 11,14,16,19 12,13,15,20 1,13,18,20 2,14,17,19 3,15,18,20 4,16,17,19

SLR 2,4,17,19 3,5,18,20 1,4,6,19 2,5,7,20 1,3,6,8 2,4,7,9 3,5,8,10 4,6,9,11 5,7,10,12 6,8,11,13 7,9,12,14 8,10,13,15 9,11,14,16 10,12,15,17 11,13,16,18 12,14,17,19 13,15,18,20 1,14,16,19 2,15,17,20 1,3,16,18

Test I

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The firs test is a smooth function of two variables: f (x) =

2 X

x2j

j=1

xj

∈ [−100, 100],

f ∗ (x) = 0

The initial position of each particle is presented in Table 3. Figure 6 illustrates the contour levels and the initial positions for test I. For each PSO algorithm (PSO-SLR, PSO-Ring and PSO-VN) 1000 runs were developed. Each algorithm performs the function evaluations required to reach a PBest ≤ 1X10−16 for every particle k. The results are shown in Table 4. The fitnes evaluations performed by the 3 PSO algorithms to reach a PBest ≤ 1X10−16 in the whole swarm are similar. In fact, there are insignifican differences for the mean, median and standard desviation values, between the 3 approaches. Table 5 presents the average function evaluations required for every particle to reach a PBest ≤ 1X10−16 . Table 5 shows that there is only a significant difference in the convergence results for particles k = 2 and k = 3 between the PSO with single-linked ring and the other PSO approaches. In the PSO-SLR, particle k = 3 converges before particle k = 2, despite that particle k = 2 is nearby to the optimum (see Table 3, because particle k = 1 belongs to the neighborhood of particle k = 3 (see Table 2). In the PSO-Ring, particle k = 1 is neighbor of both particles, k = 2 and k = 3; thereby there is not an advantage for any particle.

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 235 Table 3. Initial positions for test I Particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

x1

x2

0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 20.0 50.0 100.0

0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 20.0 50.0 100.0

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Table 4. Convergence results for test I Mean Median Min Max Std. Desv.

SLR 6149.72 6080 4520 9240 684.9089

Ring 6178.28 6100 4460 9460 699.5135

VN 6135.46 6060 4660 9040 651.7433

Finally, particle k = 1 belongs only to the neighborhood of particle k = 2 in the PSO-VN algorithm. This difference in the particle’s convergence due to the neighborhood applied is shows in the next test. 4.1.2.

Test II

The test is a sinusoidal version of the function used in test 1 (see Figure 7):

f (x) =

2 X

0.1 ∗ x2j + 10 ∗ (1 − cos(2 ∗ xj ))

j=1

xj

∈ [−100, 100],

f ∗ (x) = 0

The initial position of each particle is presented in Table 6 (see Figure 8). Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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236

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce

Figure 6. Initial positions for test I. For each PSO algorithm (PSO-SLR, PSO-Ring and PSO-VN) 1000 runs were developed. Each algorithm performs the function evaluations required to reach a PBest ≤ 1X10−16 for every particle k. The results are shown in Table 7. The fitnes evaluations performed by the PSO-SLR and PSO-VN algorithms to reach a PBest ≤ 1X10−16 in the whole swarm are similar. There is a small difference between the PSO-Ring and the other PSO algorithms. Table 8 presents the average function evaluations required for every particle to reach a PBest ≤ 1X10−16 . Table 8 shows that there are several differences in the convergence results between the 3 algorithms. For instance, particles k = 2, k = 3, k = 19 and k = 20 are located in the same contour level. Nevertheless, particles k = 3 and k = 20 converges before particle k = 2 and k = 19 in the PSO-SLR, because particle k = 1 belongs to the neighborhood of particles k = 3 and k = 20. In the PSO-Ring, particle k = 1 is neighbor of the 4 particles,

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 237 Table 5. Average convergence results of each particle for test I

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Particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

SLR 0 182.539 176.566 189.604 189.093 193.025 198.822 199.424 203.559 201.734 203.963 205.105 208.825 211.329 211.363 213.955 213.514 217.571 248.259 271.798

Ring 0 162.779 179.178 186.978 192.423 194.388 199.292 200.079 203.027 202.237 203.148 204.91 208.006 207.983 209.024 214.398 217.432 225.785 247.549 271.96

VN 0 160.103 181.46 186.352 190.528 194.537 196.003 198.591 202.01 204.896 204.584 204.622 209.086 208.459 213.969 212.922 211.616 220.575 246.466 272.04

Figure 7. Test function II.

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238

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 6. Initial positions for test II Particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

x1

x2

0 3.5 3.5 9.5 9.5 24.5 24.5 51.5 51.5 99.5 100.0 -99.5 -51.5 -51.5 -24.5 -24.5 -9.5 -9.5 -3.5 -3.5

0 3.5 -3.5 9.5 -9.5 24.5 -24.5 51.5 -51.5 99.5 -100.0 -99.5 -51.5 51.5 -24.5 24.5 -9.5 9.5 -3.5 3.5

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Table 7. Convergence results for test II Mean Median Min Max Std. Desv.

SLR 8137.62 8060 6540 11500 656.6663889

Ring 8203.78 8160 6240 10880 681.234021

VN 8100.26 8030 6500 10920 635.6026456

thereby there is not an advantage for any particle. In the PSO-VN algorithm, particle k = 2 converges faster than the other 3 particles, because it has particle k = 1 in its neighborhood. The tests show that there are insignifican differences in the whole swarm convergence for the 3 PSO approaches. Nevertheless, there are several differences in the convergence of the particles due to the applied neighborhood.

4.2.

Global Optimization Benchmark

The benchmark is composed by 8 minimization problems. There are unimodal and multimodal functions. The whole benchmark is scalable to any dimension nx . These global optimization problems were summarized by Engelbrecht [4]. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 239

Figure 8. Initial positions for test II.

Ackley q P nx 2 −0.2 n1 j=1 xj

f (x) = 20 + e − 20e −e xj

1 nx

P

j=1

∈ [−30, 30],

x

nx cos(2πxj )

f ∗ (x) = 0

Griewank n

n

j=1

j=1 ∗

x x Y 1 X x2j − cos f (x) = 1 + 4000

xj

∈ [−600, 600],



xj √ j



f (x) = 0

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240

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 8. Average convergence results of each particle for test II Particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

SLR 0 298.306 222.545 308.396 238.775 317.075 263.937 331.256 298.395 343.81 325.524 348.54 343.82 343.579 328.579 332.422 311.702 238.431 299.618 223.556

Ring 0 225.299 224.776 291.54 280.979 318.263 314.212 338.879 339.768 356.428 355.905 355.437 342.09 341.502 314.854 319.428 283.039 294.183 223.093 225.873

VN 0 222.875 303.633 238.499 237.763 297.957 324.26 317.984 306.138 337.51 342.434 340.622 313.637 335.81 338.771 331.696 238.947 307.844 322.332 305.351

Hyperellipsoid nx X

f (x) =

j 2 x2j

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j=1

f ∗ (x) = 0

∈ [−1, 1],

xj Quadric f (x) =

j nx X X j=1

xj

k=1

xj

!2

∈ [−100, 100],

f ∗ (x) = 0

Rastrigin f (x) =

nx X (x2j − 10cos(2πxj ) + 10) j=1

xj

∈ [−5.12, 5.12],

f ∗ (x) = 0

Rosenbrock f (x) =

nX x −1

[100(xj+1 − x2j )2 + (1 − xj )2 ]

j=1

xj

∈ [−30, 30],

f ∗ (x) = 0

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 241 Schwefel f (x) =

nx X j=1

xj

q xj sin( |xj |) + 418.9829nx

∈ [−500, 500],

f ∗ (x) = 0

Spherical f (x) =

nx X

x2j

j=1

xj

4.3.

∈ [−100, 100],

f ∗ (x) = 0

Parameters

Van den Bergh analyzed PSO’s convergence [12] and provided a model to obtain convergent trajectories 0.5(c1 + c2) < w. In his work, Van den Bergh used the standard parameter values proposed by Clerc and Kennedy [2], and also applied by Bratton and Kennedy [1]: w = 0.7298 c1 = 1.49618

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c2 = 1.49618 In all the experiments a population of size m = 100 was used. 50 runs were performed for every benchmark problem. Two set of experiments were performed: dimension nx = 20 and dimension nx = 30. In the firs set, 200, 0000 fitnes evaluations were applied to solve the benchmark in dimension 20. Similarly, 300, 000 fitnes evaluations were applied to solve the benchmark in dimension 30.

4.4.

Results

In Table 9, the results for dimension nx = 20 for every PSO structure are presented. The proposed structure, singly-linked ring, outperforms the mean values of the 50 runs performed by the ring and von Neumann structures as it is shown in Table 9. Besides, the best values obtained by the singly-linked ring are better than the other structures in 6 out of 8 benchmark problem. Only, in the Griewank function, the von Neumann structure obtains the best result. In similar way, the ring structure obtains the best result in the Rastrigin function. Also, the standard deviation results obtained by the singly-linked structure for the benchmark are smaller than the von Neumann and ring structures. In some functions, the results obtained by two structures seem very similar. Therefore, a statistical analysis is applied to the experimental results and the best structure is reported. A hypothesis proof is performed on the mean values. Two comparison sets were applied: ring vs singly-linked and von Neumann vs singly-linked. The null hypothesis H0 supposes that the mean value of the two structures is equivalent µ1 = µ2 . The alternative hypothesis Ha supposes that the mean value of the singly-linked

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242

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 9. Results for the benchmark functions for dimension 20

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Function

Statistic Best Median Ackley Mean Worst Std. Dev. Best Median Griewank Mean Worst Std. Dev. Best Median H-ellipsoid Mean Worst Std. Dev. Best Median Quadric Mean Worst Std. Dev. Best Median Rastrigin Mean Worst Std. Dev. Best Median Rosenbrock Mean Worst Std. Dev. Best Median Schwefel Mean Worst Std. Dev. Best Median Spherical Mean Worst Std. Dev.

Ring Von N. SLR 0.106261 0.002845 0.002435 1.861881 1.647229 1.156019 1.750684 1.377504 1.029661 2.922855 2.816154 2.317138 6.7002E-01 8.6686E-01 8.1298E-01 0.014591 0.005901 0.009909 0.078323 0.054906 0.050264 0.096952 0.070609 0.060522 0.451276 0.250851 0.142776 7.4805E-02 5.0278E-02 3.8161E-02 1.4498E-06 1.0674E-07 4.298E-10 2.7103E-05 8.1694E-06 1.2746E-06 1.0444E-04 1.1113E-04 1.5103E-05 0.001206 0.001957 1.2335E-04 2.1578E-04 3.3502E-04 3.3421E-05 0.003947 3.4728E-04 4.1009E-05 0.154515 0.040149 0.037536 0.784072 0.189055 0.172961 9.651544 1.486739 1.617738 1.6967E+00 3.4585E-01 3.1058E-01 6.968484 9.093364 7.200471 13.330697 15.597391 13.659383 13.802695 16.902551 13.548400 25.537836 35.818615 19.138254 3.9479E+00 5.4289E+00 3.1153E+00 5.105592 7.192253 4.171506 17.811417 16.701331 15.811971 22.354720 17.203611 15.190398 150.4778 57.258483 19.364013 2.3274E+01 6.4784E+00 3.4239E+00 237.5104 236.8783 0.177308 1023.2296 1046.452566 849.6372 1035.079311 984.638802 820.375450 1836.8720 1719.1183 1664.9198 4.1372E+02 3.6995E+02 3.4302E+02 1.0801E-04 8.9328E-06 1.9372E-07 0.004016 9.6368E-04 1.9008E-04 0.011241 0.010830 0.003881 0.125470 0.164913 0.030161 2.0102E-02 2.7882E-02 8.0861E-03

is the best µ1 < µ2 . H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 < 0

(4.1)

Since 50 runs were performed, a Normal distribution approximation is assumed to perform a hypothesis test for the difference of means. The hypothesis test applied in this Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 243 chapter assumes that the sample variances are different. A test statistic is calculated: T =

¯1 − X ¯ ) − (µ1 − µ2 ) (X q 22 ∼ N (0, 1) s22 s1 + m1 m2

(4.2)

where s1 and s2 are the standard deviations, and m1 and m2 are the sample sizes. In statistical hypothesis testing, a test statistic T is a numerical summary of a set of data that reduces the data to one or a small number of values that can be used to perform a hypothesis test. The test statistic T is compared against a Zα with a significanc level 1 − α = 95%. If T < Zα H0 is rejected, and Ha is accepted statistically. Table 10 shows the statistical comparison between singly-linked and ring structures for the benchmark in dimension 20. Table 10. Singly-linked vs Ring in dimension 20

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Function Ackley Griewank Hyperellipsoid Quadric Rastrigin Rosenbrock Schwefel Spherical

Z0.05 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854

T -4.8394 -3.0674 -2.8932 -2.5051 -0.3575 -2.1534 -2.8249 -2.4018

Reject H0 H0 H0 H0 Ha H0 H0 H0

The results shown in Table 10 reject the null hypothesis µ1 = µ2 in 7 out of 8 benchmark functions. Only in the Rastrigin function the statistical test rejects the alternative hypothesis µ1 < µ2 . Therefore, the singly-linked structure is better than the ring structure in 7 out of 8 benchmark functions and both are statistically equivalent in the Rastrigin function. Table 11 shows the statistical comparison between singly-linked and von Neumann structures. The results accept the alternative hypothesis µ1 < µ2 in 6 out of 8 benchmark function. Only in two functions, Griewank and Quadric, the singly-linked and the von Neumann structures are statistically equivalent. Now, the same experiments are performed for the 3 structures in dimension nx = 30. Table 4.4. shows the results of the 3 structures in the benchmark. The singly-linked ring structure has the best statistical results in the benchmark functions. Only, in the Rosenbrock and Schwefel functions, the singly-linked ring structure is outperformed in some statistical measures by the Von Neuman structure.

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244

A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 11. Singly-linked vs von Neumann in dimension 20 Function Ackley Griewank Hyperellipsoid Quadric Rastrigin Rosenbrock Schwefel Spherical

Z0.05 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854

T -2.0696 -1.1300 -2.0164 -0.2448 -3.7891 -1.9427 -2.3023 -1.6925

Reject H0 Ha H0 Ha H0 H0 H0 H0

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The same statistical test for the mean, applied in dimension 20, is performed in dimension 30. First, we compare the singly-linked structure against the ring structure. Table 13 shows the statistical criteria for every benchmark function. The singly-linked structure is statistically better than the ring structure in 5 out of 8 benchmark functions. The singly-linked and ring structures are statistically equivalent in 3 functions in dimension 30: Quadric, Rastrigin and Rosenbrock. Table 14 shows the statistical test between the singly-linked and von Neumann structures in dimension 30. The singly-linked outperforms von Neumann structure in 6 out of 8 functions. The von Neumann and singly-linked are statistically equivalent in the Ackley and Schwefel functions. The experiments and its statistical tests indicate a superiority of the singly-linked ring structure about the ring and von Neumann structures. In some problems the results are statistically equivalent, but in general the PSO with a singly linked neighborhood outperforms the PSO with double link.

5.

Conclusion

In the PSO algorithm the communication between particles is essential. The information is transmitted according to an interaction structure (neighborhood), which can be global or local. The neighborhood affects the transmission speed and influence the PSO convergence. This chapter introduces a new neighborhood called singly-linked ring structure. Its main property is a single link between two particles. The proposed structure is compared against two neighborhoods which are the state-of-the-art: ring structure and von Neumann structure. These structures possess a mutual attraction between two particles. Two controled tests were applied to illustrate the global and individual convergence of the PSO algorithm applying the 3 neighborhoods. The convergence results show that the global convergence is similar for the 3 neighborhoods. On the other hand, the individual (particle) convergences are different for the 3 PSO approaches. A well-known set of global optimization problems is used to compare the 3 structures with the same PSO parameters. A statistical test was performed for every experiment to compare the mean values of the 3 structures. In most functions, the singly-linked ring is statistically better than the ring and von Neumann structures.

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A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 245 Table 12. Results for the benchmark functions for dimension 30

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Function

Statistic Best Median Ackley Mean Worst Std. Dev. Best Median Griewank Mean Worst Std. Dev. Best Median H-ellipsoid Mean Worst Std. Dev. Best Median Quadric Mean Worst Std. Dev. Best Median Rastrigin Mean Worst Std. Dev. Best Median Rosenbrock Mean Worst Std. Dev. Best Median Schwefel Mean Worst Std. Dev. Best Median Spherical Mean Worst Std. Dev.

Ring 2.129412 2.945031 2.967015 3.798993 4.3466E-01 0.958151 1.160741 1.178839 1.555304 1.2614E-01 0.050003 0.259791 0.313978 0.778620 2.0299E-01 467.2965 2251.7695 2518.5067 8795.6470 1.6240E+03 21.242383 32.088686 33.849417 50.898040 6.7739E+00 101.1194 219.7404 237.0977 811.4891 1.4339E+02 1022.0243 2506.0064 2532.6517 3647.1278 6.0123E+02 6.336205 18.383593 21.704584 92.265745 1.6495E+01

Von N. 1.858810 2.882489 2.873179 3.777151 4.3165E-01 1.061010 1.230401 1.241576 1.624297 1.3724E-01 0.027305 0.231684 0.283577 0.898880 1.9627E-01 715.8750 3218.8427 3936.0056 15742.1755 3.2254E+03 20.752083 39.328396 40.800588 69.222017 1.0309E+01 62.506414 248.7504 288.9030 852.4223 1.7805E+02 1046.2670 2237.8938 2258.6393 3654.3896 6.3905E+02 5.313214 15.688612 21.017348 71.566811 1.5287E+01

SLR 1.786844 2.770760 2.768877 3.523681 4.2654E-01 0.716385 1.128607 1.135765 1.481560 1.1787E-01 0.024706 0.179639 0.220057 0.774072 1.6164E-01 323.0117 1878.7285 2448.2081 7087.6590 1.6033E+03 17.829354 32.068386 33.616933 47.599280 6.2658E+00 70.237681 174.0442 224.8409 744.0114 1.3960E+02 791.1548 2402.1312 2328.1955 3271.0951 5.7821E+02 1.522795 14.695662 15.869342 48.290015 9.9346E+00

Although the singly-linked ring outperforms the two other structures the test is not conclusive and more experiments are being performed.

