Numerical Optimization with Computational Errors [1st ed.] 331930920X, 978-3-319-30920-0, 978-3-319-30921-7, 3319309218

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Numerical Optimization with Computational Errors [1st ed.]
 331930920X, 978-3-319-30920-0, 978-3-319-30921-7, 3319309218

Table of contents :
Front Matter....Pages i-ix
Introduction....Pages 1-9
Subgradient Projection Algorithm....Pages 11-40
The Mirror Descent Algorithm....Pages 41-58
Gradient Algorithm with a Smooth Objective Function....Pages 59-72
An Extension of the Gradient Algorithm....Pages 73-84
Weiszfeld’s Method....Pages 85-103
The Extragradient Method for Convex Optimization....Pages 105-118
A Projected Subgradient Method for Nonsmooth Problems....Pages 119-136
Proximal Point Method in Hilbert Spaces....Pages 137-147
Proximal Point Methods in Metric Spaces....Pages 149-168
Maximal Monotone Operators and the Proximal Point Algorithm....Pages 169-181
The Extragradient Method for Solving Variational Inequalities....Pages 183-203
A Common Solution of a Family of Variational Inequalities....Pages 205-224
Continuous Subgradient Method....Pages 225-238
Penalty Methods....Pages 239-264
Newton’s Method....Pages 265-296
Back Matter....Pages 297-304

Citation preview

Springer Optimization and Its Applications  108

Alexander J. Zaslavski

Numerical Optimization with Computational Errors

Springer Optimization and Its Applications VOLUME 108 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches.

More information about this series at http://www.springer.com/series/7393

Alexander J. Zaslavski

Numerical Optimization with Computational Errors

123

Alexander J. Zaslavski Department of Mathematics The Technion – Israel Institute of Technology Haifa, Israel

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-319-30920-0 ISBN 978-3-319-30921-7 (eBook) DOI 10.1007/978-3-319-30921-7 Library of Congress Control Number: 2016934410 Mathematics Subject Classification (2010): 47H09, 49M30, 65K10 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

The book is devoted to the study of approximate solutions of optimization problems in the presence of computational errors. We present a number of results on the convergence behavior of algorithms in a Hilbert space, which are known as important tools for solving optimization problems and variational inequalities. According to the results known in the literature, these algorithms should converge to a solution. In this book, we study these algorithms taking into account computational errors which are always present in practice. In this case the convergence to a solution does not take place. We show that our algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. In practice it is sufficient to find a good approximate solution instead of constructing a minimizing sequence. On the other hand, in practice computations can induce numerical errors and if one uses optimization methods to solve minimization problems these methods usually provide only approximate solutions of the problems. Our main goal is, for a known computational error, to find out what an approximate solution can be obtained and how many iterates one needs for this. This monograph contains 16 chapters. Chapter 1 is an introduction. In Chap. 2, we study the subgradient projection algorithm for minimization of convex and nonsmooth functions. The mirror descent algorithm is considered in Chap. 3. The gradient projection algorithm for minimization of convex and smooth functions is analyzed in Chap. 4. In Chap. 5, we consider its extension which is used for solving linear inverse problems arising in signal/image processing. The convergence of Weiszfeld’s method in the presence of computational errors is discussed in Chap. 6. In Chap. 7, we solve constrained convex minimization problems using the extragradient method. Chapter 8 is devoted to a generalized projected subgradient method for minimization of a convex function over a set which is not necessarily convex. In Chap. 9, we study the convergence of a proximal point method in a Hilbert space under the presence of computational errors. Chapter 10 is devoted to the local convergence of a proximal point method in a metric space under the presence of computational errors. In Chap. 11, we study the convergence of a proximal point method to a solution of the inclusion induced by a maximal monotone operator, under the presence of computational errors. In Chap. 12, the v

vi

Preface

convergence of the subgradient method for solving variational inequalities is proved under the presence of computational errors. The convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems, under the presence of computational errors, is shown in Chap. 13. In Chap. 14, we study continuous subgradient method. Penalty methods are studied in Chap. 15. Chapter 16 is devoted to Newton’s method. The results of Chaps. 2–6, 14, and 16 are new. The results of other chapters were obtained and published during the last 5 years. The author believes that this book will be useful for researchers interested in the optimization theory and its applications. Rishon LeZion, Israel October 19, 2015

Alexander J. Zaslavski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Subgradient Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Mirror Descent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Proximal Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 6 8

2

Subgradient Projection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Convex Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Subgradient Algorithm on Unbounded Sets . . . . . . . . . . . . . . . . . . . . . . . 2.6 Proof of Theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Zero-Sum Games with Two-Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Proof of Proposition 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Subgradient Algorithm for Zero-Sum Games . . . . . . . . . . . . . . . . . . . . . 2.10 Proof of Theorem 2.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 17 19 20 22 25 29 35 39

3

The Mirror Descent Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Optimization on Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Optimization on Unbounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Zero-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 44 47 48 49 53

4

Gradient Algorithm with a Smooth Objective Function . . . . . . . . . . . . . . . 4.1 Optimization on Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 An Auxiliary Result and the Proof of Proposition 4.1 . . . . . . . . . . . . 4.3 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Optimization on Unbounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 61 62 66 68 vii

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Contents

5

An Extension of the Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries and the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 75 78 82

6

Weiszfeld’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Basic Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Proof of Theorem 6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 91 94 98

7

The Extragradient Method for Convex Optimization . . . . . . . . . . . . . . . . . . 7.1 Preliminaries and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 109 113 115

8

A Projected Subgradient Method for Nonsmooth Problems . . . . . . . . . . 8.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 122 126 131

9

Proximal Point Method in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Proof of Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 140 144 146

10

Proximal Point Methods in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Preliminaries and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Proof of Theorem 10.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 An Auxiliary Result for Theorem 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Proof of Theorem 10.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Well-Posed Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 156 158 160 161 163 164 167

11

Maximal Monotone Operators and the Proximal Point Algorithm . . . 11.1 Preliminaries and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Proof of Theorem 11.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Proofs of Theorems 11.3, 11.5, 11.6, and 11.7 . . . . . . . . . . . . . . . . . . . .

169 169 173 175 176

Contents

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12

The Extragradient Method for Solving Variational Inequalities . . . . . . 12.1 Preliminaries and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Proof of Theorem 12.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Finite-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 A Convergence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 189 193 196 198

13

A Common Solution of a Family of Variational Inequalities . . . . . . . . . . 13.1 Preliminaries and the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Proof of Theorem 13.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 210 212 221

14

Continuous Subgradient Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Bochner Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Convergence Analysis for Continuous Subgradient Method . . . . . 14.3 An Auxiliary Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Proof of Theorem 14.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Continuous Subgradient Projection Method . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 228 229 231

15

Penalty Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 An Estimation of Exact Penalty in Constrained Optimization . . . . 15.2 Proof of Theorem 15.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Infinite-Dimensional Inequality-Constrained Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Proofs of Auxiliary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Proof of Theorem 15.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Proof of Theorem 15.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 243

Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Pre-differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Convergence of Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Proof of Theorem 16.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 An Auxiliary Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Proof of Theorem 16.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Pre-differentiable Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.9 Newton’s Method for Solving Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 Auxiliary Results for Theorem 16.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.11 Proof of Theorem 16.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 269 272 274 279 281 285 287 292 294 296

16

246 253 255 258 260

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Chapter 1

Introduction

In this book we study behavior of algorithms for constrained convex minimization problems in a Hilbert space. Our goal is to obtain a good approximate solution of the problem in the presence of computational errors. We show that the algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. In this section we discuss several algorithms which are studied in the book.

1.1 Subgradient Projection Method In Chap. 2 we study the subgradient projection algorithm for minimization of convex and nonsmooth functions and for computing the saddle points of convex–concave functions, under the presence of computational errors. It should be mentioned that the subgradient projection algorithm is one of the most important tools in the optimization theory and its applications. See, for example, [1–3, 12, 30, 44, 51, 79, 89, 92, 95, 96, 105, 108, 109, 112] and the references mentioned therein. We use this method for constrained minimization problems in Hilbert spaces equipped with an inner product denoted by h; i which induces a complete norm k  k. For every z 2 R1 denote by bzc the largest integer which does not exceed z: bzc D maxfi 2 R1 W i is an integer and i  zg. Let X be a Hilbert space. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: For each x 2 X and each nonempty set E  X set d.x; E/ D inffkx  yk W y 2 Eg:

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_1

1

2

1 Introduction

Let C be a nonempty closed convex subset of X, U be an open convex subset of X such that C  U and let f W U ! R1 be a convex function. For each x 2 U set [84] @f .x/ D fl 2 X W f .y/  f .x/  hl; y  xi for all y 2 Ug which is called the subdifferential of the function f at the point x [84]. Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /; jf .x/  f .y/j  Lkx  yk for all x; y 2 U: It is not difficult to see that for each x 2 U, ; 6D @f .x/  BX .0; L/: For every nonempty closed convex set D  X and every x 2 X there is a unique point PD .x/ 2 D satisfying kx  PD .x/k D inffkx  yk W y 2 Dg: We consider the minimization problem f .z/ ! min; z 2 C: Suppose that ı 2 .0; 1 is a computational error produced by our computer system and that fak g1 kD0  .0; 1/. Let us describe our algorithm. Subgradient Projection Algorithm Initialization: select an arbitrary x0 2 U. Iterative step: given a current iteration vector xt 2 U calculate t 2 @f .xt / C BX .0; ı/ and the next iteration vector xtC1 2 U such that kxtC1  PC .xt  at t /k  ı: In Chap. 2 we prove the following result (see Theorem 2.4). Theorem 1.1. Let ı 2 .0; 1, fak g1 kD0  .0; 1/ and let x 2 C satisfies f .x /  f .x/ for all x 2 C:

1.1 Subgradient Projection Method

3

1 Assume that fxt g1 tD0  U, ft gtD0  X,

kx0 k  M0 C 1 and that for each integer t  0, t 2 @f .xt / C BX .0; ı/ and kxtC1  PC .xt  at t /k  ı: Then for each natural number T, T X

at .f .xt /  f .x //

tD0

 21 kx  x0 k2 C ı.T C 1/.4M0 C 1/ Cı.2M0 C 1/

T X

at C 21 .L C 1/2

tD0

T X

a2t :

tD0

Moreover, for each natural number T, 9 8 0 1 !1 T T = < X X at at xt A  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x / max f @ ; : tD0

1

2

T X

tD0

!1 at

2

kx  x0 k C

tD0

T X

!1 at

ı.T C 1/.4M0 C 1/

tD0 1

Cı.2M0 C 1/ C 2

T X tD0

!1 at

.L C 1/2

T X

a2t :

tD0

We are interested Pin an optimal choice of at , t D 0; 1; : : : . Let T be a natural number and AT D TtD0 at be given. It is shown in Chap. 2 that the best choice is ai D .T C 1/1 AT , i D 0; : : : ; T. Let T be a natural number and at D a, t D 0; : : : ; T. It is shown in Chap. 2 that best choice of a > 0 is a D .2ı.4M0 C 1//1=2 .L C 1/1 : Now we can think about the best choice of T. It is not difficult to see that it should be at the same order as bı 1 c.

4

1 Introduction

1.2 The Mirror Descent Method Let X be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. We use the notation introduced in the previous section. Let C be a nonempty closed convex subset of X, U be an open convex subset of X such that C  U and let f W U ! R1 be a convex function. Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /; jf .x/  f .y/j  Lkx  yk for all x; y 2 U: It is not difficult to see that for each x 2 U, ; 6D @f .x/  BX .0; L/: For each nonempty set D  X and each function h W D ! R1 put inf.h; D/ D inffh.y/ W y 2 Dg and argminfh.y/ W y 2 Dg D fy 2 D W h.y/ D inf.hI D/g: In Chap. 3 we study the convergence of the mirror descent algorithm under the presence of computational errors. This method was introduced by Nemirovsky and Yudin for solving convex optimization problems [90]. Here we use a derivation of this algorithm proposed by Beck and Teboulle [19]. We consider the minimization problem f .z/ ! min; z 2 C: Suppose that ı 2 .0; 1 is a computational error produced by our computer system and that fak g1 kD0  .0; 1/. We describe the inexact version of the mirror descent algorithm. Mirror Descent Algorithm Initialization: select an arbitrary x0 2 U. Iterative step: given a current iteration vector xt 2 U calculate t 2 @f .xt / C BX .0; ı/; define gt .x/ D ht ; xi C .2at /1 kx  xt k2 ; x 2 X

1.2 The Mirror Descent Method

5

and calculate the next iteration vector xtC1 2 U such that BX .xtC1 ; ı/ \ argminfgt .y/ W y 2 Cg 6D ;: Note that gt is a convex bounded from below function on X which possesses a minimizer on C. In Chap. 3 we prove the following result (see Theorem 3.1). Theorem 1.2. Let ı 2 .0; 1, fak g1 kD0  .0; 1/ and let x 2 C satisfies f .x /  f .x/ for all x 2 C: 1 Assume that fxt g1 tD0  U, ft gtD0  X,

kx0 k  M0 C 1 and that for each integer t  0, t 2 @f .xt / C BX .0; ı/ and BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg 6D ;: Then for each natural number T, T X

at .f .xt /  f .x //

tD0

 21 .2M0 C 1/2 C ı.2M0 C L C 2/

T X

at

tD0

Cı.T C 1/.8M0 C 8/ C 21 .L C 1/2

T X

a2t :

tD0

Moreover, for each natural number T, 0 f@

T X tD0

!1 at

T X tD0

1 at xt A  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x /

6

1 Introduction

1

 2 .2M0 C 1/

2

T X

!1 C ı.2M0 C L C 2/

at

tD0 T X

Cı.T C 1/.8M0 C 8/

!1 at

1

C 2 .L C 1/

2

T X

tD0

tD0

a2t

T X

!1 at

:

tD0

We are interested Pin an optimal choice of at , t D 0; 1; : : : . Let T be a natural number and AT D TtD0 at be given. It is shown in Chap. 3 that the best choice is at D .T C 1/1 AT , i D 0; : : : ; T. Let T be a natural number and at D a, t D 0; : : : ; T. It is shown in Chap. 3 that the best choice of a > 0 a D .16ı.M0 C 1//1=2 .L C 1/1 : If we think about the best choice of T, it is clear that it should be at the same order as bı 1 c.

1.3 Proximal Point Method In Chap. 9 we analyze the behavior of the proximal point method in a Hilbert space which is an important tool in the optimization theory. See, for example, [9, 15, 16, 29, 31, 34, 36, 53, 55, 69, 70, 77, 81, 87, 103, 104, 106, 107, 111, 113] and the references mentioned therein. Let X be a Hilbert space equipped with an inner product h; i which induces the norm k  k. For each function g W X ! R1 [ f1g set inf.g/ D inffg.y/ W y 2 Xg: Suppose that f W X ! R1 [ f1g is a convex lower semicontinuous function and a is a positive constant such that dom.f / WD fx 2 X W f .x/ < 1g 6D ;; f .x/  a for all x 2 X and that lim f .x/ D 1:

kxk!1

It is not difficult to see that the set Argmin.f / WD fz 2 X W f .z/ D inf.f /g 6D ;:

1.3 Proximal Point Method

7

Let a point x 2 Argmin.f / and let M be any positive number such that M > inf.f / C 4: It is clear that there exists a number M0 > 1 such that f .z/ > M C 4 for all z 2 X satisfying kzk  M0  1: Evidently, kx k < M0  1: Assume that 0 < 1 < 2  M02 =2: The following theorem is the main result of Chap. 9. Theorem 1.3. Let k 2 Œ1 ; 2 ; k D 0; 1; : : : ;  2 .0; 1, a natural number L satisfy L > 2.4M02 C 1/2 1 and let a positive number  satisfy 1=2

 1=2 .L C 1/.21 1 C 8M0 1

/  1 and .L C 1/  =4:

Assume that a sequence fxk g1 kD0  X satisfies f .x0 /  M and f .xkC1 / C 21 k kxkC1  xk k2  inf.f C 21 k k  xk k2 / C  for all integers k  0. Then for all integers k > L, f .xk /  inf.f / C :

8

1 Introduction

By this theorem, for a given  > 0, we obtain  2 X satisfying f ./  inf.f / C  doing bc1 1 c iterations with the computational error  D c2 2 , where the constant c1 > 0 depends only on M0 ; 2 and the constant c2 > 0 depends only on M0 ; L; 1 ; 2 .