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A. Eduardo Mu˜noz-Zavala, A. Hern´andez-Aguirre and E. Ra´ul Villa-Diharce Table 13. Singly-linked vs Ring in dimension 30 Function Ackley Griewank Hyperellipsoid Quadric Rastrigin Rosenbrock Schwefel Spherical

Z0.05 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854

T -2.2743 -1.7631 -2.5594 -0.2151 -0.1538 -0.4651 -1.7331 -2.1428

Reject H0 H0 H0 Ha Ha Ha H0 H0

Table 14. Singly-linked vs von Neumann in dimension 30 Function Ackley Griewank Hyperellipsoid Quadric Rastrigin Rosenbrock Schwefel Spherical

N -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854 -1.644854

Z0.05 -1.2013 -3.9886 -1.7665 -2.9062 -3.8442 -2.0021 0.5707 -1.9966

T Ha H0 H0 H0 H0 H0 Ha H0

Reject

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References [1] Bratton, D; Kennedy, J. Definin a standard for particle swarm optimization. In Proceedings of the IEEE Swarm Intelligence Symposium. IEEE Press, April 2007, pages 120–127. [2] Clerc, M; Kennedy, J. The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, February 2002, 6(1):58–73. [3] Eberhart, R; Dobbins, R; Simpson, P. Computational Intelligence PC Tools. Academic Press Professional, 1996. [4] Engelbrecht, A. Fundamentals of Computational Swarm Intelligence. John Wiley and Sons, 2005. [5] Kennedy, J; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference On Neural Networks. IEEE Press, November 1995, pages 1942–1948. [6] Kennedy, J; Eberhart, R. The Particle Swarm: Social Adaptation in InformationProcessing Systems. McGraw-Hill, 1999. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

A New Neighborhood Topology for the Particle Swarm Optimization Algorithm 247 [7] Kennedy, J; Mendes, R. Population structure and particle swarm performance. In Proceedings of the 2002 Congress on Evolutionary Computation, CEC2002. IEEE, May 2002, pages 1671–1676. [8] Mendes, R. Population Topologies and Their Influence in Particle Swarm Performance. PhD thesis, Escola de Engenharia, Universidade do Minho, 2004. [9] Mendes, R; Kennedy, J; Neves, J. The fully informed particle swarm: Simpler, maybe better. IEEE Transactions on Evolutionary Computation, June 2004, 8(3):204–210. [10] Mendes, R; Neves, J. What makes a successful society? experiments with population topologies in particle swarms. In Advances in Artificial Intelligence. XVII Brazilian Symposium on Artificial Intelligence - SBIA’04. Springer Verlag, September - October 2004, pages 346–355. [11] Mohai, A; Rui, M; Ward, C; Posthoff, C. Neighborhood re-structuring in particle swarm optimization. In The 18th Australian Joint Conference on Artificial Intelligence. Springer Verlag, December 2005, pages 776–785.

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[12] van den Bergh; F. An Analysis of Particle Swarm Optimizers. PhD thesis, University of Pretoria, South Africa, 2002.

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In: Particle Swarm Optimization Editor: Andrea E. Olsson, pp. 249-298

ISBN 978-1-61668-527-0 c 2011 Nova Science Publishers, Inc.

Chapter 11

PSO A SSISTED M ULTIUSER D ETECTION FOR DS-CDMA C OMMUNICATION S YSTEMS Taufik Abr˜ao1,∗, Leonardo Dagui de Oliveira2,†, Bruno Augusto Ang´elico2,‡, and Paul Jean Etienne Jeszensky2,§ 1 DEEL - UEL – Electrical Eng. Dept., State University of Londrina - PR, Brazil. 2 PTC - EPUSP – Telecommunication and Control Engineering Department Escola Politecnica da Universidade de Sao Paulo, Brazil.

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Abstract In this chapter, a heuristic perspective for the multiuser detection problem in the uplink of direct sequence code division multiple access (DS-CDMA) systems is discussed. In particular, the particle swarm optimization multiuser detector (PSO-M U D) is analyzed regarding several figure of merit, such as symbol error rate, near-far and channel error estimation robustness, and computational complexity aspects. The PSO-M U D is extensively evaluated and characterized under different channel scenarios: additive white Gaussian noise (AWGN), single input single/multiple output (SISO/SIMO) fla Rayleigh, and frequency selective (multipath) Rayleigh channels. Although literature presents single-objective (SOO) and multi-objective optimization (MOO) approaches to deal with multiuser detection problem, in this chapter the single-objective optimization criterion is extensively used, since its application requirement is simpler than the MOO, and its performance results for the proposed optimization problem are quite satisfactory. Nevertheless, the MOO is shortly addressed as an alternative approach. Furthermore, the complexity × performance trade-off of the PSO-M U D is carefully analyzed via Monte-Carlo simulation (MCS), and the complexity reduction concerning the optimum multiuser detector (M U D) is quantified Simulation results show that, after convergence, the performance reached by the PSO-M U D is much better than the conventional detector (CD), and somewhat close to the single user bound (SuB), having computational complexity substantially lower than OM U D. E-mail address: E-mail address: ‡ E-mail address: § E-mail address:

∗ †

[email protected] . T. Abr˜ao is an Associate Professor at DEEL - UEL. [email protected]. L.D. de Oliveira is a PhD student at PTC - EPUSP. [email protected]. B.A. Ang´elico is a PhD student at PTC - EPUSP. [email protected]. P.J.E. Jeszensky is a Full Professor at PTC - EPUSP.

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

Keywords: computational complexity; multiple access wireless communication systems; DS-CDMA; multiuser detection; particle swarm optimization; SIMO; (single-)multipleobjective optimization.

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1.

Introduction

The present and next generations of wireless communications demand high data transmission rates and good quality of service, so as to provide multimedia applications, such as video, audio, Internet access, among others. The big challenge in these systems can be summarized as: how we can improve efficien y, capacity and deal with spectrum scarcity? Combining different forms of diversity in multiple access systems is the key to deal with these limitations. In addition, different modulation formats can be efficientl and adaptively adopted, such as binary and quadrature phase shift keying (BPSK and QPSK, respectively), high order modulation (M -QAM), depending on the quality of the channel. Error correcting codes are also indispensable. Furthermore, multiuser detection (M U D) in multiple access systems, such as DS-CDMA, is another alternative to deal with these limitations. In a DS-CDMA system, simple solutions like just a conventional detector may not provide a desirable quality of service, once the system capacity is strongly affected by multiple access interference (MAI). The capacity of a DS-CDMA system under practical scenarios is limited mainly by the MAI, self-interference (SI), near-far ratio (NFR) and fading channel effects. In order to reduce fading impact, the conventional receiver (Rake receiver) explores the path diversity, but it is not able to mitigate neither the MAI nor the NFR effects [1, 2]. In this context, multiuser detection emerged as a solution to overcome the MAI [2]. The best performance is acquired by the optimum multiuser detection (OM U D), based on the log-likelihood function (LLF) [2]. However, this is achieved at the cost of huge computational complexity, which increases exponentially with the number of users sharing the same channel. Under multiuser detection perspective the primary concern to increase system capacity consists in achieving high performance with relative low complexity increment. Since the maximum-likelihood (ML) approach is prohibitive for large systems i.e., large number of users sharing the same spectrum in a DS-CDMA systems, the analysis of suboptimal M U D and the respective complexity aspects is of paramount interest. As a result, in the last two decades, a variety of promising multiuser detectors with low complexity and suboptimum performance were proposed: from the linear detectors, subtractive interference cancelling [1, 2] approaches, to sphere decoding, semidefinit programming (SDP) [3], and more recently heuristic methods. The latter methods have been used for solving different detection models and obtaining near-maximum likelihood (near-ML) performance at cost of polynomial computational complexity. Among heuristic multiuser detection methods are included evolutionary programming (EP) [4], specially the genetic algorithm (GA) [5, 6], particle swarm optimization (PSO) [7, 8, 9], and the deterministic local search (LS) methods [10, 11, 12, 13, 14]. The suboptimal multiuser detection based on semidefinit relaxation (SDR-M U D) [15, 16] and heuristic M U Ds [5, 6, 11] approaches were initially proposed to work on low-order modulation, usually BPSK/QPSK signals. For instance, in SDR-M U D, the optimal maximum likelihood (ML) detection problem is carried out by relaxing the associ-

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ated combinatorial programming problem into an SDP problem with both the objective and the constraint functions being convex functions of continuous variables. Heuristic, deterministic local search methods, and semidefinit relaxation M U D approaches have been shown to yield near-optimal performance under BPSK/QPSK modulation formats [3, 15, 16, 5, 6, 11]. In fourth generation (4G) wideband wireless communication systems, multiple-inputmultiple-output (MIMO) schemes and high-order modulation formats, such as M ary phase shift keying (M -PSK) or M ary quadrature amplitude modulation (M -QAM), are often adopted in order to increase the throughput or the system capacity. Therefore, there has been recently much interest in extending the near-optimum low-complexity detection approaches to detect high-order modulated signals [10, 17, 18, 19, 7, 9, 20, 21, 22, 23]. For instance, SDR-M U D approaches, previously applied to single-input-single-output 64−QAM multicarrier CDMA systems [18], were applied to high-order QAM constellations in MIMO systems in [17, 19, 20, 21], with M -PSK constellation in [22], and in coded MIMO systems with pulse-amplitude modulation (PAM) constellations were investigated in [23]. Although the M -ary combinatorial problem associated with the optimal M -QAM ML detection can be solved by SDP-M U D relaxation methods, simulation results show that, when M is relatively large (M ≥ 32), the performance obtained with SDP relaxation methods becomes far from the ML performance. The reason is that the detection error probabilities of the sequence of binary variables associated with each QAM symbol are unbalanced. Some of the binary variables are more robust to detection errors than others, and recently different strategies has been investigated to exploit this characteristic. In one of them, the strategy consists in detecting those binary variables which are more robust to detection errors firs [21], and, with a multistage approach, decisions are made successively, only considering those binary variables which can be detected with higher accuracy in each stage. Based on these decisions, the original problem is reduced to a smaller-sized SDP problem for the undetermined binary variables. This process continues until all binary variables are determined. However, the drawback of this approach is an unavoidable increasing on the multiuser detector complexity associated with a delay increment to detect the symbol of all users. Finally, heuristic procedures can be adopted in order to achieve good fi ure of merit in multiuser detection problem, accomplishing excellent performance × complexity tradeoffs. Embracing heuristic procedures enable to achieve near-optimum low-complexity multiuser detection under all different scenarios, from low to high-order symbol modulation schemes, combined with fading and multiple input multiple output channels. The local search is an optimization method that consists of searches in a previously established neighborhood [14]. It is a quite simple search method. For the local search algorithm it is important to restrict the neighborhood and to choose a good start point in order to fin a valid solution with low complexity. The degree k in the k−opt local search (k-opt LS) can be chosen taken into account all the m bits mapping a symbol, and evaluating all k Hamming distance candidate-vectors from the current solution that belongs to a constellation of size M = 2m symbols. It is naturally advantageous to use Gray mapping to enumerate all the k−Hamming candidate-vectors. The search complexity increases exponentially with the degree k, becoming too computationally expensive to adopt k ≥ 2 for high-order modulation (M ≥ 8) with large number of users. In [11], the SISO DS-CDMA

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

M U D problem with BPSK modulation had been solved efficientl using the 1-opt LS. But when the modulation order increases, the k-opt LS approach becomes inefficient suffering of lack of diversification As a consequence, there are a performance degradation and an increasing in complexity [10]. In the 1-opt LS multiuser detector, all unitary Hamming distance candidate-vectors from the current best solution (obtained from previous iterations) are evaluated. If a better candidate-vector is found, the current best solution is updated, and a new iteration takes place. The search terminates when there is no improvement. Basically, three advantages make the 1-opt LS algorithm a natural choice in order to reach an efficien solution for the M U D problem under BPSK and QPSK modulation formats: a) absence of input parameters; b) simple stop criterion, avoiding prior calculation; c) simple strategy, low complexity with possible additional simplification [12, 11]. However, there are few works dealing with complex and realistic system configura tions. High-order modulation heuristic M U Ds in SISO or MIMO systems were previously addressed in [7, 9, 10]. In [10] a heuristic technique was applied to near-optimum asynchronous DS-CDMA multiuser detection problem under 16−QAM modulation and SISO multipath channels. Previous results on literature [11, 12, 8, 4, 13] suggest that evolutionary algorithms and particle swarm optimization have similar performance, and that a simple local search heuristic optimization is enough to solve the M U D problem with loworder modulation (BPSK [11] and QPSK). However, for high-order modulation formats, the LS-M U D does not achieve good performances due to a lack of search diversity, whereas the PSO-M U D has been shown to be more efficien for solving the optimization problem under M -QAM modulation [10]. This chapter is organized as follows. The DS-CDMA system model, with description of the optimum detection metric, considering low and high order modulation, singleand multi-path, and single-input-multiple-output channels is revised in Section 2.. Additionally, decoupling log-likelihood function (LLF) containing only real values suitable for high-order modulation formats, a revision and contextualization of the swarm intelligence method applied to the multiuser detection, and single/multiple-objective optimization approaches for M U D problem as well, are addressed in this section. Section 3. presents the PSO multiuser detector model. The swarm optimization procedures, including an extensive analysis on the PSO input parameters optimization are carried out in Section 4.. Exhaustive numerical analysis for several figure of merit under different channel and system scenarios, is carried out in Section 5.. In order to evaluate the feasibility of the swarm heuristic technique in solving efficientl the M U D problem, a complexity analysis, in terms of number of operations, is pointed out in Section 6.. Finally, the main conclusions of this chapter are highlighted in Section 7..

2.

System Model

In this Section, a single-cell DS-CDMA system model is described for AWGN, fla Rayleigh and multipath Rayleigh channels, considering different modulation schemes, such as binary/quadrature phase shift keying (BPSK/QPSK), and 16-quadrature amplitude modulation (16-QAM), and single or multiple antennas at the base station receiver (uplink).

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After describing the conventional detection approach, based on matched filte bank followed by a maximum ratio combining (MRC) rule, subsequent subsection explains how to apply the maximum-likelihood approach through the OM U D in order to achieve the best known uncoded detection performance at cost of a huge complexity increment. The heuristic multiuser detection model, with single- and multiple objective optimization approaches, is discussed in subsequent subsections as a promising alternative to reach the near-optimum detection performance with much lower complexity than OM U D.

2.1.

DS-CDMA

In this section, a single-cell asynchronous multiple access DS-CDMA system model with high order modulation and multiple antennas at the base station receiver (reverse link) is described under selective fading channels as well. Hence, the system model is generic enough to allow describing additive white Gaussian noise (AWGN), fla or synchronous channels, binary modulation formats and single-antenna receiver, as particular cases of the system model discussed in the sequel. The base-band transmitted signal of the kth user is described as [24] r ∞ Ek X (i) (1) sk (t) = dk gk (t − iT ), T i=−∞

(i)

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where Ek is the symbol energy, and T is the symbol duration. Each symbol dk , k = 1, . . . , K is taken independently and with equal probability from a complex alphabet set A (i) of cardinality M = 2m in a squared constellation points, i.e., dk ∈ A ⊂ C, where C is the set of complex numbers. Figure 1 shows the modulation formats considered. The normalized spreading sequence for the k-th user is given by N −1 1 X gk (t) = √ ak (n)p(t − nTc ), N n=0

0 ≤ t ≤ T,

(2)

where ak (n) is a random sequence with N chips assuming the values {±1}, p(t) is the pulse shaping, assumed rectangular with unitary amplitude and duration Tc , with Tc being the chip interval. The processing gain is given by N = T /Tc , as illustrated in Figure 2. The equivalent base-band received signal at qth receive antenna, q = 1, 2, . . . , Q, containing I symbols for each user in multipath fading channel can be expressed by rq (t) =

K X L I−1 X X

(i)

(i)

Ak dk gk (t − nT − τq,k,ℓ )hq,k,ℓ ejϕq,k,ℓ + ηq (t),

(3)

i=0 k=1 ℓ=1

with Ak =

q

Ek T , L being the number of channel paths, admitted equal for all K users, τq,k,ℓ delay1 for the signal of the kth user, ℓth path at qth receive antenna, ejϕq,k,ℓ is the

is the total respective received phase carrier; ηq (t) is the additive white Gaussian noise with bilateral 1

Considering the asynchronism among the users and random delays for different paths.

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al. ℑ ℑ

(0)

(1)

−1

1 ℜ BPSK

(1101)

(1001)

(1100)

(1000)

−3

(00)

−1 (11)

QPSK

(0101)

(0000)

(0100) 3

1

(1110)

(1010)

(1111)

(1011)

1 ℜ (10)

1

(0001)

−1

ℑ

(01)

3

−1

−3

(0010)

(0110)

(0011)

(0111)

ℜ

16−QAM

Figure 1. Some modulation formats with Gray mapping.

i

i+1

i+2

Tc Spread Sequence

t

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

N Tc Information Symbols

t

Spread Signal

t

Figure 2. Standard spreading with N = 10. (i)

power spectral density equal to N0 /2, and hq,k,ℓ is the complex channel coefficien for the ith symbol, define as (i)

(i)

(i)

hq,k,ℓ = γq,k,ℓ ejθq,k,ℓ , (i)

(4)

where the module γq,k,ℓ is a random variable characterized by a Rayleigh distribution and (i)

the phase θq,k,ℓ is a random variable modeled by the uniform distribution U[0, 2π]. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

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Generally, a slow and frequency selective channel2 is assumed. The expression in (3) is quite general and includes some special and important cases: if Q = 1, a SISO system (i) is obtained; if L = 1, the channel becomes non-selective (flat Rayleigh; if hq,k,ℓ = 1, it results in the AWGN channel; moreover, if τq,k,ℓ = 0, a synchronous DS-CDMA system is characterized. At the base station, the received signal passes through a matched filte bank (CD), with D ≤ L branches (fingers per antenna of each user. When D ≥ 1, CD is known as Rake receiver. Assuming perfect carrier phase estimation, after despreading the resultant signal is given by 1 = T

(i) yq,k,ℓ

Z

(i+1)T

nT

(i)

(i)

(i)

(i)

(i)

rq (t)gk (t − τq,k,ℓ )dt = Ak hq,k,ℓ dk + SIq,k,ℓ + Iq,k,ℓ + ηeq,k,ℓ . (5)

The firs term is the signal of interest, the second corresponds to the self-interference (SI), the third to the multiple-access interference (MAI) and the last one corresponds to the filtere AWGN. To elaborate more about (5), the SI on the ℓ-th correlator output of the k-th user from q-th antenna is given by

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(i) SIq,k,ℓ

Ak = T

D X

Z

ϕ eq,k,d

e

d=1, d6=ℓ

T

(i )

(i )

± dk ± gk (t − τq,k,ℓ )gk (t − τq,k,d )dt, hq,k,d

0

(6)

where ϕ eq,k,d = ϕq,k,d − ϕq,k,ℓ is the relative phase between the carriers of desired and self interfering signals at qth receive antenna, and the index (i± ) associated to the symbol of (i ) interest dk ± represents the i-th, or (i − 1)-th or even (i + 1)-th symbol, depending on the relative delay among the desired (index ℓ) and self interfering (index d) signals. Besides, when slow (or very slow) fading channels are assumed, the complex channel coefficient (i± ) hold fi ed at least for a couple of symbol period, hence the short time dependence hq,k,d (i)

(i )

± ≡ hq,k,d . could be eliminated, resulting in hq,k,d Similarly to SI, the MAI term on the ℓ-th correlator of the k-th user, at qth receive antenna, is given by

(i) Iq,k,ℓ

1 = T

K X

Au

u=1, u6=k

D X d=1

ϕ eq,u,d

e

·

Z

0

T

(i )

± ±) hq,u,d d(i gu (t − τq,u,d )gk (t − τq,k,ℓ )dt, u

(7)

with ϕ eq,u,d = ϕq,u,d − ϕq,k,ℓ . The cross-correlation element between the kth user, ℓth path and uth user, dth path, at qth receive antenna, can be identifie as ρqk,ℓ,u,d

1 = T

Z

T

gk (t − τq,k,ℓ )gu (t − τq,u,d )dt.