1.4 Variational Inequalities In Chap. 12 we are interested in solving of variational inequalities. The studies of gradient-type methods and variational inequalities are important topics in optimization theory. See, for example, [3, 5, 12, 30, 31, 37–39, 44, 52, 54, 59, 68, 71– 74, 93, 129] and the references mentioned therein. Let .X; h; i/ be a Hilbert space with an inner product h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set B.x; r/ D fy 2 X W kx  yk  rg: Let C be a nonempty closed convex subset of X. Consider a mapping f W X ! X. We say that the mapping f is monotone on C if hf .x/  f .y/; x  yi  0 for all x; y 2 C: We say that f is pseudo-monotone on C if for each x; y 2 C the inequality hf .y/; x  yi  0 implies that hf .x/; x  yi  0: Clearly, if f is monotone on C, then f is pseudo-monotone on C. Denote by S the set of all x 2 C such that hf .x/; y  xi  0 for all y 2 C: We suppose that S 6D ;: For each  > 0 denote by S the set of all x 2 C such that hf .x/; y  xi  ky  xk   for all y 2 C:

1.4 Variational Inequalities

9

In Chap. 12, we present examples which provide simple and clear estimations for the sets S in some important cases. These examples show that elements of S can be considered as -approximate solutions of the variational inequality. In Chap. 12, in order to solve the variational inequality (to find x 2 S), we use the algorithm known in the literature as the extragradient method [75]. In each iteration of this algorithm, in order to get the next iterate xkC1 , two orthogonal projections onto C are calculated, according to the following iterative step. Given the current iterate xk calculate yk D PC .xk  k f .xk // and then xkC1 D PC .xk  k f .yk //; where k is some positive number. It is known that this algorithm generates sequences which converge to an element of S. In Chap. 12, we study the behavior of the sequences generated by the algorithm taking into account computational errors which are always present in practice. Namely, in practice the algorithm generates 1 sequences fxk g1 kD0 and fyk gkD0 such that for each integer k  0, kyk  PC .xk  k f .xk //k  ı and kxkC1  PC .xk  k f .yk //k  ı; with a constant ı > 0 which depends only on our computer system. Surely, in this situation one cannot expect that the sequence fxk g1 kD0 converges to the set S. The goal is to understand what subset of C attracts all sequences fxk g1 kD0 generated by the algorithm. The main result of Chap. 12 (Theorem 12.2) shows that this subset of C is the set S with some  > 0 depending on ı. The examples considered in Chap. 12 show that one cannot expect to find an attracting set smaller than S , whose elements can be considered as approximate solutions of the variational inequality.

Chapter 2

Subgradient Projection Algorithm

In this chapter we study the subgradient projection algorithm for minimization of convex and nonsmooth functions and for computing the saddle points of convex– concave functions, under the presence of computational errors. We show that our algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

2.1 Preliminaries The subgradient projection algorithm is one of the most important tools in the optimization theory and its applications. See, for example, [1–3, 12, 30, 44, 51, 79, 89, 92, 95, 96, 105, 108, 109, 112] and the references mentioned therein. In this chapter we use this method for constrained minimization problems in Hilbert spaces equipped with an inner product denoted by h; i which induces a complete norm k  k. For every z 2 R1 denote by bzc the largest integer which does not exceed z: bzc D maxfi 2 R1 W i is an integer and i  zg. Let X be a Hilbert space. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: For each x 2 X and each nonempty set E  X set d.x; E/ D inffkx  yk W y 2 Eg: Let C be a nonempty closed convex subset of X, U be an open convex subset of X such that C  U and let f W U ! R1 be a convex function. Recall that for each x 2 U, © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_2

11

12

2 Subgradient Projection Algorithm

@f .x/ D fl 2 X W f .y/  f .x/  hl; y  xi for all y 2 Ug:

(2.1)

Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /;

(2.2)

jf .x/  f .y/j  Lkx  yk for all x; y 2 U:

(2.3)

In view of (2.1) and (2.3), for each x 2 U, ; 6D @f .x/  BX .0; L/:

(2.4)

It is easy to see that the following result is true. Lemma 2.1. Let z; y0 ; y1 2 X. Then kz  y0 k2  kz  y1 k2  ky0  y1 k2 D 2hz  y1 ; y1  y0 i: The next result is given in [13, 14]. Lemma 2.2. Let D be a nonempty closed convex subset of X. Then for each x 2 X there is a unique point PD .x/ 2 D satisfying kx  PD .x/k D inffkx  yk W y 2 Dg: Moreover, kPD .x/  PD .y/jj  kx  yk for all x; y 2 X and for each x 2 X and each z 2 D, hz  PD .x/; x  PD .x/i  0; kz  PD .x/k2 C kx  PD .x/k2  kz  xk2 : Lemma 2.3. Let A > 0 and n  2 be an integer. Then the minimization problem n X

a2i ! min

iD1

a D .a1 ; : : : ; an / 2 Rn and

n X

ai D A

iD1

has a unique solution a D .a1 ; : : : ; an / where ai D n1 A, i D 1; : : : ; n.

2.2 A Convex Minimization Problem

13

Proof. Clearly, the minimization problem has a solution a D .a1 ; : : : ; an / 2 Rn : Then an D A 

n1 X

ai

iD1

and .a1 ; : : : ; an1 / is a minimizer of the function .a1 ; : : : ; an1 / WD

n1 X

a2i

C A

iD1

n1 X

!2 ai

; .a1 ; : : : ; an1 / 2 Rn1 :

iD1

It is clear that for all i D 1; : : : ; n  1, 0 D .@ =@ai /.a1 ; : : : ; an1 / D 2ai  2 A 

n1 X

! ai

D 2ai  2an :

iD1

Thus ai D an for all i D 1; : : : ; n1 and ai D n1 A for all i D 1; : : : ; n. Lemma 2.3 is proved.

2.2 A Convex Minimization Problem Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. Let us describe our algorithm. Subgradient Projection Algorithm Initialization: select an arbitrary x0 2 U. Iterative step: given a current iteration vector xt 2 U calculate t 2 @f .xt / C BX .0; ı/ and the next iteration vector xtC1 2 U such that kxtC1  PC .xt  at t /k  ı: In this chapter we prove the following result. Theorem 2.4. Let ı 2 .0; 1, fak g1 kD0  .0; 1/ and let x 2 C

(2.5)

14

2 Subgradient Projection Algorithm

satisfies f .x /  f .x/ for all x 2 C:

(2.6)

1 Assume that fxt g1 t D 0  U, ft gt D 0  X,

kx0 k  M0 C 1

(2.7)

t 2 @f .xt / C BX .0; ı/

(2.8)

kxt C 1  PC .xt  at t /k  ı:

(2.9)

and that for each integer t  0,

and

Then for each natural number T, T X

at .f .xt /  f .x //

tD0

 21 kx  x0 k2 C ı.T C 1/.4M0 C 1/ C ı.2M0 C 1/

T X

at C 21 .L C 1/2

tD0

T X

a2t :

(2.10)

tD0

Moreover, for each natural number T, 0 f@

T X

!1 at

tD0

1

2

T X

1 at xt A  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x /

tD0 T X

!1 at

2

kx  x0 k C

tD0

T X

!1 at

ı.T C 1/.4M0 C 1/

tD0

1

C ı.2M0 C 1/ C 2

T X tD0

!1 at

.L C 1/2

T X

a2t :

(2.11)

tD0

Theorem 2.4 is proved in Sect. 2.4. We are interestedPin an optimal choice of at , t D 0; 1; : : : . Let T be a natural number and AT D TtD 0 at be given. By Theorem 2.4, in order to make the best choice of at ; t D 0; : : : ; T, we need to minimize the function 2 1 .a0 ; : : : ; aT / WD 21 A1 T kx  x0 k C AT ı.T C 1/.4M0 C 1/

2.2 A Convex Minimization Problem

15

2 C ı.2M0 C 1/ C 21 A1 T .L C 1/

T X

a2t

tD0

on the set ( a D .a0 ; : : : ; aT / 2 RTC1 W ai  0; i D 0; : : : ; T;

T X

) ai D AT :

iD0

By Lemma 2.3, this function has a unique minimizer a D .a0 ; : : : ; aT / where ai D .T C 1/1 AT , i D 0; : : : ; T. This is the best choice of at , t D 0; 1; : : : ; T. Theorem 2.4 implies the following result. Theorem 2.5. Let ı 2 .0; 1, a > 0 and let x 2 C satisfies f .x /  f .x/ for all x 2 C: 1 Assume that fxt g1 t D 0  U, ft gt D 0  X,

kx0 k  M0 C 1 and that for each integer t  0, t 2 @f .xt / C BX .0; ı/ and kxtC1  PC .xt  at /k  ı: Then for each natural number T, f

1

.T C 1/

T X

! xt  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x /

tD0

 21 .T C 1/1 a1 .2M0 C 1/2 C a1 ı.4M0 C 1/ C ı.2M0 C 1/ C 21 .L C 1/2 a: Now we will find the best a > 0. Since T can be arbitrary large, we need to find a minimizer of the function .a/ WD a1 ı.4M0 C 1/ C 21 .L C 1/2 a; a 2 .0; 1/:

16

2 Subgradient Projection Algorithm

Clearly, the minimizer a satisfies a1 ı.4M0 C 1/ D 21 .L C 1/2 a and a D .2ı.4M0 C 1//1=2 .L C 1/1 and the minimal value of is .2ı.4M0 C 1//1=2 .L C 1/:

(2.12)

Theorem 2.5 implies the following result. Theorem 2.6. Let ı 2 .0; 1, a D .2ı.4M0 C 1//1=2 .L C 1/1 ; x 2 C satisfies f .x /  f .x/ for all x 2 C: 1 Assume that fxt g1 t D 0  U, ft gt D 0  X,

kx0 k  M0 C 1 and that for each integer t  0, t 2 @f .xt / C BX .0; ı/ and kxtC1  PC .xt  at /k  ı: Then for each natural number T, f

1

.T C 1/

T X

! xt  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x /

tD0

 21 .T C 1/1 .2M0 C 1/2 .L C 1/.2ı.4M0 C 1//1=2 C ı.2M0 C 1/ C 21 .2ı.4M0 C 1//1=2 .L C 1/ C ı.4M0 C 1/.L C 1/.2ı.4M0 C 1//1=2 : Now we can think about the best choice of T. It is clear that it should be at the same order as bı 1 c. Putting T D bı 1 c, we obtain that

2.3 The Main Lemma

.T C 1/1

f

T X

17

! xt  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x /

tD0

 21 .2M0 C 1/2 .L C 1/.8M0 C 2/1=2 ı 1=2 C ı.2M0 C 1/ C .8M0 C 2/1=2 .L C 1/ı 1=2 C .4M0 C 1/.L C 1/.8M0 C 2/1=2 ı 1=2 : Note that in the theorems above ı is the computational error produced by our computer system. In view of the inequality above, which has the right-hand side bounded by c1 ı 1=2 with a constant c1 > 0, we conclude that after T D bı 1 c iterations we obtain a point  2 U such that BX .; ı/ \ C 6D ; and f ./  f .x / C c1 ı 1=2 ; where the constant c1 > 0 depends only on L and M0 .

2.3 The Main Lemma We use the notation and definitions introduced in Sect. 2.1. Lemma 2.7. Let ı 2 .0; 1, a > 0 and let z 2 C:

(2.13)

x 2 U \ BX .0; M0 C 1/;

(2.14)

 2 @f .x/ C BX .0; ı/

(2.15)

u2U

(2.16)

ku  PC .x  a/k  ı:

(2.17)

Assume that

and that

satisfies

18

2 Subgradient Projection Algorithm

Then a. f .x/  f .z//  21 kz  xk2  21 kz  uk2 C ı.4M0 C 1 C a.2M0 C 1// C 21 a2 .L C 1/2 : Proof. In view of (2.15), there exists l 2 @f .x/

(2.18)

kl  k  ı:

(2.19)

such that

By Lemmas 2.1 and 2.2 and (2.13), 0  hz  PC .x  a/; PC .x  a/  .x  a/i D hz  PC .x  a/; PC .x  a/  xi Cha; z  PC .x  a/i D 21 Œkz  xk2  kz  PC .x  a/k2  kx  PC .x  a/k2  Cha; z  xi C ha; x  PC .x  a/i:

(2.20)

Clearly, jha; x  PC .x  a/ij  21 .kak2 C kx  PC .x  a/k2 /:

(2.21)

It follows from (2.20) and (2.21) that 0  21 Œkz  xk2  kz  PC .x  a/k2  kx  PC .x  a/k2  Cha; z  xi C 21 a2 kk2 C 21 kx  PC .x  a/k2  21 kz  xk2  21 kz  PC .x  a/k2 C 21 a2 kk2 C ha; z  xi: (2.22) Relations (2.2), (2.13), and (2.17) imply that jkz  PC .x  a/k2  kz  uk2 j D jkz  PC .x  a/k  kz  ukj.kz  PC .x  a/k C kz  uk/  ku  PC .x  a/k.4M0 C 1/  .4M0 C 1/ı: By (2.2), (2.13), (2.14), and (2.19), ha; z  xi D hal; z  xi C ha.  l/; z  xi

(2.23)

2.4

Proof of Theorem 2.4

19

 hal; z  xi C ak  lkkz  xk  hal; z  xi C aı.2M0 C 1/:

(2.24)

It follows from (2.4), (2.18), (2.19), (2.22), (2.23), and (2.24) that 0  21 kz  xk2  21 kz  PC .x  a/k2 C 21 a2 kk2 C ha; z  xi  21 kz  xk2  21 kz  uk2 C ı.4M0 C 1/ C 21 a2 .L C 1/2 C hal; z  xi C aı.2M0 C 1/:

(2.25)

By (2.1), (2.18), and (2.25), a.f .z/  f .x//  hal; z  xi and a.f .x/  f .z//  hal; x  zi  21 kz  xk2  21 kz  uk2 C ı.4M0 C 1/ C 21 a2 .L C 1/2 C aı.2M0 C 1/: This completes the proof of Lemma 2.7.

2.4 Proof of Theorem 2.4 It is clear that kxt k  M0 C 1; t D 0; 1; : : : : Let t  0 be an integer. Applying Lemma 2.7 with z D x ; a D at ; x D xt ;  D t ; u D xtC1 we obtain that at .f .xt /  f .x //  21 kx  xt k2  21 kx  xtC1 k2 C ı.4M0 C 1 C at .2M0 C 1// C 21 a2t .L C 1/2 : (2.26)

20

2 Subgradient Projection Algorithm

By (2.26), for each natural number T, T X

at .f .xt /  f .x //

tD0



T X

.21 kx  xt k2  21 kx  xtC1 k2

tD0

C ı.4M0 C 1/ C at .2M0 C 1/ı C 21 a2t .L C 1/2 /  21 kx  x0 k2 C ı.T C 1/.4M0 C 1/ C ı.2M0 C 1/

T X

at C 21 .L C 1/2

tD0

T X

a2t :

tD0

Thus (2.10) is true. Evidently, (2.10) implies (2.11). Theorem 2.4 is proved.

2.5 Subgradient Algorithm on Unbounded Sets We use the notation and definitions introduced in Sect. 2.1. Let X be a Hilbert space with an inner product h; i, D be a nonempty closed convex subset of X, V be an open convex subset of X such that D  V;

(2.27)

and f W V ! R1 be a convex function which is Lipschitz on all bounded subsets of V. Set Dmin D fx 2 D W f .x/  f .y/ for all y 2 Dg:

(2.28)

Dmin 6D ;:

(2.29)

We suppose that

We will prove the following result. Theorem 2.8. Let ı 2 .0; 1, M > 0 satisfy Dmin \ BX .0; M/ 6D ;; M0  4M C 4;

(2.30) (2.31)

L > 0 satisfy jf .v1 /  f .v2 /j  Lkv1  v2 k for all v1 ; v2 2 V \ BX .0; M0 C 2/;

(2.32)

2.5 Subgradient Algorithm on Unbounded Sets

0 < 0  1  .L C 1/1 ;

21

(2.33)

0 D 201 ı.4M0 C 1/ C 2ı.2M0 C 1/ C 21 .L C 1/2 (2.34) and let n0 D b01 .2M C 2/2 01 c:

(2.35)

1 Assume that fxt g1 t D 0  V, ft gt D 0  X,

fat g1 t D 0  Œ0 ; 1 ; kx0 k  M

(2.36) (2.37)

and that for each integer t  0, t 2 @f .xt / C BX .0; ı/

(2.38)

kxtC1  PD .xt  at t /k  ı:

(2.39)

and

Then there exists an integer q 2 Œ1; n0 C 1 such that kxi k  3M C 2; i D 0; : : : ; q and f .xq /  f .x/ C 0 for all x 2 D: We are interested in the best choice of at ; t D 0; 1; : : : . Assume for simplicity that 1 D 0 . In order to meet our goal we need to minimize the function 2 1 ı.4M0 C 1/ C 2.L C 1/2 ;  2 .0; 1/: This function has a minimizer  D .ı.4M0 C 1//1=2 .L C 1/1 ; the minimal value of 0 is 2ı.2M0 C 1/ C 4.ı.4M0 C 1//1=2 .L C 1/ and n0 D bc where  D .2.ı.4M0 C 1//1=2 .L C 1/1 /1 .2M C 2/2 .2ı.2M0 C 1/

22

2 Subgradient Projection Algorithm

C 4.ı.4M0 C 1//1=2 .L C 1/1 /  ı 1=2 .4M0 C 1/1=2 .L C 1/.2M C 2/2 .4L C 4/1 .4M0 C 1/1=2 ı 1=2 D ı 1 .4M0 C 1/1 .2M C 2/2 41 : Note that in the theorem above ı is the computational error produced by our computer system. In view of the inequality above, in order to obtain a good approximate solution we need bc1 ı 1 c C 1 iterations, where c1 D 41 .4M0 C 1/1 .2M C 1/2 : As a result, we obtain a point  2 V such that BX .; ı/ \ D 6D ; and f ./  infff .x/ W x 2 Dg C c2 ı 1=2 ; where the constant c2 > 0 depends only on L and M0 .

2.6 Proof of Theorem 2.8 By (2.30) there exists z 2 Dmin \ BX .0; M/:

(2.40)

Assume that T is a natural number and that f .xt /  f .z/ > 0 ; t D 1; : : : ; T:

(2.41)

Lemma 2.2, (2.36), (2.37), (2.39), and (2.40) imply that kx1  zk  kx1  PD .x0  a0 0 /k C kPD .x0  a0 0 /  zk  ı C kx0  zk C a0 k0 k  1 C 2M C 1 k0 k:

(2.42)

In view of (2.32), (2.37), and (2.38), 0 2 @f .x0 / C BX .0; 1/  BX .0; L/ C 1; k0 k  L C 1:

(2.43)

2.6

Proof of Theorem 2.8

23

It follows from (2.33), (2.40), (2.42), and (2.43) that kx1  zk  2M C 2;

(2.44)

kx1 k  3M C 2:

(2.45)

Set U D V \ fv 2 X W kvk < M0 C 2g

(2.46)

C D D \ BX .0; M0 /:

(2.47)

and

By induction we show that for every integer t 2 Œ1; T, kxt  zk  2M C 2;

(2.48)

f .xt /  f .z/  .20 /1 .kz  xt k2  kz  xtC1 k2 / C 01 ı.4M0 C 1/ C ı.2M0 C 1/ C 21 1 .L C 1/2 :

(2.49)

In view of (2.44), (2.48) holds for t D 1. Assume that an integer t 2 Œ1; T and that (2.48) holds. It follows from (2.31), (2.40), (2.46), (2.47), and (2.48) that z 2 C  BX .0; M0 /; xt 2 U \ BX .0; M0 C 1/:

(2.50) (2.51)

Relation (2.39) implies that xtC1 2 V satisfies kxtC1  PD .xt  at t /k  1:

(2.52)

By (2.32), (2.38), and (2.51), t 2 @f .xt / C BX .0; 1/  BX .0; L C 1/:

(2.53)

It follows from (2.33), (2.36), (2.40), (2.48), (2.53), and Lemma 2.2 that kz  PD .xt  at t /k  kz  xt C at t k  kz  xt k C kt kat  2M C 3; kPD .xt  at t /k  3M C 3:

(2.54)

24

2 Subgradient Projection Algorithm

In view of (2.47) and (2.54), PD .xt  at t / 2 C;

(2.55)

PD .xt  at t / D PC .xt  at t /:

(2.56)

and

Relations (2.44), (2.52), and (2.54) imply that kxtC1 k  3M C 4; xtC1 2 U:

(2.57)

By (2.32), (2.38), (2.39), (2.46), (2.47), (2.50), (2.51), (2.55), (2.56), (2.57), and Lemma 2.7 which holds with x D xt ; a D at ;  D t ; u D xtC1 ; we have at .f .xt /  f .z//  21 kz  xt k2  21 kz  xtC1 k2 C ı.4M0 C 1 C at .2M0 C 1// C 21 a2t .L C 1/2 : The relation above, (2.34) and (2.36) imply that f .xt /  f .z/  .20 /1 kz  xt k2  .20 /1 kz  xtC1 k2 C 01 ı.4M0 C 1/ C .2M0 C 1/ı C 21 1 .L C 1/2 :

(2.58)

In view of (2.41), (2.58) and the inclusion t 2 Œ1; T, kz  xt k2  kz  xtC1 k2  0; kz  xtC1 k  kz  xt k  2M C 2:

(2.59)

Therefore we assumed that (2.48) is true and showed that (2.58) and (2.59) hold. Hence by induction we showed that (2.49) holds for all t D 1; : : : ; T and (2.48) holds for all t D 1; : : : ; T C 1. It follows from (2.49) which holds for all t D 1; : : : ; T, (2.41) and (2.44) that T0 < T.minff .xt / W t D 1; : : : ; Tg  f .z// 

T X tD1

.f .xt /  f .z//

2.7 Zero-Sum Games with Two-Players

 .20 /1

25

T X .kz  xt k2  kz  xtC1 k2 / tD1

C T01 ı.4M0

C 1/ C T.2M0 C 1/ı C 21 T1 .L C 1/2

 .20 /1 .2M C 2/2 C T01 ı.4M0 C 1/ C T.2M0 C 1/ı C 21 T1 .L C 1/2 : Together with (2.34) and (2.35) this implies that 0 < .20 T/1 .2M C 2/2 C 01 ı.4M0 C 1/ C .2M0 C 1/ı C 21 1 .L C 1/2 ; 21 0 < .20 T/1 .2M C 2/2 ; T < 01 .2M C 2/2 01  n0 C 1: Thus we have shown that if an integer T satisfies (2.41), then T  n0 and kz  xt k  2M C 2; t D 1; : : : ; T C 1; kxt k  3M C 2; t D 0; : : : ; T C 1: This implies that there exists an integer q 2 Œ1; n0 C 1 such that kxt k  3M C 2; t D 0; : : : ; q and f .xq /  f .z/  0 : Theorem 2.8 is proved.

2.7 Zero-Sum Games with Two-Players We use the notation and definitions introduced in Sect. 2.1. Let X; Y be Hilbert spaces, C be a nonempty closed convex subset of X, D be a nonempty closed convex subset of Y, U be an open convex subset of X, and V be an open convex subset of Y such that C  U; D  V

(2.60)

26

2 Subgradient Projection Algorithm

and let a function f W U  V ! R1 possess the following properties: (i) for each v 2 V, the function f .; v/ W U ! R1 is convex; (ii) for each u 2 U, the function f .u; / W V ! R1 is concave. Assume that a function W R1 ! Œ0; 1/ is bounded on all bounded sets and positive numbers M0 ; L satisfy C  BX .0; M0 /; D  BY .0; M0 /;

(2.61)

jf .u; v1 /  f .u; v2 /j  Lkv1  v2 k for all u 2 U and all v1 ; v2 2 V; (2.62) jf .u1 ; v/  f .u2 ; v/j  Lku1  u2 k for all v 2 V and all u1 ; u2 2 U: (2.63) Let x 2 C and y 2 D

(2.64)

f .x ; y/  f .x ; y /  f .x; y /

(2.65)

satisfy

for each x 2 C and each y 2 D. In the next section we prove the following result. Proposition 2.9. Let T be a natural number, ı 2 .0; 1, fat gTtD 0  .0; 1/ TC1 and let fbt gTtD 0  .0; 1/. Assume that fxt gTC1 t D 0  U, fyt gt D 0  V, for each t 2 f0; : : : ; T C 1g, B.xt ; ı/ \ C 6D ;; B.yt ; ı/ \ D 6D ;;

(2.66)

for each z 2 C and each t 2 f0; : : : ; Tg, at .f .xt ; yt /  f .z; yt //  .kz  xt k/  .kz  xtC1 k/ C bt

(2.67)

and that for each v 2 D and each t 2 f0; : : : ; Tg, at .f .xt ; v/  f .xt ; yt //  .kv  yt k/  .kv  ytC1 k/ C bt :

(2.68)

Let xO T D

T X

!1 ai

iD0

yO T D

T X iD0

!1 ai

T X

at xt ;

tD0 T X tD0

at yt :

(2.69)

2.7 Zero-Sum Games with Two-Players

27

Then B.OxT ; ı/ \ C 6D ;; B.OyT ; ı/ \ D 6D ;;

(2.70)

ˇ ˇ !1 T ˇ X ˇ X ˇ T ˇ ˇ at at f .xt ; yt /  f .x ; y /ˇˇ ˇ ˇ tD0 ˇ tD0 

T X

!1 at

tD0

T X

bt C

tD0

T X

!1 at

supf .u/ W u 2 Œ0; 2M0 C 1g; (2.71)

tD0

ˇ ˇ !1 T ˇ ˇ T X X ˇ ˇ ˇf .OxT ; yO T /  at at f .xt ; yt /ˇˇ ˇ ˇ ˇ tD0 tD0 

T X

!1 at

tD0

C

T X

T X

bt C Lı

tD0

!1 at

supf .u/ W u 2 Œ0; 2M0 C 1g;

(2.72)

tD0

and for each z 2 C and each v 2 D, f .z; yO T /  f .OxT ; yO T / 2

T X

!1

2

T X

supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0

!1 at

tD0

C2

C2

tD0

(2.73)

!1 supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0 T X

bt  Lı;

tD0

f .OxT ; v/  f .OxT ; yO T / T X

T X

!1 at

T X

bt C Lı:

(2.74)

tD0

Corollary 2.10. Suppose that all the assumptions of Proposition 2.9 hold and that xQ 2 C; yQ 2 D satisfy kOxT  xQ k  ı; kOyT  yQ k  ı:

(2.75)

28

2 Subgradient Projection Algorithm

Then jf .Qx; yQ /  f .OxT ; yO T /j  2Lı and for each z 2 C and each v 2 D, f .z; yQ /  f .Qx; yQ / 2

T X

!1

2

T X

supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0

!1

tD0

C2

!1

C2

supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0 T X

bt  4Lı;

tD0

f .Qx; v/  f .Qx; yQ / T X

T X

at

!1 at

tD0

T X

bt C 4Lı:

tD0

Proof. In view of (2.62), (2.63), and (2.75), jf .Qx; yQ /  f .OxT ; yO T /j  jf .Qx; yQ /  f .Qx; yO T /j C jf .Qx; yO T /  f .OxT ; yO T /j  LkQy  yO T k C LkQx  xO T k  2Lı and (2.76) holds. Let z 2 C and v 2 D. Relations (2.62), (2.63), and (2.75) imply that jf .z; yQ /  f .z; yO T /j  Lı; jf .Qx; v/  f .OxT ; v/j  Lı: By the relation above, (2.73), (2.74), and (2.75), f .z; yQ /  f .z; yO T /  Lı  f .OxT ; yO T /  2

T X

!1 at

supf .s/ W s 2 Œ0; 2M0 C 1g

tD0

2

T X tD0

!1 at

T X tD0

bt  2Lı

(2.76)

2.8

Proof of Proposition 2.9

29

!1

T X

 f .Qx; yQ /  2

supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0

2

T X

!1 at

tD0

T X

bt  4Lı;

tD0

f .Qx; v/  f .OxT ; v/ C Lı  f .OxT ; yO T / C 2

T X

!1 supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0

C2

T X

!1 at

tD0

T X

bt C 2Lı

tD0

 f .Qx; yQ / C 2

T X

!1

at

supf .s/ W s 2 Œ0; 2M0 C 1g

tD0

C2

T X tD0

!1 at

T X

bt C 4Lı:

tD0

This completes the proof of Corollary 2.10.

2.8 Proof of Proposition 2.9 It is clear that (2.70) is true. Let t 2 f0; : : : ; Tg. By (2.65), (2.67), and (2.68), at .f .xt ; yt /  f .x ; y //  at .f .xt ; yt /  f .x ; yt //  .kx  xt k/  .kx  xtC1 k/ C bt ;

(2.77)

at .f .x ; y /  f .xt ; yt //  at .f .xt ; y /  f .xt ; yt //  .ky  yt k/  .ky  ytC1 k/ C bt :

(2.78)

30

2 Subgradient Projection Algorithm

In view of (2.77) and (2.78), T X

at f .xt ; yt / 

tD0

T X

at f .x ; y /

tD0



T X

. .kx  xt k/  .kx  xtC1 k// C

tD0

T X

bt

tD0

 .kx  x0 k/ C

T X

bt ;

(2.79)

tD0 T X

at f .x ; y / 

tD0

T X

at f .xt ; yt /

tD0



T X

. .ky  yt k/  .ky  ytC1 k// C

tD0

T X

bt

tD0

 .ky  y0 k/ C

T X

bt :

(2.80)

tD0

Relations (2.61), (2.64), (2.66), (2.79), and (2.80) imply that ˇ ˇ ˇ ˇ T !1 T X ˇ ˇ X ˇ ˇ a a f .x ; y /  f .x ; y / t t t t   ˇ ˇ ˇ ˇ tD0 tD0 

T X

!1 at

tD0

T X tD0

bt C

T X

!1 at

supf .s/ W s 2 Œ0; 2M0 C 1g: (2.81)

tD0

By (2.70), there exists zT 2 C

(2.82)

kzT  xO T k  ı:

(2.83)

such that

In view of (2.82), we apply (2.67) with z D zT and obtain that for all t D 0; : : : ; T, at .f .xt ; yt /  f .zT ; yt //  .kzT  xt k/  .kzT  xtC1 k/ C bt :

(2.84)

2.8

Proof of Proposition 2.9

31

It follows from (2.63) and (2.83) that for all t D 0; : : : ; T, jf .zT ; yt /  f .OxT ; yt /j  LkzT  xO T k  Lı:

(2.85)

By (2.84) and (2.85), for all t D 0; : : : ; T, at .f .xt ; yt /  f .OxT ; yt //  at .f .xt ; yt /  f .zT ; yt // C at Lı  .kzT  xt k/  .kzT  xtC1 k/ C bt C at Lı:

(2.86)

Combined with (2.61), (2.66), and (2.82) this implies that T X

at f .xt ; yt / 

tD0



T X

at f .OxT ; yt /

tD0 T X

. .kzT  xt k/  .kzT  xtC1 k// C

tD0

T X

bt C

tD0

 .kzT  x0 k/ C

T X

bt C

tD0

T X

T X

at Lı

tD0

at Lı

tD0

 supf .s/ W s 2 Œ0; 2M0 C 1g C

T X

bt C

tD0

T X

at Lı:

(2.87)

tD0

Property (ii) and (2.69) imply that T X

at f .OxT ; yt / D

tD0

T X

! ai

iD0



T X

!