(8)

0

2

Slow channel: channel coefficient were admitted constant along the symbol period T ; and frequency selective condition is hold: T1c >> (∆B)c , the coherence bandwidth of the channel. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

Considering a maximal ratio combining (MRC) rule3 with diversity order equal to DQ for each user, the M −level complex decision variable is given by (i)

ζk

=

Q X D X

(i)

(i)

yq,k,ℓ · wq,k,ℓ

(9)

k = 1, . . . , K,

q=1 ℓ=1

(i)

(i)

b(i)

(i)

(i)

where the MRC weights wq,k,ℓ = γ bq,k,ℓ e−j θq,k,ℓ , with γ bq,k,l and θbq,k,ℓ been a channel amplitude and phase estimation, respectively. For EGC rule, the channel amplitude weights b(i)

results all identical4 , but the channel phases need to be estimated yet, wq,k,ℓ = e−j θq,k,ℓ . After that, at each symbol interval, decisions are made on the in-phase and quadrature (i) (i) components5 of ζk by scaling it into the constellation limits obtaining ζ˘k , and choosing the complex symbol with minimum Euclidean distance regarding √ the scaled decision variable. Alternatively, this procedure can be replaced by separate M −level quantizers qtz acting on the in-phase and quadrature terms separately, such that (i)

 n o  n o (i) (i) (i),CD = qtz ℜ ζ˘k + j qtz ℑ ζ˘k , dbk Areal

k = 1, . . . , K,

Aimag

(10)

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

where Areal and Aimag is the real and imaginary value sets, respectively, from the complex alphabet set A, and ℜ{·} and ℑ{·} representing the real and imaginary operators, respectively. Figure 3 illustrates the general structure of signals for SIMO uplink multipath M -QAM DS-CDMA system with MRC Rake receiver.

2.2.

Optimum Detection

The optimum maximum likelihood multiuser detector was introduced in 1984 by S. Verd´u [2] and is based on log-likelihood function. The OM U D estimates the symbols for all K users by choosing the symbol combination associated with the minimal distance metric among all possible symbol combinations in the M = 2m constellation points. In the asynchronous multipath channel scenario, the one-shot asynchronous channel approach is adopted, where a configuratio with K asynchronous users, I symbols and D branches is equivalent to a synchronous scenario with KID virtual users. Furthermore, in order to avoid handling complex-valued variables in high-order squared modulation formats, henceforward the alphabet set is re-arranged as Areal = Aimag = Y ⊂ √ (i) Z of cardinality M , e.g., 16−QAM (m = 4): dk ∈ Y = {±1, ±3}. 3

In the absence of interference, MRC is the optimal combining scheme (regardless of fading statistics) but comes at the expense of complexity increasing, since it requires knowledge of all channel fading parameters. Since knowledge of channel fading amplitudes is needed for MRC, this scheme can be used in conjunction with unequal energy signals, such as M-QAM [25]. Other combining rules include: equal gain combining (EGC), selected combining (SC), switched combining, empirical combining (EC) and hybrid schemes. 4 However, the signal-noise ratio (SNR) relation among EGC and MRC always holds, SNREGC < MRC SNR . 5 Note that, for BPSK, only the in-phase term is presented.

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

PSO Assisted Multiuser Detection for DS-CDMA Communication Systems 1

d1 (t)

S/P m

user 1

fDpl

user K

AK gK (t)

QPSK, M-QAM Mapping

1

h1,1,L

sK (t)

Q antenna

η(t)

user 1

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(i)

s1

Channel

DS/CDMA Tx

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(·)

g1 (t − τq,1,1 ) (i)

R Ts 0

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(i)

sb1

decisor

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dK (t)

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1

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CSI

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b1

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(·)

yq,1,D

D branches

user K

hQ,1,L hQ,1,1 Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

K (i) sK

user K DS/CDMA Tx

hQ,K,L hQ,K,1

Q

(i)

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(·)

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(i)

sbK

decisor

q th antenna

MRC, user K

g1 (t − τq,1,D )

gK (t − τq,K,D )

b)

Figure 3. Uplink base-band DS-CDMA system model with Conventional receiver. a) K users transmitters; b) SIMO channel and Conventional receiver with Q multiple receive antennas. The OM U D is based on the maximum likelihood criterion [2] that chooses the vector of symbols dp , formally define in (16), which maximizes the metric dopt = arg

max

dp

∈Y 2KID



f Ω dp





,

(11)

where f (Ω(·)) is a single or multi-objective function (see Subsection 2.4.) that takes into account some combination rule over Q receive signals, considering the basic log-likelihood function, to be define formally in (14). In general form, the multi-objective cost function Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

258

Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

is define as: f (dp ) = [f1 (dp ), . . . , fQ (dp )] = [Ω1 (dp ), . . . , ΩQ (dp )] ,

(12)

where Q log-likelihood objective functions are separately applied to the pth vectorcandidate dp . On the other hand, in a SIMO channel, the single-objective function generally is written as a combination of the LLFs from all receive antennas, given by 

f Ω dp




=

Q X q=1

(13)


Ωq dp .

Assuming the more general case considered in this chapter, i.e., K asynchronous users in a SIMO multipath Rayleigh channel with diversity D ≤ L, the LLF can be define as a decoupled optimization problem with only real-valued variables, such that (14)

⊤ ⊤ ⊤ Ωq (dp ) = 2d⊤ p Wq yq − dp Wq RWq dp ,

with definition yq :=



ℜ{yq } ℑ{yq } dp :=

 

;

Wq :=

ℜ{dp } ℑ{dp }



;



ℜ{AH} −ℑ{AH} ℑ{AH} ℜ{AH} R :=



R 0 0 R





;

(15)

,

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where yq ∈ R2KID×1 , Wq ∈ R2KID×2KID , dp ∈ Y 2KID×1 , R ∈ R2KID×2KID . The

vector dp ∈ Y KID×1 in Equation (15) is define as (1)

(1)

(1)

(1)

(I)

(I)

(I)

(I)

dp = [(d1 · · · d1 ) · · · (dK · · · dK ) · · · (d1 · · · d1 ) · · · (dK · · · dK )]⊤ . | {z } | {z } | {z } | {z } D times

D times

D times

(16)

D times

In addition, the yq ∈ CKID×1 is the despread signal in Equation (5) for a given q, in a vector notation, described as i h (I) (I) (I) (I) (1) (1) (1) (1) yq = (yq,1,1 · · · yq,1,D ) · · · (yq,K,1 · · · yq,K,D ) · · · (yq,1,1 · · · yq,1,D ) · · · (yq,K,1 · · · yq,K,D ) . (17)

Matrices H and A are the coefficient and amplitudes diagonal matrices, and R represents the block-tridiagonal, block-Toeplitz cross-correlation matrix, composed by the submatrices R[1] and R[0], such that [2] 

  R=  

R[0] R[1]⊤ 0 R[1] R[0] R[1]⊤ 0 R[1] R[0] ... ... ... 0 0 0

... 0 0 ... 0 0 ... 0 0 ... ... ... . . . R[1] R[0]



  ,  

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PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

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with R[0] and R[1] being KD matrices with elements   1, , ρq ρa,b [0] =  k,ℓ,u,l q ρu,l,k,ℓ ,  0, ρa,b [1] = ρqu,l,k,ℓ ,

if (k = u) and (ℓ = l) if (k < u) or (k = u and ℓ < l) if (k > u) or (k = u and ℓ > l), if k ≥ u , if k < u

(19)

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where a = (k − 1)D + ℓ, b = (u − 1)D + l and k, u = 1, 2, . . . , K; ℓ, l = 1, 2, . . . , D; the correlation function ρqk,ℓ,u,d is calculated through Equation (8). The evaluation in (11) can either be extended along the whole message, where all symbols of the transmitted vector for all K users are jointly detected (vector ML approach) or the decisions can be taken considering the optimal single symbol detection of all K multiuser signals (symbol ML approach). In the synchronous case, the symbol ML approach with I = 1 is considered, whereas in the asynchronous case the vector ML approach is adopted with I = 7 (I must be, at least, equal to three (I ≥ 3)). The vector dp in (15) belongs to a discrete set with size depending on M , K, I and D. Hence, the optimization problem posed by (11) can be solved directly using a m−dimensional (m = log2 M ) search method. Therefore, the associated combinatorial problem strictly requires an exhaustive search in AKID possibilities of d, or equivalently an exhaustive search in Y 2KID possibilities of dp for the decoupled optimization problem with only real-valued variables. As a result, the maximum likelihood detector has a complexity that increases exponentially with the modulation order, number of users, symbols and branches, becoming prohibitive even for a moderate product values m K I D, i.e., even for a BPSK modulation format, medium system loading (K/N ), small number of symbols I and correlators from D branches.

2.3.

Multiuser Detection: A Heuristic Perspective

The maximization of (11) is a combinatorial optimization problem, i.e., the set of possible arguments comprises a finit set. Combinatorial optimization problems can always be solved by exhaustive search, through the cost function computation for every possible argument, following by the selection of the candidate-vector which maximizes the loglikelihood function [2]. OM U D is the exhaustive search realization for the combinatorial multiuser detection problem, which the selection of the optimum vector dopt can be done in O(2mKID ) operations. The equivalent optimum symbol vector in Equation (11) is estimated in order to maximize the sequence transmission probability given that r(t) was received, where r(t) is extended for all message. In [26] it was demonstrated that multiuser detection problem results in a nondeterministic polynomial-time hard (NP-hard) complexity, that is, no algorithm is known for OM U D whose computational complexity results polynomial in terms of the simultaneous number of users K, detected symbols I and processed branches D, regardless of the cross-correlation matrix, R. Hence, the OM U D is impractical to implement. Thenceforth, a great variety of suboptimal approaches trying to solve efficientl the M U D problem have been proposed and characterized in the literature: from linear multiuser

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detectors [2, 27] to heuristic multiuser detectors [28, 5]. Alternatives to OM U D into the class of linear multiuser detectors include the Decorrelator [29], and MMSE [30]. Besides, the classic non-linear multiuser detectors include the interference cancellation (IC) M U D [31] and zero-forcing decision feedback (ZFDF) [32]. The drawback with (non-)linear, ZFDF, and hybrid cancellers sub-optimal MuDs is that they fail in approaching the ML performance under realistic channel and system scenarios while feasibility with relatively low complexity is held. On the other hand, heuristic near-optimum M U D approaches rise as a promising alternative, saving search-time and complexity. Among them, the genetic algorithm-based multiuser detector (GA-M U D), which was initially proposed in [28] for synchronous AWGN DS-CDMA, the evolutionary programming (EP-M U D) [33], ant colony optimization (ACO-M U D) [34], Tabu search (TS-M U D) [35], local search (LS-M U D) [13], and the particle swarm optimization multiuser detector (PSO-M U D). GA and EP-MuDs were extensively compared under different channels scenarios in [6]. For most of the practical cases, M U Ds based on heuristic techniques result in almost optimum performance, i.e., very close to the performance reached by the OM U D, with the advantage of smaller computational cost and delay time per detected symbol, hence an attractive trade-off between convergence speed, complexity and performance. The promising characteristics of PSO under multiuser detection problem perspective, such as simplicity regards to GA-M U D approach, while keeping excellent performance ×complexity trade-off, have attracted much attention recently on the development of heuristic swarm approach for multiuser detection over a widely practical channel and system scenarios. Thereby, more research is necessary in order to explore the potentiality of swarm intelligence when applied to solve efficientl the multiuser detection problem. In the next subsection single-objective optimization and multi-objective optimization approaches, suitable for DS-CDMA systems under SISO and SIMO channels, are described.

2.4.

Weighting Multi-objective Optimization

In presence of spatial diversity (Q receive antennas), the multiuser detection problem can be modelled through single or multi-objective optimization (SOO and MOO, respectively) approaches. This subsection brings some arguments in order to adopt SOO or alternatively MOO approach. Since the maximum-likelihood (or optimum multiuser) detector has a complexity that increases exponentially with the number of users, and therefore it is not practical to be implemented, heuristic approach rise as a possible solution. In the SIMO context, two types of objective function could be considered in order to implement Equation (11). The firs one is the linearly combined Q-LLFs antenna-diversity-aided strategy (LC Q-LLFs), in which each of K-bits vector-candidates on the qth receive antenna is linearly combined considering all the Q receive antennas. Thus, the fitnes value of the pth K-bits vectorcandidate and the associated objective function can be described as [36] Q

  f (dp ) = Ω(dp ) ,

Ω (dp ) =

1 X Ωq (dp ), Q

p = 1, 2, . . . , P.

q=1

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PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

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where P is the population size of PSO strategy. Note that assuming the channel fading associated with different receive antennas been independent, then Ωq (dp ) 6= Ωz (dp ), for q 6= z. As a result, under deep fading condition in some of the Q antennas, the data estimation corresponding to different antennas may result unequal. Note that Equation (13) is a direct realization of the LC Q-LLFs strategy. The second heuristic objective function is a weighting multi-objective (WO) version, suggested in [37], which takes into account independent and combined log-likelihood functions, such that   f (dp ) = Ω1 (dp ), . . . , ΩQ (dp ), Ω(dp ) = [f1 (dp ), . . . , fQ (dp ), fQ+1 (dp )] ,

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p = 1, 2, . . . , P,

(21)

where (Q + 1) log-likelihood objective functions are separately applied to the pth K−bits vector-candidate dp (or equivalently to the pth 2K−bits real-values vector-candidate dp in Equation (14)). The firs Q fitnes values are the LLFs of (14), which is related to each of Q receive antennas. The (Q + 1)th fitnes value is the LC Q-LLFs given by (20), i.e., fQ+1 (dp ) = Ω(dp ). Indeed, due to the independent fading on different receive antennas, in most cases, it is impossible to fin a vector-candidate that results the best optimal6 for Ωq (dp ), ∀q. It is clear that the implementation of weighting multi-objective function in (21) implies a significan complexity increment respect to LC Q-LLFs objective function, Equation (20). However, numerical results have been indicated that the correspondent performance improvement is only marginal for most of realistic channels and practical system scenarios [37, 38]. Hence, in several situations, the best performance×complexity trade-off lies on a single-objective function described by LC Q-LLFs of Equation (20). In the next section, PSO heuristic approach is explored in order to solve (11) efficientl . Thus, with these Q + 1 objective functions, the real-values vector-candidates in (14) is explored by the PSO-M U D algorithm, according to the single-objective optimization approach of Equation (20) and, alternatively, can be implemented using the WO Q-LLFs optimization approach described by (21).

3.

PSO Multiuser Detectors

Particle swarm optimization (PSO) was developed after some researchers have analyzed birds behavior and discern that the advantage obtained through their group life could be explored as a tool for a heuristic search. Considering this new concept of interaction among individuals, in 1995 J. Kennedy and R. Eberhart developed a new heuristic search based on a particle swarm [39]. The PSO principle is the movement of a group of particles, randomly distributed in the search space, each one with its own position and velocity. The position of each particle is modifie by the application of velocity in order to reach a better performance [39]. The interaction among particles is inserted in the calculation of particle velocity. In the following subsections, two discrete PSO algorithm versions adapted to the SIMO M U D problem are described in details. 6

That satisfie simultaneously all LLFs.

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3.1.

Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

Discrete Swarm Optimization Algorithm

Discrete or, in several cases, binary PSO [40] is suitable to deal with digital information detection/decoding. Hence, binary PSO is adopted in this chapter. The PSO’ particle selection for evolving is based on the highest fitnes values obtained through (20) or alternatively (21). Decisions under LC Q-LLFs strategy are based on a single entity by combining the information from Q receive antennas. On the other hand, decisions under WO Q-LLFs strategy can be established from a single or multiple entities decisions perspective. If the decision rule is based on the highest fitne s values from Q + 1 cost functions, a single-objective weighting optimization approach is obtained. However, considering a subset of Q + 1 independent solutions to evolve to the next iteration, a multiobjective particle swarm optimization can be applied. In this chapter, only SOO-based decisions procedures were implemented and compared. So, in order to evaluate (20) or, alternatively (21), over all particle-candidates, it is necessary to calculate the particle velocity and its respective position. For simplicity, in this subsection a SISO channel (Q = 1 antenna) is assumed. Accordingly, each candidatevector define like di has its binary representation, bp [t], of size mKI, used for the velocity calculation, and the pth PSO particle position at instant (iteration) t is represented by the mKI × 1 binary vector  r  ⊤ r r r r bp [t] = [b1p b2p · · · brp · · · bKI p ] ; bp = bp,1 · · · bp,ν · · · bp,m ; bp,ν ∈ {0, 1}, (22) (i)

where each binary vector brp is associated with one dk symbol in Equation (16). Each particle has a velocity represented by

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best vp [t + 1] = ω · vp [t] + φ1 · Up1 [t](bbest p [t] − bp [t]) + φ2 · Up2 [t](bg [t] − bp [t]), (23)

where ω is the inertial weight; Up1 [t] and Up2 [t] are diagonal matrices with dimension mKI, whose elements are random variables with uniform distribution U ∈ [0, 1]; bbest g [t] and bbest [t] are the best global position and the best local positions found until the tth p iteration, respectively; φ1 and φ2 are weight factors (acceleration coefficients regarding the best individual and the best global positions influence in the velocity update, respectively. For M U D optimization with binary representation, each element of bp [t] in (23) just assumes “0” or “1” values. Hence, a discrete mode for the position choice is carried out inserting a probabilistic decision step based on threshold, depending on the velocity. Several functions have this characteristic, such as the sigmoid function [40] r S(vp,ν [t]) =

1 r [t] −vp,ν

1+e

,

(24)

r [t] is the rth element of the pth particle velocity vector, vr where vp,ν = p   r r r vp,1 · · · vp,ν · · · vp,m , and the selection of the future particle position is obtained through the statement r if urp,ν [t] < S(vp,ν [t]),

otherwise,

brp,ν [t + 1] = 1; brp,ν [t + 1] = 0,

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PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

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where brp,ν [t] is an element of bp [t] (see Equation (22)), and urp,ν [t] is a random variable with uniform distribution U ∈ [0, 1]. After obtaining a new particle position bp [t + 1], it is mapped back into its correspondent symbol vector dp [t + 1], and further in the real form dp [t + 1], for the evaluation of the objective function in (13). In order to obtain further diversity for the search universe, the Vmax factor is added to the PSO model, Equation (23), which is responsible for limiting the velocity in the range [±Vmax ]. The insertion of this factor in the velocity calculation enables the algorithm to escape from possible local optima. The likelihood of a bit change increases as the particle velocity crosses the limits established by [±Vmax ], as shown in Table 1. Table 1. Minimum bit change probability as a function of Vmax . Vmax 1 − S(Vmax )

1 0.2690

2 0.1192

3 0.0474

4 0.0180

5 0.0067

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Population size P is typically in the range of 10 to 40 [41]. However, considering the multiuser detection problem operating under different optimization scenarios, such as AWGN and multipath channels, high order modulation (M -QAM), and multiple antennas, the size of PSO population must be optimized in order to obtain efficien y under all these different scenarios. Taking into account the background of [8], the PSO’s population size for the M U D problem is set to j p k P = 10 0.3454 π(mKI − 1) + 2 . (26) Algorithm 1 describes the pseudo-code for the PSO implementation. Next, we discuss the WO Q-LLFs optimization version for the SIMO PSO-M U D.

3.2.

WO Q-LLF Selection for SIMO PSO-M U D

In the weighting multi-objective PSO (WO-PSO) M U D algorithm, at each iteration the new particle velocity is calculated weighting the contribution of the particle position associated to each receive antenna, based on multi-objective function in (21), resulting vp [t + 1] = ω · vp [t] + +

Q+1 X q=1



(27)



[t] − bp [t] , [t] − bp [t] + φq2 · Uqp2 [t] bbest,q φq1 · Uqp1 [t] bbest,q g p

where now bbest,q [t] and bbest,q [t] are the best global and the best local particle positions, g p respectively, found so far (t-th iteration) in the qth antenna (q = 1, . . . , Q), as well as consideringPall combined antennas (q = Q + 1). Besides, the positive acceleration coefficient q q satisfy Q+1 q=1 φ1 + φ2 = C, where C is a real constant, generally assumed equal to 4 [42]. In order to obtain fast convergence without losing a certain exploration and exploitation capabilities, φ2 could be increased, being chosen for the single-carrier BPSK DS-CDMA

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

Algorithm 1 PSO Algorithm for the M U D Problem b Input: dCD , P, G, ω, φ1 , φ2 , Vmax ; Output: d begin 1. initialize firs population: t = 0; e where B e contains (P − 1) particles randomly B[0] = bCD ∪ B, generated; best CD bbest ; p [0] = bp [0] and bg [0] = b vp [0] = 0: null initial velocity; 2. while t ≤ G a. calculate Ω(dp [t]), ∀bp [t] ∈ B[t] using (13); b. update velocity vp [t], p = 1, . . . , P, through (23); c. update best positions: for p = 1, . . . , P best if Ω(dp [t]) > Ω(dbest p [t]), bp [t + 1] ← bp [t] best best else bp [t + 1] ← bp [t] end h i if ∃ bp [t] such that Ω(dp [t]) > Ω(dbest [t]) ∧ g   Ω(dp [t]) ≥ Ω(dj [t]), j 6= p , bbest g [t + 1] ← bp [t] best else bbest g [t + 1] ← bg [t] d. Evolve to a new swarm population B[t + 1], using (25); e. set t = t + 1. end b = bbest [G]; b b map b 3. b −→ d. g end − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −−

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dCD : CD output. P: Population size. G: number of swarm iterations. For each dp there is a bp associated.

multiuser detection problem [8] from the range: φ2 ∈ [2; 10] (while φ1 = 2), resulting in an intensificatio search for the best global position. A comparative study considering combined multi-fitnes functions particle swarm optimization (WOPSO) and standard PSO S/MIMO multicarrier CDMA M U D, both under single-objective decision approach, was carried out in [37]. In summary, simulation results have been shown the capabilities of both schemes to scape from local solutions, thanks to a balance between exploration and exploitation, resulting similar M U D performance for both approaches. Note that in (28), the (Q + 1)th cost function and the respective positions bbest,Q+1 [t], p and bbest,Q+1 [t] positions allow the exploration capability of the WO PSO SIMO, while g the others Q positions, bbest,q [t] and bbest,q [t], q = 1, . . . , Q bring additional exploitation p g capability. In order to balance these capabilities, we set φQ+1 = φQ+1 = 21 (exploration), 1 2 1 and φq1 = φq2 = 2Q , 1 ≥ q ≥ Q (exploitation). Indeed, for each iteration of the WOPSO SIMO algorithm, there is a set containing (Q + 1) best particle positions, bbest,q [t], where the firs Q positions are associated to the p Q end-points of visited Pareto front Fp∗ [43], and the (Q + 1)th one is the position in Fp∗

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PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

265

that maximizes Ω(bp ) in (20). In the same way, there is another set containing (Q + 1) best global positions, bbest,q [t], where the firs Q positions are Q end-points of Fg∗ , and the g (Q + 1)th one is the position in Fg∗ that maximizes Ω(dp ). However, in order to maintain the WOPSO optimization process as simple as possible7 , specially in scenarios where
the number of receive antennas (Q) increases or, mainly when both system loading K N and Q are large, just the Q + 1 best particles obtained evaluating directly (21) is considered to evolve for the next iteration. After the WOPSO iterations b = bbest,Q+1 terminate, the fina estimation vector is determined by the vector b [G], which g is associated with the particle position that maximizes Ω(dp ). The pseudo-code for the SIMO DS-CDMA strategy is described in Algorithm 2.

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4.

PSO-M U D Parameters Optimization

In this section, the PSO -M U D parameters optimization is carried out in order to improve the PSO algorithm complexity × performance trade-off, allowing fast convergence with relative high quality of the solutions found. A firs analysis of the PSO parameters gives raise to the following behaviors: ω is responsible for creating an inertia of the particles, inducing them to keep the movement towards the last directions of their velocities; φ1 aims to guide the particles to each individual best position, inserting diversificatio in the search; φ2 leads all particles towards the best global position, hence intensifying the search and reducing the convergence time; Vmax inserts perturbation limits in the movement of the particles, allowing more or less diversificatio in the algorithm. For the M U D application, the parameters of PSO can be optimized according to the considered system. The capacity of changing the parameters individually is an advantage, since the algorithm can be easily adjusted to suit problems requiring more or less diversifi cation and intensification Numerical results obtained from Monte-Carlo simulations show that PSO is sensitive to system conditions, and some particular concerns must be taken into account according to the DS-CDMA communication system and channel scenarios. Recent published works applying PSO to M U D usually assumes conventional values for PSO input parameters, such [44], or optimized values only for specif c system and channel scenarios, such [8] for fla Rayleigh channel, [10] for multipath and high-order modulation, and [45] for multicarrier CDMA systems as well. In the following, a survey on the optimization process, displaying the general behavior of PSO and considering various system and channel scenarios is exhibited. Furthermore, the PSO is optimized also for systems with high-order modulation. For all analysis, the initial velocity of all particles is null, i.e., v[0] = 0.

4.1.

Vmax Optimization

AWGN channels: Figure 4 indicates that the performance is unchangeable for Vmax ≥ 2. However, for a increasing loading, the adoption of Vmax = 3 results in convergence problems as well (nor shown here). Analyzing Table 1, the higher the number of users in the 7

Note that the algorithm has to be implemented in real time using digital signal processing platforms.

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Algorithm 2 WOPSO SIMO DS-CDMA

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b Input: dCD , P, G, ω, φ1 , φ2 , Vmax ; Output: d begin 1. initialize firs population: t = 0; vp [0] = 0: null initial velocity; e with (P − 1) randomly generated; B[0] = bCD ∪ B, best,q bp [0] = bp [0] and bbest,q [0] = bCD , ∀ q. g 2. while t ≤ G a. calculate Ω(dp [t]), ∀ bp [t] ∈ B[t], Eq. (14), linearly combining the LLF of all antenna according (20); b. update velocity vp [t], p = 1, . . . , P, Eq. (28) c. update best positions: for p = 1, . . . , P for q = 1, . . . , Q [t + 1] ← bp [t] [t]), then bbest,q if fq (dp [t]) > fq (dbest,q p p best,q else bp [t + 1] ← bbest,q [t]; p end best,Q+1 if fQ+1 (dp [t]) > fQ+1 (dbest [t + 1] ← bp [t]; p [t]), then bp best,Q+1 else bbest,Q+1 [t + 1] ← b [t]; p p end end for q = 1, . . . , Q   if ∃ dp [t], such that fq (dp [t]) > fq (dj [t]), ∀j 6= p ∧ h i fq (dp [t]) > fq (dbest,q [t]) , then bbest,q [t + 1] ← bp [t] g g else bbest,q [t + 1] ← bbest,q [t] g g end   if ∃ dp [t], such that fQ+1 (dp [t]) > fQ+1 (dj [t]), ∀j 6= p ∧   f (bp [t]) > f (bgbest,Q+1 [t]) , then bbest,Q+1 [t + 1] ← bp [t] g best,Q+1 else bbest,Q+1 [t + 1] ← b [t] g g end end d. new swarm population B[t + 1], Eq. (25); e. set t = t + 1. end b = bgbest,Q+1 [G]; b b map b 3. b −→ d. end − − − − − − − − − − − − − − − − − − − − − − − − − − −− dCD : CD output; P: swarm population size; G : number of swarm generations. For each dp there is a bp associated.

system, the bigger the average number of bit changes, which can result slow convergence. Hence, a good choice is Vmax = 4, corroborating with previous literature results [42]. Rayleigh channels. In [8], the best performance × complexity trade-off for BPSK PSOM U D algorithm was obtained setting Vmax = 4. Herein, Figure 5 shows that the convergence is already achieved for Vmax > 2 in a medium loading condition. However, to avoid lack of intensificatio for full loading, Vmax = 4 is adopted as a good alternative.

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Figure 4. Vmax optimization in AWGN channel with BPSK modulation. Average BER × Vmax for Eb /N0 = 7 dB, K = 15, φ1 = 6, φ2 = 1, ω = 1, and G = 40. −1

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Figure 5. Vmax optimization in fla Rayleigh channels with BPSK modulation. Average BER × Vmax for Eb /N0 = 22 dB, K = 15, φ1 = 2, φ2 = 10, ω = 1, and G = 40.

4.2.

φ1 and φ2 Optimization

AWGN channels: Under AWGN channels, two situations are verifie regarding φ1 : for high values of φ1 , the algorithm tends to converge slowly, but achieving better performance. Values lower than 3 results in a better convergence speed, but with a poor performance, as Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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can be seen in Figure 6. Moreover, the performance gap becomes evident with an increase in the number of users. Considering φ2 , Figures 6 and 7 show different convergence and performance behavior when φ2 increases over the interval [1.0; 10.0]. Figure 7.(a) shows that the performance of PSO-M U D degrades with the increasing number of users, being more evident for high values of φ2 . Instead of the large number of required iterations for convergence with φ2 = 1, Figure 7.(b), the performance improvement justifie the adoption of such value. Therefore, for the AWGN channels, a good choice for the acceleration coefficient seems to be φ1 = 6 and φ2 ∈ [1; 2]. CD φ1 = 2; φ2 = 2 φ1 = 2; φ2 = 10 φ1 = 4; φ2 = 4 −2

10

BERAvg

φ1 = 6; φ2 = 1 φ1 = 6; φ2 = 2 φ1 = 10; φ2 = 1 φ1 = 10; φ2 = 2

−3

10

SuB (BPSK)

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0

5

10

15

Iterations

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25

30

Figure 6. φ1 and φ2 optimization in AWGN channel with BPSK modulation, Eb /N0 = 7 dB, K = 15, Vmax = 4, and ω = 1. Rayleigh channels: Distinctly from AWGN case, for Rayleigh channels the performance improvement caused by φ1 increment is no longer evident, and its value can be reduced without performance losses, as can be seen in Figure 8. Therefore, a good choice seems to be φ1 = 2, achieving a reasonable convergence rate. Figure 9.(a) illustrates different convergence performances achieved with φ1 = 2 and φ2 ∈ [1; 15] for medium system loading and medium-high Eb /N0 . Even for high system loading, the PSO performance is quite similar for different values of φ2 , as observed in Figure 9 (b). Hence, considering the performance × complexity trade-off, a reasonable choice for φ2 under fla Rayleigh channels is φ2 = 10.

4.3.

ω Optimization

It is worth noting that relatively larger value for the inertial weight PSO input parameter, ω, is helpful for global optimum, and lesser influence by the best global and local positions, while a relatively smaller value for ω is helpful for course convergence, i.e., smaller inertial Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

PSO Assisted Multiuser Detection for DS-CDMA Communication Systems −1

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Iterations for convergence

CD

φ2 = 1

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φ2 = 2 φ2 = 3 −2

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φ2 = 5 φ2 = 10

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20 10 0 5

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K [users]

(a)

(b)

Figure 7. φ2 optimization in AWGN channel with BPSK modulation, Eb /N0 = 7 dB, φ1 = 6, and ω = 1; (a) Average BER × K; (b) iterations needed for achieving convergence as a function of the number of users. −1

10

CD

φ1 = 0.5 φ1 = 2.0 φ1 = 4.0

BERAvg

φ1 = 8.0 −2

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10

SuB (BPSK)

−3

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Figure 8. φ1 optimization in fla Rayleigh channels with BPSK modulation, Eb /N0 = 22 dB, K = 15, φ2 = 10, Vmax = 4, and ω = 1. weight encourages the local exploration [41, 46] as the particles are more attracted towards best bbest p [t] and bg [t]. AWGN channels: Figure 10 offers some insight on parameter ω optimization process under AWGN channels. It is observed that the adoption of high values of ω implies in fast convergence, but this means a lack of search diversity, and the algorithm can easily be trapped in some local optimum, whereas small values for ω result in slow convergence due to excessive bit changes. The optimized value is ω = 1, which achieves a better performance ×

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−1

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φ2 = 1

φ2 = 2

φ2 = 2 φ2 = 4

−2

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φ2 = 4

BERAvg

BERAvg

CD

−1

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φ2 = 7

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φ2 = 10

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φ2 = 15

SuB (BPSK)

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(a)

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Figure 9. φ2 optimization in fla Rayleigh channels with BPSK modulation, Vmax = 4, ω = 1, and φ1 = 2; (a) convergence performance with Eb /N0 = 22 dB and K = 15; (b) average BER × K with Eb /N0 = 20 dB, G = 30 iterations. complexity, as shows Figure 10. CD

ω = 0.50 −2

ω = 0.75

BERAvg

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10

ω = 1.00 ω = 1.25 ω = 2.50

−3

SuB (BPSK)

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Figure 10. ω optimization under AWGN channel with BPSK modulation, considering Eb /N0 = 7 dB, K = 15, φ1 = 6, φ2 = 2, and Vmax = 4. Flat channels: Similarly to AWGN case, Figure 11 shows the convergence of the PSO scheme for different values of ω. It is evident that the best performance × complexity trade-off is accomplished with ω = 1. Many research papers have been proposed new strategies for PSO principle in order to improve its performance and reduce its complexity, for instance, in [47] the authors have been discussed adaptive nonlinear inertia weight in order to improve PSO convergence.

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CD

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Figure 11. ω optimization under fla Rayleigh channels with BPSK modulation, Eb /N0 = 22 dB, K = 15, φ1 = 2, φ2 = 10 and Vmax = 4. However, the results obtained by PSO for both channels show that no further specialized strategy is necessary, since the conventional PSO works well to solve the M U D DS-CDMA problem in several practical scenarios.

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4.4.

φ1 and φ2 Optimization under High-order Modulation

The optimization process for systems with high-order modulation is quite similar compared to BPSK process. Only the case with Rayleigh channel is evaluated, since it represents a more realistic condition. Special attention is given for φ1 and φ2 optimization, which are different from values obtained with BPSK modulation. The optimization of the inertial weight and maximum velocity achieve analogous results, and here their optimized values, obtained by simulation, are ω = 1 and Vmax = 4 for both QPSK and 16-QAM modulations. Under fla Rayleigh channels, and with these ω and Vmax optimized values, the acceleration factors could be optimized for QPSK modulation (Figure 12) and 16-QAM (Figure 13). Again, under 16-QAM modulation, the PSO-M U D requires more intensification once the search becomes more complex due to each symbol to contain 4 bits. Figure 13 shows the convergence curves for different values of φ1 and φ2 . The performance gap is more evident with an increasing number of users and Eb /N0 . Analyzing this result, the chosen values are φ1 = 6 and φ2 = 1.

4.5.

Optimization for Systems with Diversity Exploration

The best range for the acceleration coefficient under resolvable multipath channels (L ≥ 2) for M U D SISO DS-CDMA problem beseems φ1 = 2 and φ2 ∈ [12; 15], as indicated by the simulation results shown on Figure 14. For medium system loading and signal-noise Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al. CD φ1 = 2; φ2 = 2 φ1 = 2; φ2 = 10 −1

10

φ1 = 4; φ2 = 4

SERAvg

φ1 = 6; φ2 = 1 φ1 = 6; φ2 = 2 φ1 = 10; φ2 = 1 φ1 = 10; φ2 = 2

SuB (QPSK)

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Figure 12. φ1 and φ2 optimization under fla Rayleigh channels for QPSK modulation, Eb /N0 = 22 dB, K = 15, ω = 1 and Vmax = 4.

0

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CD φ1 = 10; φ2 = 2 φ1 = 6; φ2 = 2 φ1 = 6; φ2 = 1

−1

φ1 = 2; φ2 = 2

SERAvg

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10

φ1 = 2; φ2 = 10

−2

10

SuB (16−QAM)

−3

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0

50

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Iterations

150

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Figure 13. φ1 and φ2 optimization under fla Rayleigh channels for 16-QAM modulation, Eb /N0 = 30 dB, K = 15, ω = 1 and Vmax = 4.

ratio (SNR) as well, Figure 14 indicates that the best values for acceleration coefficient are φ1 = 2 and φ2 = 15, allowing the combination of fast convergence and near-optimum performance achievement. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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−1

10

CD φ1 = 2; φ2 = 2

BERAvg

φ1 = 2; φ2 = 6 φ1 = 2; φ2 = 10 −2

10

φ1 = 2; φ2 = 15 φ1 = 6; φ2 = 2 φ1 = 6; φ2 = 10

−3

10

SuB (BPSK) 0

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30

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40

45

50

Figure 14. φ1 and φ2 optimization under Rayleigh channels with path diversity (L = D = 2) for BPSK modulation, Eb /N0 = 22 dB, K = 15, ω = 1, Vmax = 4.

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4.6.

Optimized Parameters for PSO-M U D

As mentioned previously, the optimized input parameters for PSO-M U D vary regarding the system and channel scenario conditions. Monte-Carlo simulations (MCS) exhibited in Section 5. adopt the values presented in Table 2 as the optimized input PSO parameters. System loading L range indicates the boundaries for K N which the input PSO parameters optimization were carried out. For system operation characterized by spatial diversity (Q > 1 receive antennas), the PSO-M U D behaviour, in terms of convergence speed and quality of solution, is very similar to that presented under multipath diversity.

Table 2. Optimized parameters for asynchronous PSO-M U D. Channel AWGN Flat Rayleigh Flat Rayleigh Flat Rayleigh Path Diversity

Modulation BPSK BPSK QPSK 16-QAM BPSK

L range [0.16; 1.00] [0.16; 1.00] [0.16; 1.00] [0.03; 0.50] [0.03; 0.50]

ω 1 1 1 1 1

φ1 6 2 4 6 2

φ2 1 10 4 1 15

Vmax 4 4 4 4 4

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5.

Numerical Results

In this section, the PSO multiuser detector performance is analyzed in terms of average bit- or symbol-error-rate (BERAvg or SERAvg ), via Monte-Carlo simulation (MCS). Several aspects are considered, such as high order modulation, perfect and imperfect channel estimation at the receiver side, NFR effect, and the impact of spatial and multipath diversity order. The results obtained are compared with theoretical single-user bound (SuB), according to Appendix A, since the OM U D computational complexity results prohibitive. Monte-Carlo Simulation setup is summarized in the Appendix B. The general system and channel parameters adopted in this chapter are described in the sequel. The analysis presented in the sequel is divided according to system and channel conditions as follows • AWGN channel with BPSK modulation; • fla Rayleigh and multipath channels with BPSK modulation, including space or time diversity; • fla Rayleigh channel with QPSK/16-QAM modulation and space diversity; In order to evaluate the impact of realistic limitations over a DS-CDMA system, four variety of simulation results are discussed in the subsequent analysis of the PSO-M U D: • Convergence curve as a function of number of iterations;

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• BER/SER performance as a function of Eb /N0 ; • BER/SER performance as a function of number of users K, i.e., robustness against system loading increasing (L = K/N ), and; • BER/SER performance as a function of the increasing in the amount of power interference, i.e., robustness against NFR.

5.1.

AWGN Channels

System conditions and PSO parameters for AWGN channel simulations are summarized in Table 3. Although DS-CDMA systems have been equipped with a power control scheme, the fast variations intrinsic to the wireless channel may result in different received power among the users. In such case, the detection of the weak users is affected and therefore their performance is degraded. Figure 15 shows that the PSO-M U D is robust to the NFR, keeping its performance and convergence rate almost constant, while the CD is strongly degraded under N F R > 0. Besides, once the DS-CDMA system is limited by multiple access interference and the CD does not eliminate this MAI, its performance does not reach the SuB of the channel and it is expected that the correspondent performance gap regards to SuB becomes bigger with the Eb /N0 increasing. Figure 16 shows this behavior for the CD, while the PSO-M U D performance remains close to the SuB performance, independently of the received Eb /N0 .

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Table 3. System conditions and PSO parameters for AWGN. Parameter

Adopted Values DS-CDMA System # Rx antennas Q=1 Spreading Sequences Random, N = 31 modulation BPSK # mobile users K ∈ [5, 30] Received SNR Eb /N0 ∈ [2; 8] dB PSO-M U D Parameters Population size, P Eq. (26) acceleration coefficient φ1 = 6; φ2 = 1 inertia weight ω=1 Maximal velocity Vmax = 4

10

−2

BERAvg

10

BERAvg

CD PSO SuB (BPSK)

−3

10

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0

10

20

Iterations (a)

30

40

−1

10

−2

10

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0

CD PSO SuB (BPSK)

10

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30

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(b)

Figure 15. Convergence performance of PSO-M U D in AWGN channel and BPSK modulation, for K = 15 and Eb /N0 = 7 dB, with (a) perfect power control; (b) interferes with N F R = 8dB. The performance exhibited in (b) considers only the weaker users.

When the number of users sharing the system increases, the multiple access interference becomes larger and the CD performance is strongly degraded, since it is not able to eliminate this MAI. Figure 17 shows the increasing MAI effect:  5 the CD performance is deteriorated for increasing system loading in the range L ∈ 31 ; 30 31 . Nevertheless, the PSO-M U D performance is kept close to SuB even under full system loading (L = K N ≈ 1), being capable of eliminating the MAI almost completely. In addition, results presented in [45] for multicarrier CDMA show that genetic algorithm and other heuristic M U Ds techniques are not able to completely reach the SuB performance too, which raises the possibility that OM U D may not approach the SuB performance under high-full system loading condition, causing a small performance loss for K > 20 users, as viewed in Figure 17. In summary, PSO-M U D optimization is still efficien even for high-full loading, accomplishing robustness against high level of multiple access interference under AWGN channels.

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7

8

Eb /N0 [dB]

Figure 16. Performance of PSO-M U D under AWGN channels for medium system loading (K = 15, N = 31) and BPSK modulation. −1

BERAvg

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10

−2

10

CD PSO SuB (BPSK)

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5

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25

30

K [users]

Figure 17. Performance of PSO in AWGN channel as a function of number of users, with Eb /N0 = 7dB.

5.2.

Rayleigh Channels

The adopted PSO-M U D parameters, as well as system and channel conditions employed in Monte Carlo simulations are summarized in Table 4. Non-line-of-sight propagation (Rayleigh) channels are adopted, including fla (L = 1) and selective channels with L = 2 Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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or L = 3 independent paths.

Table 4. System, channel and PSO-M U D parameters for fading channels performance analysis. Adopted Values DS-CDMA System # Rx antennas Q = 1, 2, 3 Spreading Sequences Random, N = 31 modulation BPSK, QPSK and 16−QAM # mobile users K ∈ [5; 31] Received SNR Eb /N0 ∈ [0; 30] dB PSO-M U D Parameters Population size, P Eq. (26) acceleration coefficient φ1 = 2, 6; φ2 = 1, 10 inertia weight ω=1 Maximal velocity Vmax = 4 Rayleigh Channel Channel state info. (CSI) perfectly known at Rx coefficien error estimates Number of paths L = 1, 2, 3

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Parameter

The results presented for AWGN show that the PSO-M U D performance is almost constant while varying the number of users, but it is expected that its speed of convergence become as slower as higher the number of users. Considering the PSO-M U D in a fla Rayleigh channel, Figure 18 exhibits this effect, showing firs the convergence for a system with K = 15 users (half system loading), 18.(a), and second for system with K = 31 users (full system loading), 18.(b). Note that there is a little loss in performance after convergence for L = 1, due to the PSO-M U D does not totally eliminate the MAI, and its convergence speed is slower, since the search becomes more intricate. However, the PSO-M U D behavior is homogenous, consistent and effective under increasing system loading scenarios. Likewise to the AWGN case, Figure 19 corroborates the effectiveness of the PSO-M U D in fla Rayleigh channels against the increasing multiple access interference. As can be seen, independently of the loading, PSO-M U D always approaches the SuB performance. It is evident that the conventional detector, based on the matched filte , is very sensitive to the number of users, caused by the direct relation of increasing number of users and MAI. Alternatively to NFR analysis carried out for AWGN channel using convergence curves for PSO-M U D, Figure 20 presents the performance as a function of received Eb /N0 for two different near-far ratio scenarios under fla Rayleigh channel. Figure 20.(a) was obtained for perfect power control, whereas Figure 20.(b) was generated considering half users with N F R = +6 dB. Again, the BERAvg performance is calculated only for the weaker users. Note the performance of the PSO-M U D is almost constant despite of the N F R = +6 dB for half of the users, illustrating the robustness of the PSO-M U D against unbalanced powers in fla fading channels.

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−1

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Figure 18. Convergence performance of PSO-M U D with BPSK modulation, fla Rayleigh channels and moderate signal-to-noise ratio (Eb /N0 = 12 dB). (a) medium system loading (K = 15); and (b) full system loading (K = 31).

−1

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BERAvg

10

CD PSO SuB (BPSK)

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Figure 19. Performance of PSO-M U D under fla Rayleigh channels for different system loading, with Eb /N0 = 25 dB and BPSK modulation. 5.2.1.

Path Diversity

DS-CDMA systems are able to explore the path diversity, since the spread of the signal may create distinguishable copies of transmitted signal at the receiver, with distinct delays and channel coefficient [24]. This can be explored, representing a diversity gain and improving the system capacity. Figure 21 shows the BERAvg convergence of PSO-M U D for different of paths, L = 1 2 and 3, when the detector explores fully the path diversity, i.e., the number of finger of conventional detector is equal the number of copies of signal received,

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

PSO Assisted Multiuser Detection for DS-CDMA Communication Systems −1

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Figure 20. Average BERAvg × Eb /N0 for fla Rayleigh channel with K = 15: (a) perfect power control; (b) N F R = +6 dB for 7 users. In scenario (b), the performance is calculated only for the weaker users. D = L. The power delay profil considered is exponential, with mean paths energy as shown in Table 5 [24]. It is worth mentioning that the mean received energy is equal for the three conditions, i.e., the resultant improvement with increasing number of paths is due the diversity gain only.

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Table 5. Three power-delay profiles for different Rayleigh fading channels used in Monte-Carlo simulations. Param. Path, ℓ τℓ E[γℓ2 ]

PD -1

1 0 1.0000

PD -2

1 0 0.8320

2 Tc 0.1680

PD -3

1 0 0.8047

2 Tc 0.1625

3 2Tc 0.0328

Note there is a performance gain with the exploration of such diversity, verifie in both the Rake receiver and PSO-M U D. Nevertheless, the Rake is still affected by MAI, Equation (7), and now by the self interference, Equation (6), since other paths interferes in the detection of signals. The PSO-M U D performance is close to SuB in all cases, exhibiting its capability of exploring path diversity and dealing with SI as well. In addition, the convergence aspects are kept for all conditions. The PSO-M U D single-objective function, Equation (13), takes into account the channel coefficien estimation of each user, and imperfect channel estimation degrades its performance. In order to raise a quantitative characterization of such aspect, the PSO-M U D is evaluate under channel error estimation, which are modelled through the continuous uniform distributions U [1 ± ǫ] centralized on the true values of the coefficients resulting (i)

(i)

γk,ℓ = U [1 ± ǫγ ] × γk,ℓ ;

(i) (i) θbk,ℓ = U [1 ± ǫθ ] × θk,ℓ ,

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(28)

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where ǫγ and ǫθ are the maximum module and phase normalized errors for the channel coefficients respectively. −1

10

CD PSO SuB (BPSK) −2

BERAvg

10

L=1 L=2 L=3 −3

10

0

5

10

15

Iterations

20

25

30

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Figure 21. Convergence performance of PSO-M U D under asynchronous Rayleigh channel for BPSK modulation, multipath slow Rayleigh channels, K = 15, Eb /N0 = 15 dB and I = 3, for L = 1, 2 and 3 paths. For a low-moderate SNR and medium system loading (L = 15/31), Figure 22 shows the performance degradation of the PSO-M U D considering BPSK modulation, fla and selective (L = 2 paths) Rayleigh channels with estimation errors of order of 10% or 25%, i.e., ǫγ = ǫθ = 0.10 or ǫγ = ǫθ = 0.25, respectively. It is observed that when channel estimation presents errors, the performance is degraded, and the higher the SNR, more evident is the gap between PSO-M U D performance and SuB. However the PSO-M U D is still much better than CD in any situation, and one can see that if the estimation error is constrained by ǫγ,θ < 0.1, a reasonable performance can be achieved. As expected from previous results for fla Rayleigh channel, Figure 23 shows the performance degradation CD for L = 1 and L = 2 as function of number of users K, and the robustness of PSO-M U D against the increasing loading. It is evident that the PSO-M U D has a superior performance than CD, for all evaluated loading, being more considerable for full loading and L = 2 paths. As can be seen, independently of the loading and number of paths, PSO-M U D always accomplishes a much better performance. 5.2.2.

Spatial Diversity

Spatial diversity is an approach which aims to exploit the different correlation between received signals at different locations, so as to avoid deep fading in the received signal. In the results presented here, two assumptions are considered when there are more than one antenna at receiver: first the average received power is equal for all antennas; and second,

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−1

10

−2

BERAvg

10

CD PSO - ǫγ = ǫθ = 0.00

−3

10

PSO - ǫγ = ǫθ = 0.10 PSO - ǫγ = ǫθ = 0.25

L=1 L=2

SuB (BPSK) −4

10

0

2

4

6

8

10

12

14

16

18

20

Eb /N0 [dB]

Figure 22. Performance of PSO-M U D with BPSK modulation, fla and multipath Rayleigh (D = L = 2) channels with error in the channel estimation.

−1

BERAvg

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10

CD PSO SuB (BPSK) L=1 L=2

−2

10

−3

10

5

10

15

20

25

30

K [users]

Figure 23. PSO-M U D performance as a function of number of users. Rayleigh channels with L = D = 1 and 2, BPSK modulation, and Eb /N0 = 15 dB are considered. the SNR at the receiver input is define as the received SNR per antenna. Therefore, there is a power gain of 3 dB when adopted Q = 2, 6 dB with Q = 3, and so on. Although the objective function calculation becomes more expensive for a larger number of antennas, one can expect that the convergence of PSO-M U D be similar, once the number of detected symbols are equal. The effect of increasing the number of receive

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antennas in the convergence curves is shown in Figure 24, where PSO-M U D works on systems with Q = 1, 2 and 3 antennas. Note that there is an improvement in the performance, given by the diversity and SNR gains, for both CD and PSO-M U D. A delay in the PSOM U D convergence is observed when more antennas are added to the receiver, caused by the larger gap that it has to surpass. Furthermore, PSO-M U D achieves the SuB performance for all the three cases. −1

10

−2

BERAvg

10

Q=1 Q=2 Q=3

−3

10

CD PSO SuB (BPSK)

−4

10

−5

10

0

5

10

15

Iterations

20

25

30

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Figure 24. PSO-M U D Convergence in fla Rayleigh channels. BPSK modulation, Eb /N0 = 12 dB, K = 15, and Q = 1, 2, 3 antennas. Regarding the effect of channel estimation errors in the performance of the PSO-M U D, Figure 25 shows this effect under similar conditions adopted in Figure 22, but now with spatial diversity, i.e., Q = 1, 2 antennas. Note that PSO-M U D reaches the SuB in both conditions with perfect channel estimation, and the improvement is more evident when the diversity gain increases. Note that, in this case, the diversity gain is higher when compared to the multipath case in Figure 22, since the average energy is equally distributed among antennas, while for path diversity is considered a realistic exponential power-delay profile Moreover, there is a SNR gain of +3 dB in the Eb /N0 for each additional receive antenna. Although there is a general performance degradation when the error in channel coefficien estimation increases, PSO-M U D still achieves much better performance than the CD under any error estimation condition, being more evident for larger number of antennas. If the number of users sharing the system increases, the MAI substantially increases too, and the performance of single-user detectors, such as conventional detector (CD), is degraded accordingly. Nevertheless, Figure 26 shows that PSO-M U D performance is robust to the system loading, and increases accordingly with the number of receive antennas, while the CD is very sensitive, being the performance gap between them more evident for Q = 2 and high loading. Again, the performance difference to Figure 23 is explained by the higher diversity gain (average energy evenly distributed among receive antennas) and the SNR gain, once each antenna has a unitary average received energy.

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Q=1 Q=2

−1

10

−2

BERAvg

10

−3

10

CD PSO : ǫγ = ǫθ = 0.00 −4

10

PSO : ǫγ = ǫθ = 0.10 PSO : ǫγ = ǫθ = 0.25 SuB (BPSK)

−5

10

0

5

10

15

Eb /N0 [dB]

20

25

30

Figure 25. Performance of PSO-M U D under fla Rayleigh channels with BPSK modulation, K = 15 and Q = 1 and 2.

−1

10

Q=1 Q=2

BERAvg

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−2

10

CD PSO SuB (BPSK)

−3

10

−4

10

5

10

15

20

25

30

K [users]

Figure 26. PSO-M U D performance × number of users K, in fla Rayleigh channel, BPSK modulation, Eb /N0 = 15dB, and spatial diversity. 5.2.3.