T X

0 @at

tD0

T X

1

!1 ai

f .OxT ; yt /A

iD0

at f .OxT ; yO T /:

(2.88)

tD0

By (2.87) and (2.88), T X

at f .xt ; yt / 

tD0



T X

at f .OxT ; yO T /

tD0 T X tD0

at f .xt ; yt / 

T X

at f .OxT ; yt /

tD0

 supf .s/ W s 2 Œ0; 2M0 C 1g C

T X tD0

bt C

T X tD0

at Lı:

(2.89)

32

2 Subgradient Projection Algorithm

By (2.70), there exists hT 2 D

(2.90)

khT  yO T k  ı:

(2.91)

such that

In view of (2.90), we apply (2.68) with v D hT and obtain that for all t D 0; : : : ; T, at .f .xt ; hT /  f .xt ; yt //  .khT  yt k/  .khT  ytC1 k/ C bt :

(2.92)

It follows from (2.62) and (2.91) that for all t D 0; : : : ; T, jf .xt ; hT /  f .xt ; yO T /j  LkhT  yO T k  Lı:

(2.93)

By (2.92) and (2.93), for all t D 0; : : : ; T, at .f .xt ; yO T /  f .xt ; yt //  at .f .xt ; hT /  f .xt ; yt // C at Lı  .khT  yt k/  .khT  ytC1 k/ C bt C at Lı:

(2.94)

In view of (2.94), T X

at f .xt ; yO T / 

tD0



T X

at f .xt ; yt /

tD0 T X

. .khT  yt k/  .khT  ytC1 k// C

tD0

T X tD0

bt C

T X

at Lı:

(2.95)

tD0

Property (i) and (2.69) imply that T X

at f .xt ; yO T / D

tD0

T X iD0



T X tD0

! ai

T X

0 @at

tD0

at f .OxT ; yO T /:

T X

!1 ai

1 f .xt ; yO T /A

iD0

(2.96)

2.8

Proof of Proposition 2.9

33

By (2.61), (2.66), (2.90), (2.95), and (2.96), T X

at f .OxT ; yO T / 

tD0

T X

at f .xt ; yt /

tD0



T X

T X

at f .xt ; yO T / 

tD0

at f .xt ; yt /

tD0

 .khT  y0 k/ C

T X

bt C

tD0

T X

at Lı

tD0

 supf .s/ W s 2 Œ0; 2M0 C 1g C

T X

bt C

tD0

T X

at Lı:

(2.97)

tD0

It follows from (2.89) and (2.97) that ˇ T ˇ T ˇX ˇ X ˇ ˇ at f .OxT ; yO T /  at f .xt ; yt /ˇ ˇ ˇ ˇ tD0

tD0

 supf .s/ W s 2 Œ0; 2M0 C 1g C

T X

bt C

tD0

T X

at Lı:

tD0

This implies (2.72). Let z 2 C. By (2.67), T X

at f .xt ; yt /  f .z; yt //

tD0



T X

Π.kz  xt k/  .kz  xtC1 k/ C

tD0

T X

bt :

(2.98)

tD0

By property (ii) and (2.69), T X

at f .z; yt / D

tD0

T X

! ai

iD0



T X tD0

!

T X

0 @at

tD0

at f .z; yO T /:

T X

!1 ai

1 f .z; yt /A

iD0

(2.99)

34

2 Subgradient Projection Algorithm

In view of (2.98) and (2.99), T X

at f .xt ; yt / 

tD0

T X

at f .z; yO T /

tD0



T X

at .f .xt ; yt /  f .z; yt //

tD0



T X

Π.kz  xt k/  .kz  xtC1 k/ C

tD0

T X

bt

tD0

 .kz  x0 k/ C

T X

bt :

(2.100)

tD0

It follows from (2.61), (2.70), and (2.72) that f .z; yO T / 

T X

!1

T X

ai

T X



at f .xt ; yt /

tD0

iD0

!1

supf .s/ W s 2 Œ0; 2M0 C 1g 

ai

iD0

T X

!1 ai

 f .OxT ; yO T /  2

bt

tD0

iD0 T X

T X

!1 supf .s/ W s 2 Œ0; 2M0 C 1g

at

tD0

2

T X

!1 at

tD0

T X

bt  Lı

tD0

and (2.73) holds. Let v 2 D. By (2.68), T X

at .f .xt ; v/  f .xt ; yt //

tD0



T X

Π.kv  yt k/  .kv  ytC1 k/ C

tD0

T X

bt :

(2.101)

tD0

By property (i) and (2.69), T X tD0

at f .xt ; v/ D

T X iD0

! ai

T X tD0

0 @at

T X iD0

!1 ai

1 f .xt ; v/A

2.9 Subgradient Algorithm for Zero-Sum Games T X



35

! at f .OxT ; v/:

(2.102)

tD0

In view of (2.101) and (2.102), T X

at f .OxT ; v/ 

tD0

T X

at f .xt ; yt /

tD0

 .kv  y0 k/ C

T X

bt :

tD0

Together with (2.61), (2.66), and (2.72) this implies that f .OxT ; v/ 

T X

!1 at

tD0

C

T X

at f .xt ; yt /

tD0

T X

!1

supf .s/ W s 2 Œ0; 2M0 C 1g C

at

tD0

 f .OxT ; yO T / C 2

T X

!1 at

tD0

!1 at

T X

T X

bt

tD0

supf .s/ W s 2 Œ0; 2M0 C 1g

tD0

C2

T X tD0

!1 at

T X

bt C Lı:

tD0

Therefore (2.74) holds. This completes the proof of Proposition 2.9.

2.9 Subgradient Algorithm for Zero-Sum Games We use the notation and definitions introduced in Sect. 2.1. Let X; Y be Hilbert spaces, C be a nonempty closed convex subset of X, D be a nonempty closed convex subset of Y, U be an open convex subset of X, and V be an open convex subset of Y such that C  U; D  V:

(2.103)

For each concave function g W V ! R1 and each x 2 V set @g.x/ D fl 2 Y W hl; y  xi  g.y/  g.x/ for all y 2 Vg:

(2.104)

36

2 Subgradient Projection Algorithm

Clearly, for each x 2 V, @g.x/ D .@.g/.x//:

(2.105)

Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /; D  BY .0; M0 /;

(2.106)

a function f W U  V ! R1 possesses the following properties: (i) for each v 2 V, the function f .; v/ W U ! R1 is convex; (ii) for each u 2 U, the function f .u; / W V ! R1 is concave, for each v 2 V, jf .u1 ; v/  f .u2 ; v/j  Lku1  u2 k for all u1 ; u2 2 U

(2.107)

and that for each u 2 U, jf .u; v1 /  f .u; v2 /j  Lkv1  v2 k for all v1 ; v2 2 V:

(2.108)

For each .; / 2 U  V, set @x f .; / D fl 2 X W f .y; /  f .; /  hl; y  i for all y 2 Ug; (2.109) @y f .; / D fl 2 Y W hl; y  i  f .; y/  f .; / for all y 2 Vg:

(2.110)

In view of properties (i) and (ii) and (2.107)–(2.110), for each  2 U and each

2 V, ; 6D @x f .; /  BX .0; L/;

(2.111)

; 6D @y f .; /  BY .0; L/:

(2.112)

Let x 2 C and y 2 D satisfy f .x ; y/  f .x ; y /  f .x; y / for each x 2 C and each y 2 D. Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. Let us describe our algorithm.

(2.113)

2.9 Subgradient Algorithm for Zero-Sum Games

37

Subgradient Projection Algorithm for Zero-Sum Games Initialization: select arbitrary x0 2 U and y0 2 V. Iterative step: given current iteration vectors xt 2 U and yt 2 V calculate t 2 @x f .xt ; yt / C BX .0; ı/;

t 2 @y f .xt ; yt / C BY .0; ı/ and the next pair of iteration vectors xtC1 2 U, ytC1 2 V such that kxtC1  PC .xt  at t /k  ı; kytC1  PD .yt C at t /k  ı: In this chapter we prove the following result. 1 Theorem 2.11. Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. Assume that fxt gt D 0  U, 1 1 1 fyt gt D 0  V, ft gt D 0  X, f t gt D 0  Y,

BX .x0 ; ı/ \ C 6D ;; BY .y0 ; ı/ \ D 6D ;

(2.114)

and that for each integer t  0, t 2 @x f .xt ; yt / C BX .0; ı/;

(2.115)

t 2 @y f .xt ; yt / C BY .0; ı/;

(2.116)

kxtC1  PC .xt  at t /k  ı

(2.117)

kytC1  PD .yt C at t /k  ı:

(2.118)

and that

Let for each natural number T, xO T D

T X iD0

!1 at

T X

at xt ; yO T D

tD0

T X

!1 at

iD0

T X

at yt :

(2.119)

tD0

Then for each natural number T, ˇ ˇ !1 T ˇ ˇ X X ˇ ˇ T ˇ at at f .xt ; yt /  f .x ; y /ˇˇ ˇ ˇ ˇ tD0 tD0  Œ21 .2M0 C 1/2 C ı.T C 1/.4M0 C 1/

T X tD0

!1 at

38

2 Subgradient Projection Algorithm T X

1

C ı.2M0 C 1/ C 2

!1 .L C 1/2

at

tD0

T X

2

 Œ2 .2M0 C 1/ C ı.T C 1/.4M0 C 1/ T X

1

C ı.2M0 C 1/ C 2

(2.120)

!1 at

tD0

!1 at

a2t ;

tD0

ˇ ˇ !1 T ˇ ˇ T X X ˇ ˇ ˇf .OxT ; yO T /  ˇ a a f .x ; y / t t t t ˇ ˇ ˇ ˇ tD0 tD0 1

T X

.L C 1/2

tD0

T X

a2t C Lı;

(2.121)

tD0

and for each natural number T, each z 2 C, and each u 2 D, f .z; yO T /  f .OxT ; yO T / T X

 .2M0 C 1/2

!1 2

at

tD0

 2ı.2M0 C 1/ 

T X

T X

.L C 1/2

tD0

f .OxT ; v/  f .OxT ; yO T / C .2M0 C 1/

T X

2

C2

at

tD0

C 2ı.2M0 C 1/ C

tD0

T X

a2t  Lı;

tD0

!1

T X

.T C 1/ı.4M0 C 1/

at

tD0

!1 at

!1

!1 at

T X

!1 at

ı.T C 1/.4M0 C 1/

tD0

.L C 1/2

T X

a2t C Lı:

tD0

We are interestedPin the optimal choice of at , t D 0; 1; : : : ; T. Let T be a natural number and AT D TtD 0 at be given. By Theorem 2.11, in order to make the best P choice of at ; t D 0; : : : ; T, we need to minimize the function TtD 0 a2t on the set ( a D .a0 ; : : : ; aT / 2 R

TC1

W ai  0; i D 0; : : : ; T;

T X

) ai D AT :

iD0

By Lemma 2.3, this function has a unique minimizer a D .a0 ; : : : ; aT / where ai D .T C 1/1 AT , i D 0; : : : ; T which is the best choice of at , t D 0; 1; : : : ; T. Now we will find the best a > 0. Let T be a natural number and at D a for all t D 0; : : : ; T. We need to choose a which is a minimizer of the function

2.10

Proof of Theorem 2.11

39

T .a/ D ..T C 1/a/1 Œ.2M0 C 1/2 C 2ı.T C 1/.4M0 C 1/ C 2ı.2M0 C 1/ C a.L C 1/2 D .2M0 C 1/2 ..T C 1/a/1 C 2ı.4M0 C 1/a1 C 2ı.2M0 C 1/ C .L C 1/2 a:

Since T can be arbitrary large, we need to find a minimizer of the function .a/ WD 2a1 ı.4M0 C 1/ C .L C 1/2 a; a 2 .0; 1/: In Sect. 2.2 we have already shown that the minimizer is a D .2ı.4M0 C 1//1=2 .L C 1/1 and the minimal value of is .8ı.4M0 C 1//1=2 .L C 1/: Now our goal is to find the best integer T > 0 which gives us an appropriate value of T .a/. Since in view of the inequalities above, this value is bounded from below by c0 ı 1=2 with the constant c0 depending on L; M0 , it is clear that in order to make the best choice of T, it should be at the same order as bı 1 c. For example, T D bı 1 c. Note that in the theorem above ı is the computational error produced by our computer system. We obtain a good approximate solution after T D bı 1 c iterations. Namely, we obtain a pair of points xO 2 U, yO 2 V such that BX .Ox; ı/ \ C 6D ;; BY .Oy; ı/ \ D 6D ; and for each z 2 C and each v 2 D, f .z; yO /  f .Ox; yO /  cı 1=2 ; f .Ox; v/  f .Ox; yO / C cı 1=2 ; where the constant c > 0 depends only on L and M0 .

2.10 Proof of Theorem 2.11 By (2.106), (2.114), (2.117), and (2.118), for all integers t  0, kxt k  M0 C 1; kyt k  M0 C 1: Let t  0 be an integer. Applying Lemma 2.7 with a D at ; x D xt ; f D f .; yt /;  D t ; u D xtC1

(2.122)

40

2 Subgradient Projection Algorithm

we obtain that for each z 2 C, at .f .xt ; yt /  f .z; yt //  21 kz  xt k2  21 kz  xtC1 k2 C ı.4M0 C 1 C at .2M0 C 1// C 21 a2t .L C 1/2 : (2.123) Applying Lemma 2.7 with a D at ; x D yt ; f D f .xt ; /;  D  t ; u D ytC1 we obtain that for each v 2 D, at .f .xt ; v/  f .xt ; yt //  21 kv  yt k2  21 kv  ytC1 k2 C ı.4M0 C 1 C at .2M0 C 1// C 21 a2t .L C 1/2 : (2.124) For all integers t  0 set bt D ı.4M0 C 1 C at .2M0 C 1// C 21 a2t .L C 1/2 and define .s/ D 21 s2 ; s 2 R1 : It is easy to see that all the assumptions of Proposition 2.9 hold and it implies Theorem 2.11.

Chapter 3

The Mirror Descent Algorithm

In this chapter we analyze the convergence of the mirror descent algorithm under the presence of computational errors. We show that the algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

3.1 Optimization on Bounded Sets Let X be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: For each x 2 X and each nonempty set E  X set d.x; E/ D inffkx  yk W y 2 Eg: Let C be a nonempty closed convex subset of X, U be an open convex subset of X such that C  U and let f W U ! R1 be a convex function. Recall that for each x 2 U, @f .x/ D fl 2 X W f .y/  f .x/  hl; y  xi for all y 2 Ug:

(3.1)

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_3

41

42

3 The Mirror Descent Algorithm

Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /; jf .x/  f .y/j  Lkx  yk for all x; y 2 U:

(3.2) (3.3)

In view of (3.1) and (3.3), for each x 2 U, ; 6D @f .x/  BX .0; L/:

(3.4)

For each nonempty set D  X and each function h W D ! R1 put inf.h; D/ D inffh.y/ W y 2 Dg and argminfh.y/ W y 2 Dg D fy 2 D W h.y/ D inf.hI D/g: We study the convergence of the mirror descent algorithm under the presence of computational errors. This method was introduced by Nemirovsky and Yudin for solving convex optimization problems [90]. Here we use a derivation of this algorithm proposed by Beck and Teboulle [19]. Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. We describe the inexact version of the mirror descent algorithm. Mirror Descent Algorithm Initialization: select an arbitrary x0 2 U. Iterative step: given a current iteration vector xt 2 U calculate t 2 @f .xt / C BX .0; ı/; define gt .x/ D ht ; xi C .2at /1 kx  xt k2 ; x 2 X and calculate the next iteration vector xtC1 2 U such that BX .xtC1 ; ı/ \ argminfgt .y/ W y 2 Cg 6D ;: Note that gt is a convex bounded from below function on X which possesses a minimizer on C. In this chapter we prove the following result. Theorem 3.1. Let ı 2 .0; 1, fak g1 kD0  .0; 1/ and let x 2 C

(3.5)

3.1 Optimization on Bounded Sets

43

satisfies f .x /  f .x/ for all x 2 C:

(3.6)

1 Assume that fxt g1 tD0  U, ft gtD0  X,

kx0 k  M0 C 1

(3.7)

t 2 @f .xt / C BX .0; ı/

(3.8)

and that for each integer t  0,

and BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg 6D ;:

(3.9)

Then for each natural number T, T X

at .f .xt /  f .x //

tD0

 21 .2M0 C 1/2 C ı.2M0 C L C 2/

T X

at

tD0

C ı.T C 1/.8M0 C 8/ C 21 .L C 1/2

T X

a2t :

(3.10)

tD0

Moreover, for each natural number T, 1 0 !1 T T X X at at xt A  f .x /; minff .xt / W t D 0; : : : ; Tg  f .x / f@ tD0

1

tD0

 2 .2M0 C 1/

2

T X

!1 C ı.2M0 C L C 2/

at

tD0

C ı.T C 1/.8M0 C 8/

T X tD0

!1 at

1

C 2 .L C 1/

2

T X tD0

a2t

T X

!1 at

:

tD0

Theorem 3.1 is proved in Sect. 3.3. We are interestedPin an optimal choice of at , t D 0; 1; : : : . Let T be a natural number and AT D TtD0 at be given. By Theorem 3.1, in order to make the best P choice of at ; t D 0; : : : ; T, we need to minimize the function TtD0 a2t on the set ) ( T X ai D AT : a D .a0 ; : : : ; aT / 2 RTC1 W ai  0; i D 0; : : : ; T; iD0

44

3 The Mirror Descent Algorithm

By Lemma 2.3, this function has a unique minimizer a D .a0 ; : : : ; aT / where ai D .T C 1/1 AT , i D 0; : : : ; T. This is the best choice of at , t D 0; 1; : : : ; T. Let T be a natural number and at D a, t D 0; : : : ; T. Now we will find the best a > 0. By Theorem 3.1, we need to choose a which is a minimizer of the function 21 ..T C 1/a/1 .2M0 C 1/2 C ı.2M0 C L C 2/ Ca1 ı.8M0 C 8/ C 21 .L C 1/2 a: Since T can be arbitrary large, we need to find a minimizer of the function .a/ WD a1 ı.16M0 C 16/ C .L C 1/2 a; a 2 .0; 1/: Clearly, the minimizer is a D .16ı.M0 C 1//1=2 .L C 1/1 and the minimal value of is 2.ı.16M0 C 16//1=2 .L C 1/: Now we can think about the best choice of T. It is clear that it should be at the same order as bı 1 c. Note that in the theorem above ı is the computational error produced by our computer system. In order to obtain a good approximate solution we need T iterations which is at the same order as bı 1 c. As a result, we obtain a point  2 U such that BX .; ı/ \ C 6D ; and f ./  f .x / C c1 ı 1=2 ; where the constant c1 > 0 depends only on L and M0 .

3.2 The Main Lemma We use the notation and definitions introduced in Sect. 3.1. Lemma 3.2. Let ı 2 .0; 1, a > 0 and let z 2 C:

(3.11)

3.2 The Main Lemma

45

Assume that x 2 U \ BX .0; M0 C 1/;

(3.12)

 2 @f .x/ C BX .0; ı/; 1

(3.13) 2

g.v/ D h; vi C .2a/ kv  xk ; v 2 X

(3.14)

u2U

(3.15)

BX .u; ı/ \ fv 2 C W g.v/ D inf.gI C/g 6D ;:

(3.16)

and that

satisfies

Then a.f .x/  f .z//  ıa.2M0 C L C 2/ C 8ı.M0 C 1/ C21 a2 .L C 1/2 C 21 kz  xk2  21 kz  uk2 : Proof. In view of (3.13), there exists l 2 @f .x/

(3.17)

kl  k  ı:

(3.18)

such that

Clearly, the function g is Frechet differentiable on X. We denote by g0 .v/ its Frechet derivative at v 2 X. It is easy to see that g0 .v/ D  C a1 .v  x/; v 2 X:

(3.19)

uO 2 BX .u; ı/ \ C

(3.20)

g.Ou/ D inf.gI C/:

(3.21)

By (3.16), there exists

such that

46

3 The Mirror Descent Algorithm

It follows from (3.19) and (3.21) that 0  hg0 .Ou/; v  uO i D h C a1 .Ou  x/; v  uO i:

(3.22)

By (3.2), (3.11), (3.12), (3.17), and (3.18), a.f .x/  f .z//  ahx  z; li D ahx  z; i C ahx  z; l  i  ahx  z; i C akx  zkkl  k  ıa.2M0 C 1/ C ahx  z; i D ıa.2M0 C 1/ C ah; x  uO i C ah; uO  zi D ıa.2M0 C 1/ C hx  uO ; ai Chz  uO ; x  uO  ai C hz  uO ; uO  xi:

(3.23)

Relations (3.11) and (3.22) imply that hz  uO ; x  uO  ai  0:

(3.24)

In view of Lemma 2.1, hz  uO ; uO  xi D 21 Œkz  xk2  kz  uO k2  kOu  xk2 :

(3.25)

It follows from (3.4), (3.13), and (3.20) that hx  uO ; ai D hx  u; ai C hu  uO ; ai  aı.L C 1/ C hx  u; ai  aı.L C 1/ C 21 kx  uk2 C 21 a2 kk2 :

(3.26)

By (3.4), (3.13), and (3.23)–(3.26), a.f .x/  f .z//  ıa.2M0 C 1/ C aı.L C 1/ C21 kx  uk2 C 21 a2 kk2 C21 kz  xk2  21 kz  uO k2  21 kOu  xk2  ıa.2M0 C L C 2/ C 21 a2 .L C 1/2 C21 kz  xk2  21 kz  uO k2 C21 kx  uk2  21 kOu  xk2 :

(3.27)

3.3

Proof of Theorem 3.1

47

In view of (3.2), (3.12), (3.16), and (3.20), jkx  uk2  kOu  xk2 j  jkx  uk  kOu  xkj.kx  uk C kOu  xk/  4ku  uO k.M0 C 1/  4.M0 C 1/ı:

(3.28)

Relations (3.2), (3.11), (3.12), and (3.20) imply that jkz  uO k2  kz  uk2 j  jkz  uO k  kz  ukj.kz  uO k C kz  uk/  ku  uO k.4M0 C 1/  .4M0 C 1/ı:

(3.29)

By (3.27), (3.28), and (3.29), a.f .x/  f .z//  ıa.2M0 C L C 2/ C C21 a2 .L C 1/2 C21 kz  xk2  21 kz  uk2 C 8.M0 C 1/ı: This completes the proof of Lemma 3.2.

3.3 Proof of Theorem 3.1 It is clear that kxt k  M0 C 1; t D 0; 1; : : : : Let t  0 be an integer. Applying Lemma 3.2 with z D x ; a D at ; x D xt ;  D t ; u D xtC1 we obtain that at .f .xt /  f .x //  21 kx  xt k2  21 kx  xtC1 k2 Cı.8.M0 C 1/ C at .2M0 C L C 2// C 21 a2t .L C 1/2 : (3.30) By (3.2), (3.5), (3.7), and (3.30), for each natural number T, T X

at .f .xt /  f .x //

tD0



T X tD0

.21 kx  xt k2  21 kx  xtC1 k2 /

48

3 The Mirror Descent Algorithm

Cı.2M0 C L C 2/

T X

at C .T C 1/ı.8M0 C 8/ C 21 .L C 1/2

tD0

T X

a2t

tD0

 21 .2M0 C 1/2 C ı.2M0 C L C 2/

T X

at

tD0

C.T C 1/ı.8M0 C 8/ C 21 .L C 1/2

T X

a2t :

tD0

Thus (3.10) is true. Evidently, (3.10) implies the last relation of the statement of Theorem 3.1. This completes the proof of Theorem 3.1

3.4 Optimization on Unbounded Sets We use the notation and definitions introduced in Sect. 3.1. Let X be a Hilbert space with an inner product h; i, D be a nonempty closed convex subset of X, V be an open convex subset of X such that D  V;

(3.31)

and f W V ! R1 be a convex function which is Lipschitz on all bounded subsets of V. Set Dmin D fx 2 D W f .x/  f .y/ for all y 2 Dg:

(3.32)

We suppose that Dmin 6D ;: We will prove the following result. Theorem 3.3. Let ı 2 .0; 1, M > 1 satisfy Dmin \ BX .0; M/ 6D ;; M0 > 80M C 6;

(3.33) (3.34)

L > 1 satisfy jf .v1 /  f .v2 /j  Lkv1  v2 k for all v1 ; v2 2 V \ BX .0; M0 C 2/;

(3.35)

0 < 0  1  .4L C 4/1 ;

(3.36)

0 D 1601 ı.M0 C 1/ C 4ı.2M0 C L C 2/ C 1 .L C 1/2

(3.37)

3.5 Proof of Theorem 3.3

49

and let n0 D b01 .2M C 1/2 01 c C 1:

(3.38)

1 Assume that fxt g1 tD0  V, ft gtD0  X,

fat g1 tD0  Œ0 ; 1 ; kx0 k  M

(3.39)

and that for each integer t  0, t 2 @f .xt / C BX .0; ı/

(3.40)

and BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg 6D ;:

(3.41)

Then there exists an integer q 2 Œ1; n0  such that f .xq /  inf.f I D/ C 0 ; kxq k  15M C 1: We are interested in the best choice of at ; t D 0; 1; : : : . Assume for simplicity that 1 D 0 . In order to meet our goal we need to minimize 0 which obtains its minimal value when 0 D .4ı.M0 C 1//1=2 .L C 1/1 and the minimal value of 0 is 4ı.2M0 C L C 2/ C 4.4ı.M0 C 1//1=2 .L C 1/; Thus 0 is at the same order as bı 1=2 c. By (3.38) and the inequalities above, n0 is at the same order as bı 1 c.

3.5 Proof of Theorem 3.3 By (3.33) there exists z 2 Dmin \ BX .0; M/:

(3.42)

Assume that T is a natural number and that f .xt /  f .z/ > 0 ; t D 1; : : : ; T:

(3.43)

50

3 The Mirror Descent Algorithm

In view of (3.41), there exists

2 BX .x1 ; ı/ \ argminfh0 ; vi C .2a0 /1 kv  x0 k2 W v 2 Dg:

(3.44)

Relations (3.42) and (3.44) imply that h0 ; i C .2a0 /1 k  x0 k2  h0 ; zi C .2a0 /1 kz  x0 k2 :

(3.45)

It follows from (3.34), (3.35), (3.39), and (3.40) that k0 k  L C 1:

(3.46)

1 a1 0  1  4.L C 1/:

(3.47)

In view of (3.36),

By (3.35), (3.39), (3.40), (3.42), and (3.45)–(3.47), .L C 1/M C .2a0 /1 .2M C 1/2  h0 ; zi C .2a0 /1 kz  x0 k2  .2a0 /1 k  x0 k2 C h0 ;  x0 i C h0 ; x0 i  .2a0 /1 k  x0 k2  .L C 1/k  x0 k  .L C 1/M: Together with (3.36) this implies that M C .2M C 1/2  k  x0 k2  21 k  x0 k; .k  x0 k  41 /2  .4M C 1/2 ; k  x0 k  8M: Together with (3.39) and (3.44) this implies that k k  9M; kx1 k  9M C 1; k  zk  10M; kx1  zk  10M C 1:

(3.48)

By induction we show that for every integer t 2 Œ1; T, kxt  zk  14M C 1; f .xt /  f .z/  ı.2M0 C L C 2/ C

(3.49) 801 ı.M0

1

C 1/ C 2 1 .L C 1/

C.20 /1 .kz  xt k2  kz  xtC1 k2 /:

2

(3.50)

3.5 Proof of Theorem 3.3

51

Set U D V \ fv 2 X W kvk < M0 C 2g

(3.51)

C D D \ BX .0; M0 /:

(3.52)

and

In view of (3.48), (3.49) holds for t D 1. Assume that an integer t 2 Œ1; T and that (3.49) holds. It follows from (3.34), (3.42) and (3.52) that z 2 C  BX .0; M0 /:

(3.53)

In view of (3.34), (3.42), and (3.49), xt 2 U \ BX .0; M0 C 1/:

(3.54)

Relations (3.35), (3.40), and (3.54) imply that t 2 @f .xt / C BX .0; ı/  BX .0; L C 1/:

(3.55)

In view of (3.41), there exists h 2 BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Dg:

(3.56)

By (3.42) and (3.56), ht ; hi C .2at /1 kh  xt k2  ht ; zi C .2at /1 kz  xt k2 :

(3.57)

In view of (3.57), ht ; zi C .2at /1 kz  xt k2  .2at /1 kh  xt k2 C ht ; h  xt i C ht ; xt i: It follows from the inequality above, (3.34), (3.36), (3.42), (3.49), and (3.55) that .L C 1/M C .2at /1 .14M C 1/2  .2at /1 kh  xt k2  .L C 1/kh  xt k  .L C 1/.15M C 1/  .2at /1 .kh  xt k2  kh  xt k/  .L C 1/.15M C 1/  .2at /1 .kh  xt k  1/2  .2at /1  .L C 1/.15M C 1/; .2at /1 C .L C 1/.16M C 1/ C .2at /1 .14M C 1/2

52

3 The Mirror Descent Algorithm

 .2at /1 .kh  xt k  1/2 ; 4.14M C 1/2  2 C 16M C .14M C 1/2  .kh  xt k  1/2 ; kh  xt k  28M C 4; khk  44M C 5 < M0 :

(3.58)

By (3.52), (3.56), and (3.58), h 2 C:

(3.59)

Relations (3.52), (3.56), and (3.59) imply that h 2 argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg and h 2 BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg:

(3.60)

It follows from (3.31), (3.35), (3.51), (3.52)–(3.55), (3.60), and Lemma 3.2 which holds with x D xt ; a D at ;  D t ; u D h that at .f .xt /  f .z//  ıat .2M0 C L C 2/ C8ı.M0 C 1/ C 21 a2t .L C 1/2 C21 kz  xt k2  21 kz  xtC1 k2 Together with the inclusion at 2 Œ0 ; 1  this implies that f .xt /  f .z/  01 8ı.M0 C 1/ C.2M0 C L C 2/ı C 21 1 .L C 1/2 C.20 /1 kz  xt k2  .20 /1 kz  xtC1 k2

(3.61)

and (3.50) holds. In view of (3.37), (3.43), (3.49), and (3.61), kz  xt k2  kz  xtC1 k2  0; kz  xtC1 k  kz  xt k  14M C 1: Hence by induction we showed that (3.50) holds for all t D 1; : : : ; T and (3.49) holds for all t D 1; : : : ; T C 1.