High-order Modulation

Regarding the new generations of wireless systems, beyond spatial and path diversity, the wireless systems explore high-order modulation formats in order to increase spectral efficien y. Indeed, Figure 27 indicates that PSO-M U D could achieve suitable convergence Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

under several modulation formats and antenna diversity. This Figure exhibits convergence for three different modulations: (a) BPSK, (b) QPSK, and (c) 16-QAM. Note that, the PSO-M U D convergence behavior is quite similar, reaching the SuB in a number of iterations proportional to the order modulation i.e., proportional to the size of search universe. It is worth mentioning, as presented in Table 2, that the PSO-M U D optimized parameters is specifi for each order modulation. −1

10

−2

−4

0

−2

10

CD PSO SuB (BPSK)

Q=1 Q=2

−5

10

10 Avg

−3

10 10

−1

SER

BERAvg

10

10

20

40

Iterations

60

80

−3

10

−4

10

−5

CD PSO SuB (QPSK)

Q=1 Q=2

0

20

40

Iterations

(a)

60

80

(b)

−1

SERAvg

10

−2

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10

−3

10

CD PSO SuB (16−QAM)

Q=1 Q=2

−4

10

0

20

40

Iterations

60

80

(c)

Figure 27. Convergence of PSO-M U D under fla Rayleigh channel, Eb /N0 = 20 dB, and (a) K = 24 users with BPSK modulation, (b) K = 12 users with QPSK modulation and (b) K = 6 users with 16-QAM modulation When a QPSK or higher-order modulation formats are considered, such as M -QAM or M -PSK with M ≥ 8, the multiuser detection becomes more complex, since each symbol carries m = log2 M bits. This implies that, if there are K users to be jointly detected, PSOM U D must deal with m2K bits if a decoupled representation and optimization procedure with only real value variables is adopted, Equation (14), since the PSO algorithm employed here has a binary representation. Hence, this may affect the performance of PSO-M U D. Nevertheless, varying Eb /N0 , no performance loss was observed, as shown in Figure 28, for Q = 1 and 2 receive antennas and medium system loading. In both receive antenna

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cases, the achieved performance of PSO-M U D is close to the QPSK SuB. On the other hand, the CD performance is severely affected with order modulation increasing, resulting in poorer performance regarding the BPSK modulation case of Figure 25.

−1

10

SERAvg

Q=1 Q=2 −2

10

−3

10

CD PSO SuB (QPSK)

−4

10

0

5

10

15

20

25

30

Eb /N0 [dB]

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Figure 28. Average SER × Eb /N0 for QPSK modulation, K = 15, Q = 1, 2 and fla Rayleigh channel. Systems with higher-order modulation tends to be more and more prejudiced by multiple access interference, since the amplitude carries more information. As expected, for 16-QAM modulation, Figure 29 shows the poorer performance of CD regarding the QPSK and BPSK modulation cases. This increasing degradation is mainly caused by distance reduction of the symbols in the constellation when order modulation increases, and by the MAI. However, even so, the performance of the PSO-M U D under medium system loading 15 ) reaches the SuB for 16-QAM, in both conditions, with Q = 1 and 2, for all (L = 31 evaluated SNR range. Finally, when the number of users increases, Figure 30 shows that for 16-QAM modulation and fla Rayleigh channels, the PSO-M U D performance degradation with φ1 = 6, φ2 = 1 is quite slight in the range (0 < L ≤ 0.5), and always much better than CD for all system loading values. The increment in performance degradation when L > 0.5 could be explained by the fact that further input parameter optimization (mainly over φ1 and φ2 ) must be accomplished in order to achieve complete convergence under P population size and G swarm iterations.

6.

Complexity Analysis

The analysis of the computational complexity is essential, since the M U D application requires real-time processing and the time requirement is hard. The OM U D complexity, as previously mentioned, grows exponentially with the number of users. Heuristic detectors, Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

SERAvg

286

10

−1

10

−2

10

−3

Q=1 Q=2

10

−4

0

CD PSO SuB (16−QAM) 5

10

15

Eb /N0 [dB]

20

25

30

Figure 29. Average SER × Eb /N0 for 16-QAM in fla Rayleigh channel with K = 15, Q = 1, 2. 0

10

CD PSO SuB (16−QAM) −1

SERAvg

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10

−2

10

−3

10

5

10

15

K [users]

20

25

Figure 30. PSO-M U D and CD performance degradation × system loading under 16-QAM modulation and fla Rayleigh channel.

such PSO-M U D related here, work in the sense of diminishing this complexity by leading the search to suitable regions of the search space avoiding useless evaluations, achieving polynomial complexity. In the sequel, an analytical and a numerical analysis is carried out, considering multiplications, additions and random number generations.

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6.1.

287

Analytical Complexity

In the sense of fully explore the complexity of M U D detectors, an analysis of the loglikelihood function is presented, followed by its implication on OM U D and PSO-M U D. The LLF considered here needs the correlation matrix, which is calculated in Equation (18). Since this calculation is necessary for both M U Ds, it is omitted here. The remain operations are detailed in the following.

6.1.1.

OM U D Complexity

The number of operations depends on the number of users K, number of symbols I, modulation order m, number of receive antennas Q and number of paths; and the PSO-M U D performance also depends on the population size P and a priori iterations G for convergence. As mentioned previously, for the OM U D, the number of operations increases exponentially with the number of users. For a generic modulation order m, the entire search space is 2mKI . It is worth mentioning that in the mathematical formulation of LLF, there are null products that can be avoided when considering a DSP implementation, simplifying the evaluation of each candidate-vector. Hence, once one cost function calculation demands [4 · (2KID)2 + 5 · (2KID)] · Q products, and [(2KID − 1) · (4KID + 1) + 1] · Q additions, and the total complexity of OM U D can be acquired multiplying these values by the size of the search universe of candidate-vectors, which results in

Mult. : (2mKI ) · [4 · (2KID)2 + 5 · (2KID)] · Q,

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Add. : (2mKI ) · [(2KID − 1) · (4KID + 1) + 1] · Q. Nevertheless, according to [11], the cost function calculation can be simplifie through a recursive approach when exists a deterministic behavior in the search. Therefore, after the firs cost function calculation, a look-up table is used, and the computational complexity of OM U D becomes Mult. : (2mKI − 1) · (3 + 4KI) · Q + [4 · (2KID)2 + 5 · (2KID)] · Q, Add. : (2mKI − 1) · [(2KI + 4) · Q − 1] + [(2KID − 1) · (4KID + 1) + 1] · Q, which is much smaller than the previous one and accomplishes identical performance.

6.1.2.

PSO-M U D Complexity

In the PSO case, the cost function calculation requires the same number of operation. In addition, PSO evolving is random and simplification from deterministic aspects presented in OM U D are not applicable here. Further operations are required for PSO principle, in order to population evolving, essentially in the velocity calculation. Considering an a priori number of generations G and a population size P, the total number of operations required

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al.

by the PSO is  Mult. : GP · 4mKI + Q · [2 · (2KID)2 + 2KID] +  +Q · 2 · (2KID)2 + 4 · (2KID) + P · [2 · (2KID)2 + 2KID] ,

Add. : G · {5mKIP + QP · [(2KID − 1)(4KID + 1) + 1] + P · (Q − 1)} + +PQ · [(2KID − 1).(4KID + 1) + 1],

RNG : G · (3mKIP) + 2KI · (P − 1), with RNG representing random number generations. Note the computational complexity of the search, which is exponential for the OM U D, becomes polynomial for the PSO-M U D.

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6.2.

Numerical Complexity

From the number of operations presented in the previous subsection, one can conclude that the analytical complexity of OM U D and PSO-M U D differs mainly by the simplification of the deterministic search and by the number of evaluated candidate-vectors. The optimum detector evaluates all possible combinations of symbols for all users, implying in the exponential term 2mKI ; however, for each candidate-vector evaluation, simplification due to its deterministic search are applied. The PSO-M U D has a polynomial number of cost function calculation, which depends directly on the population size and the number of a priori iterations, but its cost function calculation can not be simplified It is expected that the number of operations to be smaller for OM U D for simple and sometimes non-practical detection cases, i.e., with synchronous transmissions, few users and low modulation order, and in absence of diversity exploitation. For more elaborated and realistic DS-CDMA systems, the PSO becomes much simpler regarding the OM U D complexity. In the sequel, a numerical analysis presents the number of operations for different system conditions. 6.2.1.

AWGN Synchronous Channel

This analysis consists of the simplest case studied in this chapter. The communication channel is only AWGN, and it has neither fading, nor spatial nor path diversities. Moreover, the uplink communication is synchronous, and a considerable simplificatio is achieved when the LLF presents I = 1. Table 6 shows the number of multiplication, additions and random number generation as a function of Eb /N0 . It is possible to see the increment in the number of operations when the Eb /N0 increases, because the number of iterations for convergence increases when the Eb /N0 increases. A comparison between M U Ds shows that, even in a simple channel and system scenarios, the PSO-M U D has a comparable complexity with OM U D, since the search universe of optimization is small in these scenarios. Moreover, it is worth mentioning that PSO-M U D has approximately the same number of multiplications and additions, while for OM U D the number of multiplications is twice the number of additions.

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Table 6. Number of operations for system in AWGN synchronous channel for a varying Eb /N0 , with K = 15 users and BPSK modulation. Eb /N0 [dB] Mult. [×105 ] Add. [×105 ] RNG.[×104 ]

6.2.2.

2 9.07 8.83 2.11

3 9.64 9.39 2.25

4 10.77 10.49 2.52

PSO-M U D 5 6 11.34 12.48 11.05 12.15 2.65 2.92

7 14.74 14.37 3.46

8 18.71 18.24 4.41

OM U D 20.68 10.83 0

Flat Rayleigh Channel

When a more realistic fla Rayleigh channel is considered, the convergence becomes faster than in AWGN channel, and the complexity reduction over OM U D is more evident when the number of users increases. With K = 15 users, Table 7 shows the number of operations when the Eb /N0 is varying, similarly to AWGN case, with BPSK modulation, Q = 1 antenna and D = 1 path.

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Table 7. Number of operations for system under flat Rayleigh channel, varying Eb /N0 , K = 15 users, and BPSK modulation. Eb /N0 [dB] Mult. [×105 ] Add. [×105 ] RNG.[×104 ]

0 4.54 4.41 1.03

5 6.81 6.62 1.57

PSO-M U D 10 15 20 7.94 9.64 9.07 7.73 9.39 8.83 1.84 2.25 2.11

25 10.21 9.94 2.38

30 10.77 10.49 2.52

OM U D 20.68 10.83 0

As expected, from simulations results, the total number of operations is hardly affected by the number of users, even for a simple system scenario with no spatial and no path diversities. It is important highlight that for K > 15 the computational complexity of OM U D is much higher than PSO-M U D, being prohibitive for real-time implementation. Table 8 shows the number of operations when the number of users is varying, with Eb /N0 = 20 dB, BPSK modulation, Q = 1 antenna and D = 1 path. 6.2.3.

Path Diversity

The exploration of path diversity is used in asynchronous systems under selective fading channels, which implies in increasing the receiver complexity, since replicas of received signal and symbols are processed and evaluated in parallel. Table 9 shows the number of operations when K is varying, with Eb /N0 = 20 dB, BPSK modulation, Q = 1 antenna and D = 2 paths. It is observed when comparing Tables 8 and 9 that the number of operations increases drastically due the diversity exploitation and asynchronous system. Although the number of symbols is three times for the asynchronous case (I = 3), the

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290

Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al. Table 8. Number of operations for system under flat Rayleigh channels for K varying, with Eb /N0 = 20 dB and BPSK modulation. K [users] Mult. [×105 ] Add. [×105 ] RNG.[×104 ]

5 0.46 0.43 0.29

10 2.58 2.48 0.88

K [users] Mult. [×105 ] Add. [×105 ]

5 0.01 0.006

10 0.46 0.24

PSO-M U D 15 20 13.04 25.91 12.71 25.40 3.06 4.62 OM U D OM U D 15 20 20.68 870.38 10.83 450.92

25 54.10 53.25 7.80

25 34561.17 17783.90

complexity increment is much higher. In the OM U D case, the number of operations is also much larger, specially for a high number of users.

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Table 9. Number of operations for system in multipath Rayleigh channel for K varying, with Eb /N0 = 20 dB and BPSK modulation. K [users] Mult. [×108 ] Add. [×108 ] RNG.[×105 ]

5 0.03 0.03 0.17

10 0.19 0.18 0.56

K [users] Mult. [×108 ] Add. [×108 ]

5 0.02 0.01

10 1.32 · 103 6.77 · 102

6.2.4.

PSO-M U D 15 20 0.57 1.91 0.57 1.90 1.17 2.94 OM U D 15 20 7 6.44 · 10 2.80 · 1012 7 3.27 · 10 1.42 · 1012

25 3.97 3.78 4.69

30 8.27 8.25 8.53

25 1.14 · 1017 5.78 · 1016

30 4.49 · 1021 2.26 · 1021

Spatial Diversity

Different from path diversity exploitation, the spatial diversity in the M U D makes the number of operations grows almost linearly with the number of receive antennas. Table 10 shows the number of operations for different number of receive antennas Q = 1, 2 and 3, with K = 15 and Eb /N0 = 12 dB.

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Table 10. Number of operations as a function of increasing receive antennas. System under flat Rayleigh channel, Eb /N0 = 12 dB, K = 15 and BPSK modulation. Q [antennas] Mult. [×106 ] Add. [×106 ] RNG.[×104 ] Mult. [×106 ] Add. [×106 ]

6.2.5.

1

2 3 PSO-M U D 0.85 1.90 3.50 0.83 1.84 3.39 1.98 2.25 2.79 OM U D 2.07 4.14 6.20 1.08 2.20 3.31

Modulation Order

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When a high-order modulation is considered, the number of operations become meaningful, since each symbol detection involves mI bits per user per path8 . As expected, the behavior observed here is that the complexity grows significantl with the modulation order. The PSO-M U D in systems with QPSK and 16-QAM modulation still presents a manageable complexity, while the OM U D results in prohibitive implementation for any of these modulations formats under realistic system and channel scenarios. Table 11 shows the number of operations when the Eb /N0 is varying, with K = 15 dB, QPSK modulation, Q = 1 antenna and D = 1 finge . Table 11. Number of operations for a system under flat Rayleigh channel, Eb /N0 varying, K = 15 and QPSK modulation. Eb /N0 [dB] Mult. [×106 ] Add. [×106 ] RNG.[×105 ]

0 1.32 1.30 0.59

5 2.03 1.99 0.91

PSO-M U D 10 15 20 2.65 2.88 3.11 2.61 2.84 3.07 1.20 1.31 1.42

25 3.43 3.37 1.56

30 4.13 4.06 1.88

OM U D 6.76 · 104 3.54 · 104 -

Concluding the numerical analysis of computational complexity, the number of operations for a M U D in a 16-QAM modulation system is exhibited in Table 12. Note the huge difference between the PSO-M U D and OM U D. It is worth mentioning that convergence of PSO-M U D becomes larger with the number of users for 16-QAM, which results in a large number of operations. 8

Or 2mI bits per user per path, if considering to avoid handling complex-valued decision variables, as described by (15) Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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Table 12. Number of operations for K varying. System under flat Rayleigh channel, Eb /N0 = 30 dB and 16-QAM modulation.

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7.

K [users] Mult. [×107 ] Add. [×107 ] RNG.[×106 ]

5 0.04 0.04 0.09

10 0.63 0.63 0.77

K [users] Mult. [×105 ] Add. [×105 ]

5 0.24 0.13

10 4.73 · 105 2.53 · 105

PSO-M U D 15 20 2.08 7.28 2.08 7.28 1.80 4.90 OM U D 15 20 11 7.26 · 10 1.00 · 1018 11 3.80 · 10 5.20 · 1017

25 19.65 19.64 10.80 25 1.31 · 1024 6.72 · 1023

Conclusion

Under multipath channels, low and high-order modulations formats and antenna deversity, the PSO algorithm shows to be efficien for SISO/SIMO M U D asynchronous DS-CDMA problem, even under the increasing of number of multipath, system loading (MAI), NFR and/or SNR. Under a variety of analysed realistic scenarios, the performance achieved by PSO-M U D always was near-optimal but with much lower complexity when compared to the OM U D. Errors in channel estimation deteriorate the PSO-M U D the performance, but this is inherent to the cost function calculation. Despite this drawback, heuristic M U D still remain superior with regard to conventional receiver, even with module and phase channel coeffi cient errors about 25% the PSO-M U D keeps much more efficien than conventional receiver with perfect channel estimation. In all evaluated system conditions, PSO-M U D resulted in small degradation performance if those errors are confine to 10% of the actual instantaneous values. Especially, for a system under medium loading, a marginal performance degradation was observed when ǫγ,θ < 0.05. Under realistic and practical systems conditions, the PSO-M U D results in much less computational complexity than OM U D. The PSO-M U D, when compared to the CD receiver, is able to reach a much better performance with a tangible and affordable computational complexity. This manageable complexity, of course, depends on the hardware resources available at the receiver side (feasible at base-station), and the system requirements such as minimum throughput, maximum admitted symbol-error-rate, and so on. Besides, the PSO-M U D DS-CDMA accomplishes a fl xible performance × complexity trade-off solutions, showing to be appropriate for low and high-order modulation formats, (a)synchronous, as well as AWGN, multipath and spatial diversity scenarios. The increasing system loading slightly deteriorates the performance of the PSO-M U D, but only under higher-order modulation. In all other scenarios, PSO-M U D presents complete robustness against MAI increasing. Finally, the complexity reduction in relation to OM U D is huge. Particularly when compared with conventional receiver the complexity increment is manageable at base-station,

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allowing the PSO-M U D implementation in the third and fourth generations of mobile communication systems.

Appendix A. Minimal Number of Trials and Single-User Performance The minimal number of trials (T R) evaluated in the each simulated point (SNR) was obtained based on the single-user bound (SuB) performance. Considering a confidenc interval, and admitting that a non-spreading and a spreading systems have the same equivalent bandwidth (BW ≈ T1s = BWspread ≈ TNc ), and thus, equivalently, both systems have the same channel response (delay spread, diversity order and so on), the SuB performance in both systems will be equivalent. So, the average symbol error rate for a single-user under M -QAM DS-CDMA system and L Rayleigh fading path channels with exponential power-delay profil and maximum ratio combining reception is found in [48, Eq. (9.26)] as SERSuB = 2α

L X

(29)

pℓ (1 − βℓ ) +

ℓ=1

α2 where: 

pℓ = 

"

  X L L 4X 1 −1 pℓ βℓ × tan pℓ − π βℓ

L Y

ℓ=1

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k=1,k6=ℓ

βℓ =

s



ℓ=1

  −1 νk  , 1− νℓ

ν ℓ gQAM , 1 + ν ℓ gQAM

gQAM =

#

  1 √ α= 1− , M 3 , 2(M − 1)

and

ν ℓ = ν ∗ℓ log2 M = mν ∗ℓ denotes the average received signal-noise ratio per symbol for the ℓth path, with ν ∗ℓ being the correspondent SNR per bit per path. Once the lower bound is defined the minimal number of trials can be define as nerrors , TR = SERSuB where the higher nerrors value, the more reliable will be the estimate of the SER obtained in MCS [49]. In this work, the minimum adopted nerrors = 100, and considering a reliable d ⊂ [0.823; 1.215] SER. Simulations interval of 95%, it is assured that the estimate SER were carried out using MATLAB v.7.3 plataform, The MathWorks, Inc.