3.6 Zero-Sum Games

53

It follows from (3.37)–(3.39), (3.42), (3.43), and (3.50) that T0 < T.minff .xt / W t D 1; : : : ; Tg  f .z// 

T X .f .xt /  f .z// tD1

 .20 /

1

T X .kz  xt k2  kz  xtC1 k2 / tD1

C8T01 ı.M0

C 1/ C T.2M0 C L C 2/ı C 21 T1 .L C 1/2

 .20 /1 .2M C 1/2 C 8T01 ı.M0 C 1/ CT.2M0 C L C 2/ı C 21 T1 .L C 1/2 ; 0 < .20 T/1 .2M C 1/2 C 801 ı.M0 C 1/ C.2M0 C L C 2/ı C 21 1 .L C 1/2 ; 21 0 < .20 T/1 .2M C 1/2 and T < 01 .2M C 1/2 01 < n0 : Thus we have shown that if an integer T  1 satisfies f .xt /  f .z/ > 0 ; t D 1; : : : ; T; then T < n0 and (3.49) holds for all t D 1; : : : ; T C 1. This implies that there exists an integer t 2 f1; : : : ; n0 g such that f .xt /  f .z/  0 ; kxt k  15M C 1: Theorem 3.3 is proved.

3.6 Zero-Sum Games We use the notation and definitions introduced in Sect. 3.1. Let X; Y be Hilbert spaces, C be a nonempty closed convex subset of X, D be a nonempty closed convex subset of Y, U be an open convex subset of X, and V be an open convex subset of Y such that C  U; D  V:

(3.62)

54

3 The Mirror Descent Algorithm

Suppose that there exist L > 0, M0 > 0 such that C  BX .0; M0 /; D  BY .0; M0 /;

(3.63)

a function f W U  V ! R1 possesses the following properties: (i) for each v 2 V, the function f .; v/ W U ! R1 is convex; (ii) for each u 2 U, the function f .u; / W V ! R1 is concave, and that for each u 2 U, jf .u; v1 /  f .u; v2 /j  Lkv1  v2 k for all v1 ; v2 2 V;

(3.64)

for each v 2 V, jf .u1 ; v/  f .u2 ; v/j  Lku1  u2 k for all u1 ; u2 2 U:

(3.65)

Recall that for each .; / 2 U  V, @x f .; / D fl 2 X W f .y; /  f .; /  hl; y  i for all y 2 Ug; @y f .; / D fl 2 Y W hl; y  i  f .; y/  f .; / for all y 2 Vg: In view of properties (i) and (ii) and (3.63)–(3.65), for each  2 U and each 2 V, ; 6D @x f .; /  BX .0; L/; ; 6D @y f .; /  BY .0; L/: Let x 2 C and y 2 D

(3.66)

f .x ; y/  f .x ; y /  f .x; y /

(3.67)

satisfy

for each x 2 C and each y 2 D. Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. Let us describe our algorithm. Mirror Descent Algorithm for Zero-Sum Games Initialization: select arbitrary x0 2 U and y0 2 V. Iterative step: given current iteration vectors xt 2 U and yt 2 V calculate t 2 @x f .xt ; yt / C BX .0; ı/;

t 2 @y f .xt ; yt / C BY .0; ı/

3.6 Zero-Sum Games

55

and the next pair of iteration vectors xtC1 2 U, ytC1 2 V such that BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg 6D ;; BY .ytC1 ; ı/ \ argminfh t ; ui C .2at /1 ku  yt k2 W u 2 Dg 6D ;: In this chapter we prove the following result. 1 Theorem 3.4. Let ı 2 .0; 1 and fak g1 kD0  .0; 1/. Assume that fxt gtD0  U, 1 1 1 fyt gtD0  V, ft gtD0  X, f t gtD0  Y,

BX .x0 ; ı/ \ C 6D ;; BY .y0 ; ı/ \ D 6D ; and that for each integer t  0, t 2 @x f .xt ; yt / C BX .0; ı/;

t 2 @y f .xt ; yt / C BY .0; ı/; BX .xtC1 ; ı/ \ argminfht ; vi C .2at /1 kv  xt k2 W v 2 Cg 6D ;; BY .ytC1 ; ı/ \ argminfh t ; ui C .2at /1 ku  yt k2 W u 2 Dg 6D ;: Let for each natural number T, xO T D

T X

!1 at

T X

iD0

at xt ; yO T D

tD0

T X

!1 at

iD0

T X

at yt :

tD0

Then for each natural number T, BX .OxT ; ı/ \ C 6D ;; BY .OyT ; ı/ \ D 6D ;; ˇ ˇ ˇ ˇ T !1 T X ˇ ˇ X ˇ at at f .xt ; yt /  f .x ; y /ˇˇ ˇ ˇ ˇ tD0 tD0

(3.68)

 Œ21 .2M0 C 1/2 C 8ı.T C 1/.M0 C 1/

T X

!1 at

tD0 1

Cı.2M0 C L C 2/ C 2

T X tD0

!1 at

ˇ ˇ !1 T ˇ ˇ T X X ˇ ˇ ˇf .OxT ; yO T /  at at f .xt ; yt /ˇˇ ˇ ˇ ˇ tD0 tD0

.L C 1/2

T X tD0

a2t ;

56

3 The Mirror Descent Algorithm

1

2

 Œ2 .2M0 C 1/ C 8ı.T C 1/.M0 C 1/

T X

!1 at

tD0 T X

1

Cı.2M0 C 2L C 2/ C 2

!1 .L C 1/2

at

tD0

T X

a2t ;

tD0

and for each natural number T, each z 2 C, and each v 2 D, f .OxT ; v/  f .OxT ; yO T / C.2M0 C 1/

2

T X

!1

T X

C

at

tD0

!1 16ı.T C 1/.M0 C 1/

at

tD0 T X

C2ı.2M0 C 2L C 2/ C

!1 .L C 1/2

at

tD0

f .z; yO T /  f .OxT ; yO T / .2M0 C 1/

2

T X

 16

tD0

2ı.2M0 C 2L C 2/ 

a2t ;

tD0

!1 at

T X

T X

!1 at

.T C 1/ı.M0 C 1/

tD0 T X

!1 at

.L C 1/2

tD0

T X

a2t :

tD0

Proof. Evidently, (3.68) holds. It is not difficult to see that kxt k  M0 C 1; kyt k  M0 C 1; t D 0; 1; : : : : Let t  0 be an integer. Applying Lemma 3.2 with a D at ; x D xt ; f D f .; yt /;  D t ; u D xtC1 we obtain that for each z 2 C, at .f .xt ; yt /  f .z; yt //  21 kz  xt k2  21 kz  xtC1 k2 Cı.8M0 C 8 C at .2M0 C L C 2// C 21 a2t .L C 1/2 : Applying Lemma 3.2 with a D at ; x D yt ; f D f .xt ; /;  D  t ; u D ytC1

3.6 Zero-Sum Games

57

we obtain that for each v 2 D, at .f .xt ; v/  f .xt ; yt //  21 kv  yt k2  21 kv  ytC1 k2 Cı.8M0 C 8 C at .2M0 C L C 2// C 21 a2t .L C 1/2 : For all integers t  0 set bt D ı.8M0 C 8 C at .2M0 C L C 2// C 21 a2t .L C 1/2 and define .s/ D 21 s2 ; s 2 R1 : It is easy to see that all the assumptions of Proposition 2.9 hold and it implies Theorem 3.4. We are interestedPin the optimal choice of at , t D 0; 1; : : : . Let T be a natural number and AT D TtD0 at be given. By Theorem 3.4, in order to make the best P choice of at ; t D 0; : : : ; T, we need to minimize the function TtD0 a2t on the set ( a D .a0 ; : : : ; aT / 2 R

TC1

W ai  0; i D 0; : : : ; T;

T X

) ai D AT :

iD0

By Lemma 2.3, this function has a unique minimizer a D .a0 ; : : : ; aT / where ai D .T C 1/1 AT , i D 0; : : : ; T which is the best choice of at , t D 0; 1; : : : ; T. Now we will find the best a > 0. Let T be a natural number and at D a for all t D 0; : : : ; T. We need to choice a which is a minimizer of the function T .a/ D ..T C 1/a/1 .2M0 C 1/2 C 2ı.2M0 C L C 2/ C16ı.T C 1/.M0 C 1/.a.T C 1//1 C a.L C 1/2 D .2M0 C 1/2 ..T C 1/a/1 C 2ı.2M0 C L C 2/ C 16ı.M0 C 1/a1 C.L C 1/2 a: Since T can be arbitrary large, we need to find a minimizer of the function .a/ WD 16a1 ı.M0 C 1/ C .L C 1/2 a; a 2 .0; 1/: This function has a minimizer a D 4.ı.M0 C 1//1=2 .L C 1/1 and the minimal value of is 8.ı.M0 C 1//1=2 .L C 1/:

58

3 The Mirror Descent Algorithm

Now our goal is to find the best T > 0 which gives us an appropriate value of T .a/. Since in view of the inequalities above, this value is bounded from below by c0 ı 1=2 with the constant c0 depending on L; M0 it is clear that in order to make the best choice of T, it should be at the same order as bı 1 c. For example, T D bı 1 c. Note that in the theorem above ı is the computational error produced by our computer system. We obtain a good approximate solution after T D bı 1 c iterations. Namely, we obtain a pair of points xO 2 U, yO 2 V such that BX .Ox; ı/ \ C 6D ;; BY .Oy; ı/ \ D 6D ; and for each z 2 C and each v 2 D, f .z; yO /  f .Ox; yO /  cı 1=2 ; f .Ox; v/  f .Ox; yO / C cı 1=2 ; where the constant c > 0 depends only on L and M0 .

Chapter 4

Gradient Algorithm with a Smooth Objective Function

In this chapter we analyze the convergence of a projected gradient algorithm with a smooth objective function under the presence of computational errors. We show that the algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

4.1 Optimization on Bounded Sets Let X be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: For each x 2 X and each nonempty set E  X put d.x; E/ D inffkx  yk W y 2 Eg: Let C be a nonempty closed convex subset of X, U be an open convex subset of X such that C  U and let f W U ! R1 be a convex continuous function. We suppose that the function f is Frechet differentiable at every point x 2 U and for every x 2 U we denote by f 0 .x/ 2 X the Frechet derivative of f at x. It is clear that for any x 2 U and any h 2 X hf 0 .x/; hi D lim t1 .f .x C th/  f .x//: t!0

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_4

(4.1)

59

60

4 Gradient Algorithm with a Smooth Objective Function

For each nonempty set D and each function g W D ! R1 set inf.gI D/ D inffg.y/ W y 2 Dg; argminfg.z/ W z 2 Dg D fz 2 D W g.z/ D inf.gI D/g:

(4.2) (4.3)

We suppose that the mapping f 0 W U ! X is Lipschitz on all bounded subsets of U. It is well known (see Lemma 2.2) that for each nonempty closed convex set D  X and each x 2 X there exists a unique point PD .x/ 2 D such that kx  PD .x/k D inffkx  yk W y 2 Dg: In this chapter we study the behavior of a projected gradient algorithm with a smooth objective function which is used for solving convex constrained minimization problems [91, 92, 98]. In the sequel we use the following proposition [91, 92, 98] which is proved in Sect. 4.2. Proposition 4.1. Assume that x; u 2 U, L > 0 and that for each v1 ; v2 2 ftxC .1  t/u W t 2 Œ0; 1g, kf 0 .v1 /  f 0 .v2 /k  Lkv1  v2 k: Then f .u/  f .x/ C hf 0 .x/; u  xi C 21 Lku  xk2 : Suppose that there exist L > 1, M0 > 0 such that C  BX .0; M0 /; jf .v1 /  f .v2 /j  Lkv1  v2 k for all v1 ; v2 2 U; 0

0

kf .v1 /  f .v2 /k  Lkv1  v2 k for all v1 ; v2 2 U: Let ı 2 .0; 1. We describe below our algorithm. Gradient Algorithm Initialization: select an arbitrary x0 2 U \ BX .0; M0 /. Iterative step: given a current iteration vector xt 2 U calculate t 2 f 0 .xt / C BX .0; ı/ and calculate the next iteration vector xtC1 2 U such that kxtC1  PC .xt  L1 t /k  ı: In this chapter we prove the following result.

(4.4) (4.5) (4.6)

4.2 An Auxiliary Result and the Proof of Proposition 4.1

61

Theorem 4.2. Let ı 2 .0; 1 and let x0 2 U \ BX .0; M0 /:

(4.7)

1 Assume that fxt g1 tD1  U, ft gtD0  X and that for each integer t  0,

kt  f 0 .xt /k  ı

(4.8)

kxtC1  PC .xt  L1 t /k  ı:

(4.9)

and

Then for each natural number T, f .xTC1 /  inf.f I C/  .2T/1 L.2M0 C 1/2 C Lı.8M0 C 8/.T C 1/

(4.10)

and minff .xt / W t D 2; : : : ; T C 1g  inf.f I C/; f

TC1 X

! T

1

xt  inf.f I C/

tD2

 .2T/1 L.2M0 C 1/2 C Lı.8M0 C 8/:

(4.11)

We are interested in an optimal choice of T. If we choose T in order to minimize the right-hand side of (4.11) we obtain that T should be at the same order as ı 1 . In this case the right-hand side of (4.11) is at the same order as ı. For example, if T D bı 1 cC1  ı 1 , then the right-hand side of (4.11) does not exceed 2Lı.4M0 C 4 C .2M0 C 1/2 /.

4.2 An Auxiliary Result and the Proof of Proposition 4.1 Proposition 4.3. Let D be a nonempty closed convex subset of X, x 2 X and y 2 D. Assume that for each z 2 D, hz  y; x  yi  0: Then y D PD .x/.

(4.12)

62

4 Gradient Algorithm with a Smooth Objective Function

Proof. Let z 2 D. By (4.12), hz  x; z  xi D hz  y C .y  x/; z  y C .y  x/i D hy  x; y  xi C 2hz  y; y  xi C hz  y; z  yi  hy  x; y  xi C hz  y; z  yi  ky  xk2 C kz  yk2 : Thus y D PD .x/. Proposition 4.3 is proved. Proof of Proposition 4.1. For each t 2 Œ0; 1 set .t/ D f .x C t.u  x//:

(4.13)

Clearly, is a differentiable function and for each t 2 Œ0; 1, 0 .t/ D hf 0 .x C t.u  x/; u  xi:

(4.14)

By (4.13), (4.14), and the proposition assumptions, f .u/  f .x/ D .1/  .0/ Z 1 Z 1 0 .t/dt D hf 0 .x C t.u  x/; u  xidt D Z

0

0

1

D 0

Z

0

hf .x/; u  xidt C Z

0

 hf .x/; u  xi C

1

0

1

hf 0 .x C t.u  x/  f 0 .x/; u  xidt

Ltku  xk2 dt

0

D hf 0 .x/; u  xi C Lku  xk2

Z

1

tdt 0

D hf 0 .x/; u  xi C Lku  xk2 =2: Proposition 4.1 is proved.

4.3 The Main Lemma We use the notation, definitions, and assumptions introduced in Sect. 4.1. Lemma 4.4. Let ı 2 .0; 1, u 2 BX .0; M0 C 1/ \ U;

(4.15)

4.3 The Main Lemma

63

 2 X satisfy k  f 0 .u/k  ı

(4.16)

kv  PC .u  L1 /k  ı:

(4.17)

and let v 2 U satisfy

Then for each x 2 U satisfying B.x; ı/ \ C 6D ;

(4.18)

the following inequalities hold: f .x/  f .v/  21 Lkx  vk2  21 Lkx  uk2  ıL.8M0 C 8/; 1

2

f .x/  f .v/  2 Lku  vk C Lhv  u; u  xi  ıL.8M0 C 12/:

(4.19) (4.20)

Proof. For each x 2 U define g.x/ D f .u/ C hf 0 .u/; x  ui C 21 Lkx  uk2 :

(4.21)

Clearly, g W U ! R1 is a convex Frechet differentiable function, for each x 2 U, g0 .x/ D f 0 .u/ C L.x  u/; lim g.x/ D 1

kxk!1

(4.22) (4.23)

and there exists x0 2 C

(4.24)

g.x0 /  g.x/ for all x 2 C:

(4.25)

such that

By (4.24) and (4.25), for all z 2 C, hg0 .x0 /; z  x0 i  0:

(4.26)

hL1 f 0 .u/ C x0  u; z  x0 i  0 for all z 2 C:

(4.27)

In view of (4.22) and (4.26),

Proposition 4.3, (4.24), and (4.27) imply that x0 D PC .u  L1 f 0 .u//:

(4.28)

64

4 Gradient Algorithm with a Smooth Objective Function

It follows from (4.16), (4.28), and Lemma 2.2 that kv  x0 k  kv  PC .u  L1 /k C kPC .u  L1 /  PC .u  L1 f 0 .u//k  ı C L1 k  f 0 .u/k  ı.1 C L1 /:

(4.29)

In view of (4.4) and (4.24), kx0 k  M0 :

(4.30)

Relations (4.4) and (4.17) imply that kvk  M0 C 1:

(4.31)

By (4.6), (4.15), (4.21), and Proposition 4.1, for all x 2 U, f .x/  f .u/ C hf 0 .u/; x  ui C 21 Lku  xk2 D g.x/:

(4.32)

Let x2U

(4.33)

B.x; ı/ \ C 6D ;:

(4.34)

satisfy

It follows from (4.5) and (4.29) that jf .x0 /  f .v/j  Lkv  x0 k  ı.L C 1/:

(4.35)

In view of (4.24) and (4.32), g.x0 /  f .x0 /:

(4.36)

By (4.21), (4.36) and convexity of f , f .x/  f .x0 /  f .x/  g.x0 / D f .x/  f .u/  hf 0 .u/; x0  ui  21 Lku  x0 k2  f .u/ C hf 0 .u/; x  ui  f .u/  hf 0 .u/; x0  ui  21 Lku  x0 k2 D hf 0 .u/; x  x0 i  21 Lku  x0 k2 :

(4.37)

Relation (4.34) implies that there exists x1 2 C

(4.38)

4.3 The Main Lemma

65

such that kx1  xk  ı:

(4.39)

By (4.5), (4.33), (4.38), and (4.39), jf .x1 /  f .x/j  Lı:

(4.40)

It follows from (4.26) (with z D x1 ) and (4.38) that 0  hg0 .x0 /; x1  x0 i D hg0 .x0 /; x1  xi C hg0 .x0 /; x  x0 i:

(4.41)

By (4.4), (4.5), (4.15), (4.22), (4.24), (4.38), and (4.39), hg0 .x0 /; x1  xi D hf 0 .u/; x1  xi C Lhx0  u; x1  xi  Lı C Lı.2M0 C 1/:

(4.42)

hg0 .x0 /; x  x0 i D hf 0 .u/ C L.x0  u/; x  x0 i:

(4.43)

In view of (4.22) and (4.24),

Relations (4.41) and (4.43) imply that hf 0 .u/; x  x0 i D hg0 .x0 /; x  x0 i  Lhx0  u; x  x0 i  hg0 .x0 /; x1  xi  Lhx0  u; x  x0 i  Lhx0  u; x  x0 i  Lı.2M0 C 2/:

(4.44)

It follows from (4.37) and (4.44) that f .x/  f .x0 /  hf 0 .u/; x  x0 i  21 Lkx0  uk2  Lı.2M0 C 2/  Lhx0  u; x  x0 i  21 Lkx0  uk2 :

(4.45)

In view of (4.45) and Lemma 2.1, f .x/  f .x0 /  Lı.2M0 C 2/  21 Lkx0  uk2 21 LŒkx  uk2  kx  x0 k2  ku  x0 k2  D 21 Lkx  x0 k2  21 Lkx  uk2  Lı.2M0 C 2/:

(4.46)

By (4.4), (4.24), (4.34), and (4.35), f .x/  f .v/  f .x/  f .x0 /  ı.L C 1/:

(4.47)

66

4 Gradient Algorithm with a Smooth Objective Function

It follows from (4.15), (4.17), (4.24), and (4.29) that jkx  x0 k2  kx  vk2 j D jkx  x0 k  kx  vkj.kx  x0 k C kx  vk/  ı.8M0 C 8/

(4.48)

and jku  x0 k2  ku  vk2 j D jku  x0 k  ku  vkj.ku  x0 k C ku  vk/  ı.8M0 C 8/:

(4.49)

In view of (4.45), f .x/  f .x0 /  Lı.2M0 C 2/ C 21 Lkx0  uk2 Lhx0  u; x0  ui  Lhx0  u; x  x0 i  Lı.2M0 C 2/ C 21 Lkx0  uk2 Lhx0  u; x  ui:

(4.50)

By (4.35), (4.46), and (4.48), f .x/  f .v/  f .x/  f .x0 /  ı.L C 1/  21 Lkx  x0 k2  21 Lkx  uk2  Lı.2M0 C 4/  21 Lkx  vk2  21 Lkx  uk2  2Lı.4M0 C 4/ and (4.19) holds. It follows from (4.15), (4.29), (4.34), (4.35), (4.49), and (4.50) that f .x/  f .v/  f .x/  f .x0 /  ı.L C 1/  Lı.2M0 C 4/ C 21 Lkx0  uk2  Lhx0  u; x  ui  Lı.2M0 C 4/ C 21 Lku  vk2  4Lı.M0 C 1/ Lhv  u; x  ui  Lı.2M0 C 4/  21 Lku  vk2  Lhv  u; x  ui  Lı.8M0 C 12/ and (4.20) holds. Lemma 4.4 is proved.

4.4 Proof of Theorem 4.2 Clearly, the function f has a minimizer on the set C. Fix z2C

(4.51)

4.4 Proof of Theorem 4.2

67

such that f .z/ D inf.f I C/:

(4.52)

kxt k  M0 C 1; t D 0; 1; : : : :

(4.53)

It is easy to see that

Let T be a natural number and t  0 be an integer. Applying Lemma 4.4 with u D xt ;  D t ; v D xtC1 ; x D z we obtain that f .z/  f .xtC1 /  21 Lkz  xtC1 k2  21 Lkz  xt k2  ıL.8M0 C 8/: This implies that Tf .z/ 

T X

f .xtC1 /

tD1



T X .21 Lkz  xtC1 k2  21 Lkz  xt k2 /  ıLT.8M0 C 8/ tD1

 21 L.kz  xTC1 k2  kz  x1 k2 /  ıLT.8M0 C 8/:

(4.54)

Let t  0 be an integer. Applying Lemma 4.4 with x D xtC1 ; u D xtC1 ;  D tC1 ; v D xtC2 ; we obtain that f .xtC1 /  f .xtC2 /  21 LkxtC2  xtC1 k2  ıL.8M0 C 8/ and t.f .xtC1 /  f .xtC2 //  21 LtkxtC2  xtC1 k2  ıLt.8M0 C 8/: We can write the relation above as tf .xtC1 /  .t C 1/f .xtC2 / C f .xtC2 /  21 LtkxtC2  xtC1 k2  ıLt.8M0 C 8/: Summing (4.55) with t D 0; : : : ; T  1 we obtain that

(4.55)

68

4 Gradient Algorithm with a Smooth Objective Function

Tf .xTC1 / C

T1 X

f .xtC2 /

tD0

D

T1 X Œtf .xtC1 /  .t C 1/f .xtC2 / C f .xtC2 / tD0



T1 T1 X X Œ21 LtkxtC2  xtC1 k2   ıL.8M0 C 8/ t: tD0

(4.56)

tD0

By (4.54) and (4.56), T.f .z/  f .xTC1 //  21 Lkz  x1 k2  LıT.8M0 C 8/  Lı.4M0 C 4/T.T  1/ and in view of (4.51) and (4.53), f .xTC1 /  f .z/  .2T/1 L.2M0 C 1/2 C Lı.8M0 C 8/.T C 1/: In view of (4.54) T.minff .xt / W t D 2; : : : ; T C 1g  f .z//; T f

TC1 X

! T

1

!

xt  f .z/

tD2



T X

f .xtC1 /  Tf .z/

tD1

 21 L.2M0 C 1/2 C LıT.8M0 C 8/: This completes the proof of Theorem 4.2.

4.5 Optimization on Unbounded Sets We use the notation and definitions introduced in Sect. 4.1. Let X be a Hilbert space with an inner product h; i which induces a complete norm k  k. Let D be a nonempty closed convex subset of X, V be an open convex subset of X such that DV

4.5 Optimization on Unbounded Sets

69

and f W V ! R1 be a convex Fréchet differentiable function which is Lipschitz on all bounded subsets of V. Set Dmin D fx 2 D W f .x/  f .y/ for all y 2 Dg:

(4.57)

Dmin 6D ;:

(4.58)

We suppose that

We will prove the following result. Theorem 4.5. Let ı 2 .0; 1, M > 0 satisfy Dmin \ BX .0; M/ 6D ;;

(4.59)

M0  4M C 8; L  1 satisfy jf .v1 /  f .v2 /j  Lkv1  v2 k for all v1 ; v2 2 V \ BX .0; M0 C 2/; (4.60) kf 0 .v1 /  f 0 .v2 /k  Lkv1  v2 k for all v1 ; v2 2 V \ BX .0; M0 C 2/; (4.61) 0 D 4Lı.2M0 C 3/

(4.62)

and let n0 D b.4ı/1 .2M C 1/2 c C 1:

(4.63)

1 Assume that fxt g1 tD0  V, ft gtD0  X,

kx0 k  M

(4.64)

kt  f 0 .xt /k  ı

(4.65)

kxtC1  PD .xt  L1 t /k  ı:

(4.66)

and that for each integer t  0,

and

Then there exists an integer q 2 Œ1; n0 C 1 such that f .xq /  inf.f I D/ C 0 ; kxi k  3M C 3; i D 0; : : : ; q: Proof. By (4.59) there exists z 2 Dmin \ BX .0; M/:

(4.67)

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4 Gradient Algorithm with a Smooth Objective Function

By (4.60), (4.64)–(4.67), and Lemma 2.2, kx1  zk  kx1  PD .x0  L1 0 /k C kPD .x0  L1 0 /  zk  ı C kx0  zk C L1 k0 k  1 C 2M C L1 .L C 1/  2M C 3:

(4.68)

In view (4.67) and (4.68), kx1 k  3M C 3:

(4.69)

Assume that an integer T  0 and that for all t D 1; : : : ; T C 1, f .xt /  f .z/ > 0 :

(4.70)

U D V \ fv 2 X W kvk < M0 C 2g

(4.71)

C D D \ BX .0; M0 /:

(4.72)

Set

and

Assume that an integer t 2 Œ0; T and that kxt  zk  2M C 3:

(4.73)

(In view of (4.64), (4.67), and (4.68), our assumption is true for t D 0.) By (4.67) and (4.72), z 2 C  BX .0; M0 /:

(4.74)

Relations (4.67), (4.71), and (4.73) imply that xt 2 U \ BX .0; M0 C 1/:

(4.75)

It follows from (4.60), (4.65), and (4.75) that t 2 f 0 .xt / C BX .0; 1/  BX .0; L C 1/:

(4.76)

By (4.67), (4.73), (4.76), and Lemma 2.2, kz  PD .xt  L1 t /k  kz  xt C L1 t k  kz  xt k C L1 kt k  2M C 5:

(4.77)

4.5 Optimization on Unbounded Sets

71

In view of (4.67) and (4.77), kPD .xt  L1 t /k  3M C 5:

(4.78)

Relations (4.72) and (4.78) imply that PD .xt  L1 t / 2 C; 1

(4.79) 1

PD .xt  L t / D PC .xt  L t /:

(4.80)

It follows from (4.66), (4.71), and (4.78) that kxtC1 k  3M C 6; xtC1 2 U:

(4.81)

By (4.65), (4.66), (4.79), (4.80), (4.81), and Lemma 4.4 applied with u D xt ;  D t ; v D xtC1 ; x D z we obtain that f .z/  f .xtC1 /  21 Lkz  xtC1 k2  21 Lkz  xt k2  Lı.8M0 C 8/:

(4.82)

By (4.62), (4.70), and (4.82), 4Lı.2M0 C 3/ D 0 < f .xtC1 /  f .z/  21 L.kz  xt k2  kz  xtC1 k2 / C Lı.8M0 C 8/: In view of (4.73) and (4.83), kz  xtC1 k  kz  xt k  2M C 3: Thus by induction we showed that for all t D 0; : : : ; T C 1 kz  xt k  2M C 3; kxt k  3M C 3 and that (4.83) holds for all t D 0; : : : ; T. It follows from (4.63), (4.64), (4.67), and (4.83) that 4.1 C T/Lı.2M0 C 3/  .1 C T/.minff .xt / W t D 1; : : : ; T C 1g  f .z//

(4.83)

72

4 Gradient Algorithm with a Smooth Objective Function



T X .f .xtC1 /  f .z// tD0

 21 L

T X .kz  xt k2  kz  xtC1 k2 / C .T C 1/Lı.8M0 C 8/; tD0

2Lı.1 C T/  21 Lkz  x0 k2  21 L.2M C 1/2 and T < .2M C 1/2 .4ı/1  n0 : Thus we assumed that an integer T  0 satisfies f .xt /  f .z/ > 0 ; t D 1; : : : ; T C 1 and showed that T  n0  1 and kxt k  3M C 3; t D 0; : : : ; T C 1: This implies that there exists a natural number q  n0 C 1 such that f .xq /  f .z/  0  4Lı.2M0 C 3/ kxt k  3M C 3; t D 0; : : : ; q: Theorem 4.2 is proved. Note that in the theorem above ı is the computational error produced by our computer system. We obtain a good approximate solution after b.4ı/1 .2MC1/2 cC 2 iterations. Namely, we obtain a point x 2 X such that BX .x; ı/ \ D 6D ; and f .x/  inf.f I D/ C 4Lı.2M0 C 3/:

Chapter 5

An Extension of the Gradient Algorithm

In this chapter we analyze the convergence of a gradient type algorithm, under the presence of computational errors, which was introduced by Beck and Teboulle [20] for solving linear inverse problems arising in signal/image processing. We show that the algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

5.1 Preliminaries and the Main Result Let X be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: Suppose that f W X ! R1 is a convex Fréchet differentiable function on X and for every x 2 X denote by f 0 .x/ 2 X the Fréchet derivative of f at x. It is clear that for any x 2 X and any h 2 X hf 0 .x/; hi D lim t1 .f .x C th/  f .x//: t!0

For each function W X ! R1 set inf. / D inff .y/ W y 2 Xg; argmin. / D argminf .z/ W z 2 Xg WD fz 2 X W .z/ D inf. /g: We suppose that the mapping f 0 W X ! X is Lipschitz on all bounded subsets of X.