B. Monte Carlo Simulation Setup A simplifie diagram of the adopted Monte Carlo simulation setup is shown in Figure 31. In the random data generator block, the transmitted data were randomly generated and all information symbols had equal probability of being selected. The transmitter block implements Equation (1), with the set of input variable (A, I, K, N, M and spreading sequence type) adjusted according to the choices stated in Section 5..

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Taufi Abr˜ao, Leonardo Dagui de Oliveira, Bruno Augusto Ang´elico et al. Random Data Generation

Begin

Random Channel

Transmitter

trial = trial + 1 Y

Finish

N

trial < TR

N

Errors?

Conventional Detector (CD)

PSO-MuD

Y

TR =

# errors SuB

SER =

erPSO TR ⋅ K ⋅ I

Increment

erPSO

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Figure 31. Monte Carlo simulation diagram.

Figure 32. GUI for M U D DS-CDMA with PSO. The channel block adds other stochastic characteristic to the model. Here the complex additive white Gaussian noise, with bilateral power spectral density equal to N0 /2, corrupts the received signal of all users. The power-delay profil for the Rayleigh fading channel can L P be adjusted by each user, admitting random delay and normalized power: E[γℓ2 ] = 1. ℓ=1

L ¯ = PE ¯ℓ , where E ¯ℓ = SNR · Then, the average SNR at the receiver input is given by: E ℓ=1

E[γℓ2 ]. For the exponential profil with 2 paths adopted in Section 5., E[γ12 ] = 0.8320 and E[γ22 ] = 0.1680 were assumed, with the respective delays uniformly distributed on the interval τk,ℓ ∈ [0; N − 1]. The symbols estimated in the conventional receiver stage are used as start point for the

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PSO Assisted Multiuser Detection for DS-CDMA Communication Systems

295

heuristic M U Ds. R . In order to better The numerical results are obtained using the software Matlab organize the simulation scenarios, a graphical user interface (GUI) was created, as shown in Figure 32.

References [1] S. Moshavi. Multi-user detection for ds-cdma communications. IEEE Communication Magazine, 34:132–136, Oct. 1996. [2] S. Verd´u. Multiuser Detection. Cambridge University Press, New York, 1998. [3] P. H. Tan and L. K. Rasmussen. The application of semidefinit programming for detection in cdma. IEEE Journal on Selected Areas in Communication, 19(8):1442– 1449, Aug. 2001. [4] T. Abr˜ao, F. Ciriaco, and P. J. E. Jeszensky. Evolutionary programming with cloning and adaptive cost function applied to multi-user ds-cdma systems. IEEE International Symposium on Spread Spectrum Techniques and Applications (ISSSTA 04), August 2004. [5] C. Erg¨un and K. Hacioglu. Multiuser detection using a genetic algorithm in cdma communications systems. IEEE Transactions on Communications, 48:1374–1382, 2000.

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[6] F. Ciriaco, T. Abr˜ao, and P. J. E. Jeszensky. Ds/cdma multiuser detection with evolutionary algorithms. Journal Of Universal Computer Science, 12(4):450–480, 2006. [7] A.A. Khan, S. Bashir, M. Naeem, and S.I. Shah. Heuristics assisted detection in high speed wireless communication systems. In IEEE Multitopic Conference, pages 1–5, Dec. 2006. [8] L. D. Oliveira, F. Ciriaco, T. Abr˜ao, and P. J. E. Jeszensky. Particle swarm and quantum particle swarm optimization applied to ds/cdma multiuser detection in fla rayleigh channels. In ISSSTA’06 - IEEE International Symposium on Spread Spectrum Techniques and Applications, pages 133–137, Manaus, Brazil, 2006. [9] H. Zhao, H. Long, and W. Wang. Pso selection of surviving nodes in qrm detection for mimo systems. In GLOBECOM - IEEE Global Telecommunications Conference, pages 1–5, Nov. 2006. [10] L.D. Oliveira, T. Abrao, P.J.E. Jeszensky, and F. Casadevall. Particle swarm optimization assisted multiuser detector for m-qam ds/cdma systems. In SIS’08 - IEEE Swarm Intelligence Symposium, pages 1–8, Sept. 2008. [11] L. de Oliveira, F.Ciriaco, T. Abrao, and P. Jeszensky. Local search multiuser detection. AEU¨ International Journal of Electronics and Communications, 63(4):259–270, April 2009. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[12] L. D. Oliveira, F. Ciriaco, T. Abr˜ao, and P. J. E. Jeszensky. Simplifie local search algorithm for multiuser detection in multipath rayleigh channels. In The 16th IST Mobile and Wireless Communications Summit, page 5 pp., Budapest, Hungary, July 2007. [13] H. S. Lim and B. Venkatesh. An efficien local search heuristics for asynchronous multiuser detection. IEEE Communications Letters, 7(6):299–301, June 2003. [14] E. H. L. Aarts and J. K. Lenstra. Local Search in Combinatorial Optimization. Princeton University Press, USA, 2003. 536pp. [15] W.-K. Ma, T. N. Davidson, K. M. Wong, Z.-Q. Luo, and P. C. Ching. Quasi-maximumlikelihood multiuser detection using semi-definit relaxation with applications to synchronous cdma. IEEE Trans. on Signal Processing, 50(4):912–922, Apr. 2002. [16] X. M. Wang, W.-S. Lu, and A. Antoniou. A near-optimal multiuserdetector for dscdma using semidefinit programming relaxation. IEEE Transactions on Signal Processing, 51(9):2446–2450, Sept. 2003. [17] A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz). Semidefinit relaxation for detection of 16-qam signaling in mimo channels. IEEE Signal Processing Letters, 12(9):653–656, Sept. 2005.

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[18] X.M. Wang and Z. Mao. Multiuser detection for mc-cdma system with m-qam using semidefinit programming relaxation. In PACRIM - IEEE Pacific Rim Conference on Communications, Computers and signal Processing, pages 530–533, Aug 2005. [19] N.D. Sidiropoulos and Z.-Q. Luo. A semidefinit relaxation approach to mimo detection for high-order qam constellations. IEEE Signal Processing Letters, 13(9):525– 528, Sept. 2006. [20] Y. Yang, C. Zhao, P. Zhou, and W. Xu. Mimo detection of 16-qam signaling based on semidefinit relaxation. IEEE Signal Processing Letters, 14(11):797 – 800, Nov 2007. [21] Zhiwei Mao, Xianmin Wang, and Xiaofeng Wang. Semidefinit programming relaxation approach for multiuser detection of qam signals. IEEE Transactions on Wireless Communications, 6(12):4275 – 4279, Dec. 2007. [22] W. K. Ma, P. C. Ching, and Z. Ding. Semidefinit relaxation based multiuser detection for m-ary psk multiuser systems. IEEE Transactions on Signal Processing, 52(10):2862–2872, Oct. 2004. [23] Zi Li, Chengkang Pan, Yueming Cai, and Youyun Xu. A novel quadratic programming model for soft-input soft-output mimo detection. IEEE Signal Processing Letters, 14(12):924–927, Dec. 2007. [24] J. Proakis. Digital Communications. McGraw-Hill, McGraw-Hill, 1989. Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

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[25] Marvin K. Simon and Mohamed-Slim Alouini. Digital Communication over Fading Channels. Wiley & Sons, New Jersey, second edition edition, 2005. [26] Sergio Verd´u. Computational complexity of optimum multiuser detection. Algorithmica, 4(1):303–312, 1989. [27] Piero Castoldi. Multiuser Detection in CDMA Mobile Terminals. Artech House, London, UK, 2002. [28] M. J. Juntti, T. Schlosser, and J. O. Lilleberg. Genetic algorithms for multiuser detection in synchronous cdma. In Proceedings of the IEEE International Symposium on Information Theory, page 492, 1997. [29] S. Verd´u. Minimum probability of error for synchronous gaussian multiple-access channels. IEEE Transactions on Information Theory, 32:85–96, 1986. [30] H. V. Poor and S. Verd´u. Probability of error in mmse multiuser detection. IEEE Transactions on Information Theory, 43(3):858–871, 1997. [31] P. Patel and J. M. Holtzman. Analysis of a single sucessive interference cancellation scheme in a ds/cdma system. IEEE Journal on Selected Areas in Communication, 12(5):796–807, 1994. [32] A. Duel-Hallen. A family of multiuser decision-feedback detectors for asynchronous cdma channels. IEEE Transactions on Communications, 43(2/3/4):421–434, 1995.

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[33] H. S. Lim, M. V. C. Rao, A. W. C Tan, and H. T. Chuah. Multiuser detection for ds-cdma systems using evolutionary programming. IEEE Communications Letters, 7(3):101–103, 2003. [34] S.L. Hijazi and B. Natarajan. Novel low-complexity ds-cdma multiuser detector based on ant colony optimization. In VTC04-Fall – IEEE 60th Vehicular Technology Conference, volume 3, pages 1939–1943, Sept 2004. [35] Peng Hui Tan and L. K. Rasmussen. Multiuser detection in cdma - a comparison of relaxations, exact, and heuristic search methods. IEEE Transactions on Wireless Communications, 3(5):1802–1809, Sept. 2004. [36] K. Yen and L. Hanzo. Antenna-diversity-assisted genetic-algorithm-based multiuser detection schemes for synchronous cdma systems. IEEE Transactions On Communications, 51(3):366–370, March 2003. [37] Taufi Abr˜ao, Fernando Ciriaco, Leonardo D. Oliveira, Bruno A. Ang´elico, Paul J. E. Jeszensky, and F. Casadevall. Weighting particle swarm optimization simo mc-cdma multiuser detectors. In IEEE International Symposium on Spread Spectrum Techniques and Applications, Bologna, Italy, Aug. 2008. [38] Taufi Abr˜ao, Fernando Ciriaco, Leonardo D. Oliveira, Bruno A. Ang´elico, Paul J. E. Jeszensky, and F. Casadevall. Weighting particle swarm, simulation annealing and local search optimization for s/mimo mc-cdma systems. In SIS 2008 - IEEE Swarm Intelligence Symposium, pages 1–7, Sept. 2008.

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[39] James Kennedy and Russell Eberhart. Particle swarm optimization. In IEEE International Conference on Neural Networks, pages 1942–1948, 1995. [40] James Kennedy and Russell Eberhart. A discrete binary version of the particle swarm algorithm. In IEEE international conference on Systems, pages 4104–4108, 1997. [41] R. Eberhart and Y. Shi. Particle swarm optimization: developments, applications and resources. In Proceedings of the 2001 Congress on Evolutionary Computation, volume 1, pages 81–86, May 2001. [42] Y. Zhao and J. Zheng. Particle swarm optimization algorithm in signal detection and blind extraction. In IEEE, editor, 7th International Symposium on Parallel Architectures, Algorithms and Networks, pages 37–41, May 2004. [43] K. Deb. Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Chichester, U.K, 2001. [44] SOO K. K., SIU Y. M., CHAN W. S., YANG L., and CHEN R. S. Particle-swarmoptimization-based multiuser detector for cdma communications. IEEE transactions on Vehicular Technology, 56(5):3006–3013, May 2007. [45] Taufi Abr˜ao, Leonardo D. de Oliveira, Fernando Ciriaco, Bruno A. Ang´elico, Paul Jean E. Jeszensky, and Fernando Jose Casadevall Palacio. S/mimo mc-cdma heuristic multiuser detectors based on single-objective optimization. Wireless Personal Communications, April 2009.

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[46] Y..H. Shi and R. C. Eberhart. Parameter selection in particle swarm optimization. In 1998 Annual Conference on Evolutionary Programming, San Diego, USA, March 1998. [47] A. Chatterjee and P. Siarry. Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Computers & Operations Research, 33(3):859–871, March 2006. [48] Marvin K. Simon and Mohamed-Slim Alouini. Digital Communication over Fading Channels. J. Wiley & Sons, Inc., second edition, 2005. [49] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan. Simulation of Communication Systems. Plenum Press, New York, 1992.

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INDEX

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A abstraction, 83 accounting, 172 achievement, 272 adaptation, x, 120, 194, 214, 298 adjustment, 39, 41, 74, 77 Alaska, 46, 48 alternative hypothesis, 241, 243 alternatives, 68, 79 amortization, 160 amplitude, 77, 251, 252, 253, 256, 285 angiotensin II, 48 animations, 176 annealing, ix, 33, 50, 127, 197, 297 antagonism, 48 antigen, 84 applied mathematics, 117 arithmetic, 56 artificial intelligence, 170, 176 ASI, 187 Asia, 87, 95 assessment, 225 assignment, 45, 83, 135 assumptions, 50, 102, 103, 104, 112, 115, 212, 280 attachment, 138 Australia, 117, 190, 223 authors, 174, 175, 176, 177, 178, 179, 180, 187, 221, 270 automata, 173 autonomy, 173, 175 averaging, 145 avoidance, x, 174, 175, 194, 203, 204, 208, 209, 210, 211, 212, 213, 215, 219, 220, 222, 223, 224

B background, 176, 205, 263 bacterium, 204 bandwidth, 255, 293 barriers, 211 behavior, vii, xi, 2, 17, 20, 21, 81, 128, 131, 134, 135, 137, 138, 146, 147, 148, 150, 151, 163, 170,

171, 173, 174, 175, 176, 177, 187, 190, 192, 194, 200, 201, 202, 203, 213, 215, 221, 227, 228, 229, 261, 265, 268, 274, 277, 284, 287, 291 Beijing, 191 Belgium, 224 benchmarking, 81 biochemistry, 197 birds, 170, 194, 261 blocks, 202 BMI, 12 bounds, 21, 52, 112 branching, 46 Brazil, 189, 249, 295 breeding, 223 building blocks, 176

C Canada, 46, 48, 222 capsule, 85 carrier, 253, 255 categorization, 175 catfish, 80, 91 CEC, 29, 48, 190 cell, 50 channels, xi, 249, 251, 252, 253, 255, 260, 261, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 285, 289, 290, 292, 293, 295, 296, 297 chaos, 179, 180, 223 chemotaxis, 50 China, 93, 94, 95, 189, 191, 221 clarity, 184 classes, 128, 153, 154, 155, 156, 158, 159, 160, 163, 194 classification, 197, 223 clavicle, 178 cloning, 72, 74, 295 clusters, 172 CNN, 64 codes, 174, 214, 250 cognition, 178 coherence, 255

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300

Index

collaboration, 29 collisions, 174, 177 column vectors, 182 commercials, 176 communication, x, 68, 69, 70, 71, 72, 74, 79, 82, 88, 92, 194, 207, 227, 228, 229, 244, 250, 251, 265, 288, 295 communication systems, 250, 251, 295 communications channel, 232 community, vii, x, 1, 2, 3, 4, 5, 29, 169, 179 competition, 157 complex numbers, 253 complexity, xi, 138, 174, 178, 204, 220, 249, 250, 251, 252, 253, 256, 259, 260, 261, 265, 266, 268, 270, 274, 285, 286, 287, 288, 289, 290, 291, 292, 297 compliance, 93, 201 components, 39, 50, 52, 61, 82, 131, 136, 137, 139, 148, 149, 161, 162, 176, 177, 178, 181, 183, 185 computation, vii, x, 1, 22, 24, 28, 62, 98, 132, 133, 144, 147, 148, 163, 169, 178, 183, 184, 185, 186, 259 computing, 200 concentrates, 148 conditioning, 221 confidence, 293 confidence interval, 293 configuration, 148, 178, 256 Congress, 29, 47, 48, 64, 65, 93, 94, 164, 188, 190, 222, 223, 224, 247, 298 connectivity, 181 consensus, 71 conservation, 33 construction, 32, 129, 133, 179 consumers, 21, 23 consumption, 23 contextualization, 252 contour, 234, 236 contradiction, 200 control, vii, 1, 2, 3, 5, 7, 10, 11, 12, 16, 17, 20, 23, 25, 26, 27, 28, 29, 30, 73, 77, 83, 93, 157, 170, 175, 176, 181, 182, 183, 184, 185, 187, 189, 197, 222, 225, 228, 274, 275, 277, 279 convergence criteria, 129 cooling, 153 correlation, 259, 280, 287 correlation function, 259 cost minimization, 44 costs, 2, 21, 28, 58, 158, 159, 160, 161 CPU, 44, 58, 62, 63, 184, 186, 209 cumulative distribution function, 154 customers, 32, 38, 41, 42, 43, 44, 45 cycles, 32

D damping, 136, 174 danger, 209

data analysis, 224 data set, 44, 88, 181 decision making, ix, 119 decisions, 52, 150, 251, 256, 259, 262 decoding, ix, 119, 120, 122, 125, 250, 262 decomposition, 11, 19 decoupling, 252 definition, viii, 24, 28, 31, 67, 68, 83, 114, 118, 140, 151 degradation, 52, 252, 280, 282, 285, 286, 292 delivery, x, 127, 160, 162 density, 22, 44, 207, 217, 219, 254, 294 derivatives, viii, 97, 98, 102, 103, 105, 117 designers, viii, 67 detection, xi, 208, 211, 219, 220, 224, 249, 250, 251, 252, 253, 259, 260, 262, 263, 264, 274, 279, 284, 288, 291, 295, 296, 297, 298 deviation, 11, 77 differentiation, 194, 199 digestion, 175 dimensionality, 128, 134, 144, 146, 147, 150 dispersion, 70, 74, 76, 77, 78, 79, 85, 91 displacement, 130, 201, 209 distributed computing, 103, 157, 220 distribution, 21, 23, 44, 45, 77, 89, 128, 153, 154, 162, 167, 172, 186, 202, 203, 215, 218, 242, 254, 262, 263 district heating, 21, 22, 23, 30 diversification, 252, 265 diversity, xi, 70, 72, 74, 76, 77, 78, 79, 86, 87, 90, 91, 94, 136, 137, 138, 149, 194, 196, 227, 228, 250, 252, 256, 258, 260, 263, 269, 273, 274, 278, 279, 280, 282, 283, 284, 288, 289, 290, 292, 293 division, xi, 135, 249 drawing, ix, 97, 179 duration, 253

E earth, 87, 88 economics, 197 elaboration, 81 elasticity, 201 electricity, 23 electromagnetic, 8, 15, 82 emotions, 175 energy, vii, 1, 2, 7, 11, 22, 25, 27, 29, 83, 174, 208, 209, 210, 253, 256, 279, 282 energy consumption, vii, 1, 2 England, 30, 188 entrapment, x, 194 environment, x, 120, 170, 173, 177, 188, 193, 194, 196, 203, 204, 208, 209, 210, 212, 213, 215, 220, 221, 225, 228 environmental protection, 197 equality, x, 193, 195, 196, 221 equilibrium, 9 error estimation, 249, 279, 282

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Index Europe, 87 evacuation, 176 evolution, viii, 34, 37, 67, 155, 170, 171, 172, 173, 175, 183, 185, 196 execution, 73, 152, 163, 179 exploitation, 39, 50, 51, 73, 78, 79, 80, 85, 86, 87, 89, 94, 106, 108, 172, 231, 263, 264, 288, 289, 290 exposure, 84 external influences, 71 extraction, 298

F

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fermentation, 222 filters, 11, 13, 16, 18 financial markets, 71 fine tuning, 137 Finland, 30 fitness, 33, 37, 43, 50, 52, 53, 54, 55, 56, 69, 72, 76, 77, 80, 82, 83, 84, 87, 88, 91, 92, 155, 170, 171, 172, 173, 181, 234, 236, 241, 260, 261, 262 flex, 175 flexibility, 176, 201, 202 fluctuations, 75, 159, 160, 161 focusing, 125 forward integration, 206 France, 1, 23, 30 freedom, 151, 161, 178 friction, 15 fulfillment, 21 function values, 103, 145, 151 fusion, 204

G gender, 83 generalization, 32, 99, 174, 182 generation, 37, 38, 50, 53, 93, 103, 124, 128, 153, 185, 251, 288 genes, 82 genetics, 83, 92 Georgia, 67 Germany, 127, 193, 200, 221 globalization, 157 good behavior, 144 GPS, 204 graph, 34, 179, 203, 204, 215 gravitational force, 134 gravity, 206 group size, 81 grouping, 46, 135 groups, 81, 82, 89, 134, 135, 138, 144, 147, 185, 194, 215, 228 growth, 204, 222 Guangzhou, 95 guidance, 198

guidelines, 90, 108, 115

H Hawaii, 47 heating, 22 Hong Kong, 94, 224 hopes, 73, 80 House, 297 hub, 72, 157, 158, 159, 160, 161, 162, 163 human behavior, 175, 190 human brain, 175 Hungary, 167, 296 hybrid, 32, 33, 49, 50, 51, 52, 54, 55, 56, 57, 63, 64, 83, 118, 157, 194, 256, 260 hybridization, 50, 51, 53, 63, 92 hypothesis, 113, 241, 242, 243 hypothesis test, 242, 243

I ideal, 88 ideals, 89 identification, 50, 195 identity, 88, 98, 201 image, 210 images, 178 immune system, 83, 84 implementation, 3, 4, 24, 32, 53, 54, 55, 56, 120, 136, 137, 172, 230, 261, 263, 287, 289, 291, 293 incidence, 179 inclusion, 175 India, 94 indication, 85, 115 induction, 197, 222 inductor, 15 industry, vii, x, 1, 2, 3, 4, 5, 29, 127, 157, 173 inequality, x, 29, 113, 114, 193, 195, 196, 199, 203, 212, 221 inertia, 4, 15, 34, 38, 40, 44, 51, 54, 74, 75, 76, 77, 78, 80, 81, 82, 84, 89, 99, 120, 124, 172, 183, 185, 186, 196, 197, 205, 206, 228, 265, 270, 275, 277, 298 infancy, 187 infinite, 37, 158, 212, 221 information exchange, 83 initial state, 129, 149, 151, 152 insertion, 263 insight, 148, 202, 269 inspiration, ix, 97, 104 integration, x, 50, 193, 207 intelligence, vii, 170, 187, 224, 260 intentions, 175 interaction, 71, 175, 176, 177, 228, 244, 261 interactions, 71, 170, 174, 175, 176, 190 interface, 190, 214, 216, 295

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova

302

Index

interference, 250, 255, 256, 260, 274, 275, 277, 279, 285, 297 interval, 32, 124, 174, 181, 253, 256, 268, 293, 294 inversion, 154 Italy, 187, 191, 297 iteration, 2, 3, 24, 54, 56, 58, 71, 75, 77, 78, 86, 90, 98, 99, 100, 101, 102, 103, 104, 108, 112, 115, 135, 153, 154, 162, 172, 185, 186, 196, 199, 205, 209, 228, 252, 262, 263, 264, 265 iterative solution, 131

J Japan, 29, 47, 125, 188, 189, 221, 225 joints, 177, 178, 201, 205

K Keynes, 127 Korea, 49, 188

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L landscape, viii, xi, 37, 67, 69, 70, 72, 80, 87, 90, 92, 155, 227, 228 language, 58 laws, vii, 1, 2, 28 leadership, 71, 228 learning, 70, 83, 84, 172, 178, 179 libido, 175 line, 23, 34, 75, 131, 132, 137, 144, 146, 147, 149, 179, 183, 185 linear model, vii, 1, 2, 3 linear systems, 22, 118, 182 links, 174 localization, 222 logistics, 32, 158 LTD, 47

M magnet, 9 MAI, 250, 255, 274, 275, 277, 279, 282, 285, 292 Malaysia, 189 manipulation, 91 manufacturing, x, 50, 127, 179, 180, 202 mapping, 36, 37, 180, 251, 254 markets, 195 mathematical programming, ix, 119 mathematics, 94 matrix, 12, 13, 24, 25, 29, 98, 101, 120, 122, 130, 133, 163, 182, 198, 201, 202, 205, 206, 221, 230, 258, 259, 287 measurement, 20, 21, 87, 180, 200 measures, 9, 57, 64, 86, 88, 90, 140, 243

mechanical properties, x, 193, 221 median, 154, 234 memory, viii, 31, 38, 39, 40, 41, 42, 43, 46, 69, 75, 83, 120, 122, 128, 155, 170, 171, 172, 207, 228 mental state, 175 mental states, 175 metaphor, 76, 78 Ministry of Education, 187 mobile communication, 293 mobile robots, x, 193, 198, 203, 204, 205, 206, 207, 208, 211, 213, 219, 221, 222, 223 model, vii, x, 2, 9, 10, 11, 13, 14, 16, 17, 18, 20, 22, 59, 60, 103, 104, 120, 134, 157, 158, 160, 161, 162, 169, 170, 177, 178, 180, 181, 182, 191, 193, 194, 196, 200, 205, 206, 207, 223, 230, 233, 241, 252, 253, 257, 263, 294, 296 modeling, x, 8, 48, 169, 171, 174, 178, 179, 187, 188, 195, 203, 204 models, 2, 176, 178, 179, 180, 189, 204, 207, 228, 250 modulations, 271, 284, 291, 292 modules, 175, 214 modulus, 38 momentum, 205 motion, ix, x, 127, 134, 163, 174, 175, 177, 193, 203, 204, 205, 206, 207, 208, 211, 215, 219, 221, 228 motor control, 174 motor system, 175 movement, 67, 69, 71, 128, 139, 144, 146, 148, 159, 171, 176, 177, 178, 188, 209, 211, 215, 221, 225, 261, 265 multidimensional, 51, 171, 177, 246 multimedia, 250 multiplication, 4, 110, 288 multiplier, 195, 198, 199, 221 muscles, 174, 175 music, 195, 197, 222 mutation, 49, 50, 52, 53, 54, 55, 56, 58, 63, 72, 74, 80, 81, 83, 89, 91, 138, 155 mutation rate, 49, 50, 55, 58

N NATO, 187 Netherlands, 118, 188, 190, 192 network, 21, 22, 23, 30, 138, 157, 181, 189, 197, 204 neural network, 173, 176, 181, 182, 187, 188, 189, 196, 197, 223, 224 Neural Network Model, 95 neural networks, 173, 176, 181, 182, 188, 196, 223 next generation, 37, 250 nodes, 35, 44, 45, 157, 295 noise, xi, 6, 20, 149, 159, 180, 249, 253, 294 North America, 87 null hypothesis, 241, 243 numerical analysis, 180, 252, 286, 288, 291

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303

Index

O objectives, vii, x, 3, 25, 27, 29, 41, 169 observations, 51, 177 operator, 38, 39, 50, 51, 52, 56, 72, 138, 224 optimization method, viii, ix, x, 3, 14, 97, 103, 119, 121, 122, 125, 140, 176, 193, 251 organ, 231 orientation, 75, 178, 201

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P Pacific, 191, 296 pairing, 90 parallelism, 179 parameter, 3, 4, 5, 29, 30, 32, 34, 41, 44, 50, 51, 55, 81, 118, 121, 129, 131, 132, 133, 136, 142, 148, 150, 152, 153, 154, 155, 163, 180, 181, 182, 183, 185, 202, 241, 268, 269, 285 parameter vectors, 185 parameters, vii, ix, xi, 1, 4, 5, 9, 10, 13, 17, 29, 34, 40, 50, 51, 52, 53, 54, 55, 56, 58, 97, 99, 102, 103, 117, 120, 124, 129, 131, 138, 142, 152, 154, 155, 158, 160, 175, 178, 179, 180, 182, 183, 184, 200, 213, 214, 227, 233, 244, 252, 256, 265, 273, 274, 275, 276, 277, 284 parents, 52, 53 Pareto, 24, 25, 26, 27, 28, 29, 264 Pareto optimal, 24, 25, 26 path planning, 203, 224 peers, 211, 212 Perth, 117, 190, 223 Pheromones, 192 physics, 192 planning, x, 160, 176, 188, 193, 203, 204, 205, 207, 208, 211, 221 PLS, 48 poor, 221, 267 poor performance, 267 population, viii, 17, 31, 33, 39, 40, 41, 43, 44, 46, 50, 51, 52, 53, 54, 56, 58, 67, 68, 69, 71, 72, 73, 74, 83, 89, 90, 92, 117, 120, 128, 142, 144, 145, 146, 155, 170, 171, 172, 173, 176, 183, 185, 186, 194, 213, 228, 229, 230, 241, 247, 261, 263, 264, 266, 285, 287, 288 population size, 44, 58, 68, 128, 142, 144, 145, 146, 155, 183, 185, 186, 261, 263, 266, 285, 287, 288 power, 21, 23, 76, 177, 195, 197, 222, 225, 254, 274, 275, 277, 279, 280, 281, 294 prediction, 2 prediction models, 2 preference, 147 prevention, 149 probability, 25, 33, 71, 72, 137, 140, 152, 153, 154, 253, 259, 263, 293, 297 probability distribution, 153, 154 problem solving, 171

problem space, viii, 67, 68, 131, 170 process control, vii, 223 producers, 21 product design, 200 production, 21, 23, 157, 194, 200, 222 productivity, vii, 1, 2 profit, 159, 160, 161, 162 program, 116 programming, ix, 34, 35, 52, 119, 120, 121, 122, 123, 124, 125, 214, 250, 251, 260, 295, 296, 297 propagation, 22, 178, 276 proposition, 106, 115 pulse, 253 pumps, 21

Q QAM, 251 quadratic programming, 296 quality of service, 65, 250 quartile, 215

R radius, 15, 22, 81, 82, 174, 207, 211, 220 random numbers, 36, 120, 130, 131, 132, 134, 147, 153, 154, 162, 179 random walk, 138, 146, 151, 153 range, 4, 34, 36, 38, 44, 51, 53, 72, 74, 81, 84, 85, 89, 115, 153, 170, 172, 175, 179, 263, 264, 271, 273, 275, 285 real time, 23, 148, 265 reality, 171, 173 reason, 77, 138, 161, 162, 204, 206, 251 reasoning, 90, 171, 188 recall, 99, 106 recalling, 103 reception, 293 reconstruction, 181, 187, 188, 189, 190, 191 redundancy, 49, 61 refining, 79, 229 region, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 137, 139, 148, 159, 172, 177, 199, 209, 211 regression, 180 regression analysis, 180 regulation, 7, 29 regulations, 23 rejection, 6, 7, 8, 11, 15, 158, 160, 161 relationship, 37, 39, 174, 201 relaxation, 250, 251, 296 reliability, 49, 50, 51, 52, 57, 58, 64 reliability systems, 50 residuals, 180 resistance, 15 resolution, 33 resources, x, 98, 127, 157, 158, 188, 292, 298

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returns, 43, 79, 84, 90, 129, 147 risk, 29 robotics, 134, 170, 187, 200, 201, 204, 208 robustness, x, xi, 10, 193, 194, 207, 249, 274, 275, 277, 280, 292 rotations, 163, 178, 207, 211 routines, 175 routing, vii, 31, 32, 45, 46, 170 Russia, 30

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S sample variance, 243 sampling, 55, 56, 78, 81, 85 satisfaction, 102, 161 saturation, 26 scaling, 128, 130, 134, 197, 256 scaling law, 134 scarcity, 250 scattering, 81 scheduling, 23, 44, 47, 152 schooling, 3, 51, 170, 175 searches, xi, 32, 44, 58, 227, 228, 251 searching, x, 34, 36, 37, 38, 41, 42, 43, 45, 46, 53, 68, 79, 81, 83, 172, 179, 194, 215, 219 security, 32, 225 sensitivity, 11, 29, 174 sensors, 175, 208, 209, 211, 212 shape, viii, 97, 162, 186 shaping, 13, 253 shares, 80 sharing, 250, 275, 282 signals, 208, 210, 250, 251, 255, 256, 257, 259, 279, 280, 296 signal-to-noise ratio, 278 significance level, 243 silhouette, 178 simulation, x, xi, 22, 30, 38, 148, 157, 158, 159, 160, 170, 171, 173, 174, 175, 176, 177, 187, 188, 193, 194, 199, 209, 211, 212, 213, 214, 220, 221, 249, 251, 264, 271, 274, 293, 294, 295, 297 sine wave, 77 Singapore, 94 smoothness, 186 social behavior, vii, x, 3, 120, 169 social influence, 71 social network, 172 software, 174, 295 space, vii, viii, ix, x, 3, 17, 24, 31, 33, 34, 36, 37, 38, 41, 42, 43, 44, 45, 46, 50, 51, 55, 56, 63, 67, 68, 76, 77, 80, 83, 100, 120, 127, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 140, 142, 145, 146, 148, 149, 151, 152, 154, 155, 159, 161, 162, 163, 169, 170, 171, 172, 177, 178, 181, 182, 186, 228, 229, 230, 231, 246, 261, 274, 286, 287 Spain, 30, 191, 222 spectrum, 250

speed, ix, 15, 55, 58, 78, 99, 119, 122, 135, 152, 175, 177, 211, 229, 231, 244, 260, 267, 273, 277, 295 stability, 6, 12, 23, 87, 175, 246 stabilization, 73, 184 standard deviation, 154, 202, 203, 241, 243 Star Wars, 176 stars, 183 statistics, 180, 215, 256 storage, 158, 159, 160, 161, 162, 180 strategies, viii, x, 4, 31, 33, 36, 37, 41, 46, 103, 107, 152, 155, 161, 193, 197, 203, 204, 221, 251, 270 strength, 84, 88, 212 surveillance, 177 swarm intelligence, vii, 169, 170, 174, 252 symbols, 251, 253, 256, 257, 259, 281, 285, 287, 288, 289, 293, 294 synthesis, vii, 1, 2, 3, 11, 12, 13, 14, 16, 17, 18, 25, 28, 45, 49 system analysis, 117

T Taiwan, 31, 189 targets, 44, 177 telecommunications, 157 temperature, 22, 34, 151, 152, 153, 175, 197, 222 territory, 82 test statistic, 243 threshold, ix, 127, 173, 185, 186, 262 time frame, 87, 89 topology, xi, 29, 69, 71, 72, 80, 81, 172, 181, 182, 227, 228, 231, 232 tracking, 2, 5, 11, 15, 16, 20, 178 trade, 89 trade-off, xi, 249, 260, 261, 265, 266, 268, 270, 292 traffic, 197 training, 176 trajectory, 100, 101, 102 transactions, 298 transformation, 9, 33, 36, 38, 178 transition, 3, 4, 72, 152 transmission, xi, 227, 244, 250, 259 transport, 118, 157, 158, 160 transport costs, 160 transportation, x, 127, 157, 158, 159, 160, 161, 176 transshipment, 157, 158 trial, 63, 89, 92, 180, 211, 294 turbulence, 81

U unconstrained minimization, 105, 117, 198 uniform, 56, 89, 124, 131, 132, 133, 136, 153, 154, 156, 172, 183, 185, 186, 254, 262, 263, 279 universe, 89, 263, 284, 287, 288 updating, ix, 37, 42, 53, 54, 56, 89, 119, 209 uplink, xi, 249, 252, 256, 288

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V walking, 177, 203 West Indies, 165 windows, 35, 44, 45, 46 wireless systems, 283 worms, 174

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Valencia, 191 validation, 11, 16 variability, 84 variables, ix, 4, 50, 63, 85, 98, 119, 120, 128, 129, 158, 159, 171, 180, 184, 195, 202, 203, 234, 251, 256, 258, 259, 262, 284, 291 variance, viii, 31, 34, 39, 46, 86, 154 vector, 34, 36, 79, 99, 101, 110, 114, 120, 121, 122, 123, 128, 129, 130, 131, 132, 133, 136, 137, 148, 151, 159, 177, 178, 181, 182, 183, 201, 202, 206, 222, 228, 257, 258, 259, 260, 262, 263, 265 vehicles, 32, 35, 41, 44, 45, 83, 170 velocity, viii, ix, x, 3, 33, 34, 36, 37, 38, 39, 40, 41, 42, 51, 53, 54, 55, 56, 67, 70, 73, 74, 75, 76, 77, 78, 79, 81, 82, 84, 85, 87, 89, 127, 128, 129, 130, 131, 132, 133, 136, 137, 138, 139, 149, 150, 158, 160, 161, 170, 171, 172, 173, 174, 175, 177, 193, 196, 197, 198, 201, 205, 206, 208, 209, 211, 212, 221, 228, 261, 262, 263, 264, 265, 266, 271, 275, 277, 287 video games, 170 vision, 175, 178, 187 visualization, 148, 215

W

Particle Swarm Optimization: Theory, Techniques and Applications : Theory, Techniques and Applications, edited by Andrea E. Olsson, Nova