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_5

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5 An Extension of the Gradient Algorithm

Let g W X ! R1 be a convex continuous function which is Lipschitz on all bounded subsets of X. Define F.x/ D f .x/ C g.x/; x 2 X: We suppose that argmin.F/ 6D ;

(5.1)

and that there exists c 2 R1 such that g.x/  c for all x 2 X:

(5.2)

For each u 2 X, each  2 X and each L > 0 define a convex function .L/

Gu; .w/ D f .u/ C h; w  ui C 21 Lkw  uk2 C g.w/; w 2 X

(5.3)

which has a minimizer. In this chapter we analyze the gradient type algorithm, which was introduced by Beck and Teboulle in [20] for solving linear inverse problems, and prove the following result. Theorem 5.1. Let ı 2 .0; 1, M  1 satisfy argmin.F/ \ BX .0; M/ 6D ;;

(5.4)

L > 1 satisfy jf .w1 /  f .w2 /j  Lkw1  w2 k for all w1 ; w2 2 BX .0; 3M C 2/

(5.5)

and kf 0 .w1 /  f 0 .w2 /k  Lkw1  w2 k for all w1 ; w2 2 X;

(5.6)

M1  3M satisfy jf .w/j; F.w/  M1 for all w 2 BX .0; 3M C 2/; 2

M2 D .8.M1 C jc j C 1 C .L C 1/ //

1=2

C 3M C 2;

(5.7) (5.8)

L0 > 1 satisfy jg.w1 /  g.w2 /j  L0 kw1  w2 k for all w1 ; w2 2 BX .0; M2 /; 0 D 2ı..M2 C 3M C 2/.2L C 3/ C L0 /

(5.9) (5.10)

5.2 Auxiliary Results

75

and let n0 D b4LM 2 01 c:

(5.11)

1 Assume that fxt g1 tD0  X, ft gtD0  X,

kx0 k  M

(5.12)

kt  f 0 .xt /k  ı

(5.13)

and that for each integer t  0,

and .L/

BX .xtC1 ; ı/ \ argmin.Gxt ;t / 6D ;:

(5.14)

Then there exists an integer q 2 Œ0; n0 C 2 such that kxi k  M2 ; i D 0; : : : ; q and F.xq /  inf.F/ C 0 : Note that in the theorem above ı is the computational error produced by our computer system. We obtain a good approximate solution after bc1 ı 1 c iterations [see (5.10) and (5.11)], where c1 > 0 is a constant which depends only on L; L0 ; M; M2 . As a result we obtain a point x 2 X such that F.x/  inf.F/ C c2 ı; where c2 > 0 is a constant which depends only on L; L0 ; M; M2 .

5.2 Auxiliary Results .L/

Lemma 5.2 ([20]). Let u;  2 X and L > 0. Then the function Gu; has a point of .L/

minimum and z 2 X is a minimizer of Gu; if and only if 0 2  C L.z  u/ C @g.z/: Proof. By (5.2) and (5.3), .L/

lim Gu; .w/ D 1:

kwk!1

76

5 An Extension of the Gradient Algorithm .L/

This implies that the function Gu; has a minimizer. Clearly, z is a minimizer of .L/

Gu; if and only if .L/

0 2 @Gu; .z/ D  C L.z  u/ C @g.z/: Lemma 5.2 is proved. Lemma 5.3. Let M0  1, L > 1 satisfy jf .w1 /  f .w2 /j  Lkw1  w2 k for all w1 ; w2 2 BX .0; M0 C 2/ and kf 0 .w1 /  f 0 .w2 /k  Lkw1  w2 k for all w1 ; w2 2 X;

(5.15)

M1  M0 satisfy jf .w/j; F.w/  M1 for all w 2 BX .0; M0 C 2/:

(5.16)

u 2 BX .0; M0 C 1/;

(5.17)

k  f 0 .u/k  1

(5.18)

Assume that

 2 X satisfies

and that v 2 X satisfies .L/

.L/

BX .v; 1/ \ fz 2 X W Gu; .z/  inf.Gu; / C 1g 6D ;:

(5.19)

Then kvk  .8.M1 C jc j C .L C 1/2 C 1//1=2 C M0 C 2: Proof. In view of (5.19), there exists vO 2 BX .v; 1/

(5.20)

such that .L/

.L/

O  inf.Gu; / C 1: Gu; .v/

(5.21)

5.2 Auxiliary Results

77

By (5.3), (5.16), (5.17), and (5.21), O f .u/ C h; vO  ui C 21 LkvO  uk2 C g.v/ .L/

.L/

D Gu; .v/ O  Gu; .u/ C 1 D F.u/ C 1  M1 C 1:

(5.22)

It follows from (5.2), (5.16), (5.17), and (5.22) that h; vO  ui C 21 LkvO  uk2  2M1 C 1 C jc j:

(5.23)

2L1 jh; vO  uij  L1 .41 kvO  uk2 C 4kk2 /:

(5.24)

It is clear that

Since the function f is Lipschitz on BX .0; M0 C 2/ relations (5.17) and (5.18) imply that kk  kf 0 .u/k C 1  L C 1: By (5.23)–(5.25), 2L1 .2M1 C 1 C jc j/  kvO  uk2  2L1 jh; vO  uij  kvO  uk2  41 kvO  uk2  4kk2  21 kvO  uk2  4.L C 1/2 : This implies that kvO  uk2  4.M1 C 1 C jc j/ C 8.L C 1/2 and kvO  uk  .4.M1 C 1 C jc j/ C 8.L C 1/2 /1=2 : Together with (5.17) and (5.20) this implies that kvk  kvk O C 1  kvO  uk C kuk C 1  .4.M1 C 1 C jc j/ C 8.L C 1/2 /1=2 C M0 C 2: Lemma 5.3 is proved.

(5.25)

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5 An Extension of the Gradient Algorithm

5.3 The Main Lemma Lemma 5.4. Let ı 2 .0; 1, M0  1, L > 1 satisfy jf .w1 /  f .w2 /j  Lkw1  w2 k for all w1 ; w2 2 BX .0; M0 C 2/

(5.26)

kf 0 .w1 /  f 0 .w2 /k  Lkw1  w2 k for all w1 ; w2 2 X;

(5.27)

and

M1  M0 satisfy jf .w/j; F.w/  M1 for all w 2 BX .0; M0 C 2/; M2 D .8.M1 C jc j C .L C 1/2 C 1//1=2 C M0 C 2

(5.28) (5.29)

and let L0 > 1 satisfy jg.w1 /  g.w2 /j  L0 kw1  w2 k for all w1 ; w2 2 BX .0; M2 /:

(5.30)

Assume that u 2 BX .0; M0 C 1/;

(5.31)

k  f 0 .u/k  ı

(5.32)

 2 X satisfies

and that v 2 X satisfies .L/

BX .v; ı/ \ argmin.Gu; / 6D ;:

(5.33)

Then for each x 2 BX .0; M0 C 1/, F.x/  F.v/  21 Lkv  xk2  21 Lku  xk2 ı..M2 C M0 C 2/.2L C 3/ C L0 / Proof. By (5.33) there exists vO 2 X such that .L/

vO 2 argmin.Gu; /

(5.34)

kv  vk O  ı:

(5.35)

and

5.3 The Main Lemma

79

In view of the assumptions of the lemma, Lemma 5.3 and (5.34), kvk; kvk O  M2 :

(5.36)

x 2 BX .0; M0 C 1/:

(5.37)

F.x/ D f .x/ C g.x/; F.v/ D f .v/ C g.v/:

(5.38)

Let

Clearly,

Proposition 4.1 and (5.31) imply that g.v/ C f .v/  f .u/ C hf 0 .u/; v  ui C 21 Lkv  uk2 C g.v/:

(5.39)

By (5.3), (5.31), (5.32), (5.36), (5.38), and (5.39), F.x/  F.v/  Œf .x/ C g.x/  Œf .v/ C g.v/  f .x/ C g.x/  Œf .u/ C hf 0 .u/; v  ui C 21 Lkv  uk2 C g.v/ D Œf .x/ C g.x/  Œf .u/ C h; v  ui C 21 Lkv  uk2 C g.v/ Ch  f 0 .u/; v  ui .L/

 Œf .x/ C g.x/  Gu; .v/  k  f 0 .u/kkv  uk .L/

 Œf .x/ C g.x/  Gu; .v/  ı.M2 C M0 C 1/:

(5.40)

It follows from (5.3) that .L/

.L/

O Gu; .v/  Gu; .v/ D h; v  ui C 21 Lkv  uk2 C g.v/ Œh; vO  ui C 21 LkvO  uk2 C g.v/ O D h; v  vi O C 21 LŒkv  uk2  kvO  uk2  C g.v/  g.v/: O

(5.41)

Relation (5.27) and (5.32) imply that kk  kf 0 .u/k C 1  L C 1:

(5.42)

In view of (5.35) and (5.42), jh; v  vij O  .L C 1/ı:

(5.43)

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5 An Extension of the Gradient Algorithm

By (5.31), (5.35), and (5.36), jkv  uk2  kvO  uk2 j  jkv  uk  kvO  ukj.kv  uk C kvO  uk/  kv  vk.2M O 2 C 2M0 C 2/  ı.2M2 C 2M0 C 2/: (5.44) In view of (5.30), (5.35), and (5.36), jg.v/  g.v/j O  L0 kv  vk O  L0 ı:

(5.45)

It follows from (5.41) and (5.43)–(5.45) that .L/

.L/

jGu; .v/  Gu; .v/j O  .L C 1/ı C 21 Lı.2M0 C 2M2 C 2/ C L0 ı  ı.L C 1 C L0 C L.M0 C M2 C 1//:

(5.46)

Relations (5.40) and (5.46) imply that F.x/  F.v/ .L/

 f .x/ C g.x/  Gu; .v/  ı.M2 C M0 C 1/ .L/

 f .x/ C g.x/  Gu; .v/ O  ı.L0 C L C 1 C .L C 1/M2 C M0 C 1//: (5.47) By the convexity of f , (5.31), (5.32), and (5.37), f .x/  f .u/ C hf 0 .u/; x  ui  f .u/ C h; x  ui  jhf 0 .u/  ; x  uij  f .u/ C h; x  ui  kf 0 .u/  kkx  uk  f .u/ C h; x  ui  ı.2M0 C 2/:

(5.48)

Lemma 5.2 and (5.34) imply that there exists l 2 @g.v/ O

(5.49)

 C L.vO  u/ C l D 0:

(5.50)

such that

In view of (5.49) and the convexity of g, g.x/  g.v/ O C hl; x  vi: O

(5.51)

5.3 The Main Lemma

81

It follows from (5.48) and (5.51) that f .x/ C g.x/  f .u/ C h; x  ui  ı.2M0 C 2/ C g.v/ O C hl; x  vi: O

(5.52)

In view of (5.3), .L/

O D f .u/ C h; vO  ui C 21 LkvO  uk2 C g.v/: O Gu; .v/

(5.53)

By (5.50), (5.52), and (5.53), .L/

O f .x/ C g.x/  Gu; .v/  h; x  vi O C hl; x  vi O  21 LkvO  uk2  ı.2M0 C 2/ D ı.2M0 C 2/ C h C l; x  vi O  21 LkvO  uk2 D 21 LkvO  uk2  LhvO  u; x  vi O  ı.2M0 C 2/ D 21 LkvO  uk2 C LhvO  u; u  xi  ı.2M0 C 2/:

(5.54)

In view of (5.35)–(5.37), jkvO  xk2  kv  xk2 j  jkvO  xk  kv  xkj.kvO  xk C kv  xk/  kvO  vk.2M2 C 2M0 C 2/  ı.2M2 C 2M0 C 2/:

(5.55)

Lemma 2.1 implies that hvO  u; u  xi D 21 ŒkvO  xk2  kvO  uk2  ku  xk2 :

(5.56)

By (5.54) and (5.56), .L/

O f .x/ C g.x/  Gu; .v/  21 LkvO  uk2 C 21 LkvO  xk2 21 LkvO  uk2  21 Lku  xk2  ı.2M0 C 2/  21 LkvO  xk2  21 Lku  xk2  ı.2M0 C 2/  21 Lkv  xk2  21 Lku  xk2  21 Lı.2M2 C 2M0 C 2/  ı.2M0 C 2/: (5.57)

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5 An Extension of the Gradient Algorithm

It follows from (5.47) and (5.57) that F.x/  F.v/  21 Lkv  xk2  21 Lku  xk2  ı.L.M2 C M0 C 1/ C2M0 C 2 C L0 C .L C 1/.M2 C M0 C 2//: Lemma 5.4 is proved.

5.4 Proof of Theorem 5.1 By (5.4), there exists z 2 argmin.F/ \ BX .0; M/:

(5.58)

In view of (5.12) and (5.58), kx0  zk  2M:

(5.59)

If f .x0 /  f .z/ C 0 , then the assertion of the theorem holds. Let f .x0 / > f .z/ C 0 : If f .x1 /  f .z/ C 0 , then in view of Lemma 5.3, kx1 k  M2 and the assertion of the theorem holds. Let f .x1 / > f .z/ C 0 : Assume that T  0 is an integer and that for all integers t D 0; : : : ; T, F.xtC1 /  F.z/ > 0 :

(5.60)

We show that for all t 2 f0; : : : ; Tg, kxt  zk  2M

(5.61)

and F.z/  F.xtC1 /  21 Lkz  xtC1 k2  21 Lkz  xt k2  ı..M2 C M0 C 2/.2L C 3/ C L0 /: (5.62) In view of (5.59), (5.61) is true for t D 0.

5.4 Proof of Theorem 5.1

83

Assume that t 2 f0; : : : ; Tg and (5.61) holds. Relations (5.58) and (5.61) imply that kxt k  3M:

(5.63)

M0 D 3M

(5.64)

M3 D .M2 C M0 C 2/.2L C 3/ C L0 :

(5.65)

Set

and

By (5.5)–(5.9), (5.13), (5.14), (5.58), (5.63), (5.64), and Lemma 5.4 applied with x D z; u D xt ;  D t ; v D xtC1 ; we have F.z/  F.xtC1 /  21 Lkz  xtC1 k2  21 Lkz  xt k2  ıM3 :

(5.66)

It follows from (5.60) and (5.66) that 0 < F.xtC1 /  F.z/  21 Lkz  xt k2  21 Lkz  xtC1 k2 C ıM3 :

(5.67)

In view of (5.10), (5.65) and (5.67), 2M  kz  xt k  kz  xtC1 k: Thus we have shown by induction that (5.62) holds for all t D 0; : : : ; T and that (5.61) holds for all t D 0; : : : ; T C 1. By (5.60), (5.62) and (5.65), T X T0 < .F.xtC1 /  F.z// tD0



T X Œ21 Lkz  xt k2  21 Lkz  xtC1 k2  C TıM3 tD0

 21 Lkz  x0 k2 C TıM3 :

(5.68)

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5 An Extension of the Gradient Algorithm

It follows from (5.10), (5.59), (5.65), and (5.68) that T0 =2  21 L.4M 2 /; T  4LM 2 01 < n0 C 1: Thus we have shown that if T  0 is an integer and (5.60) holds for all t D 0; : : : ; T, then (5.61) holds for all t D 0; : : : ; T C 1 and T < n0 C 1. This implies that there exists an integer q 2 f1; : : : ; n0 C 2g such that kxi k  3M; i D 0; : : : ; q; F.xq /  F.z/ C 0 : Theorem 5.1 is proved.

Chapter 6

Weiszfeld’s Method

In this chapter we analyze the behavior of Weiszfeld’s method for solving the Fermat–Weber location problem. We show that the algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

6.1 The Description of the Problem Let X be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. If x 2 X and h is a real-valued function defined in a neighborhood of x which is Frechet differentiable at x, then its Frechet derivative at x is denoted by h0 .x/ 2 X. For each x 2 X and each r > 0 set BX .x; r/ D fy 2 X W kx  yk  rg: Let a 2 X. The function g.x/ D kx  ak, x 2 X is convex. For every x 2 X n fag, g is Frechet differentiable at x and g0 .x/ D kx  ak1 .x  a/: It is easy to see that @g.a/ D BX .0; 1/: Recall that the definition of the subdifferential is given in Sect. 2.1.

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_6

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86

6 Weiszfeld’s Method

In this chapter we assume that X is the finite-dimensional Euclidean space Rn , for each pair x D .x1 ; : : : ; xn /; y D .y1 ; : : : ; yn / 2 Rn , hx; yi D

n X

xi yi ;

iD1

m is a natural number, !i > 0; i D 1; : : : ; m and that A D fai 2 Rn W i D 1; : : : ; mg; where ai1 6D ai2 for all i1 ; i2 2 f1; : : : ; mg satisfying i1 6D i2 . Set f .x/ D

m X

!i kx  ai k for all x 2 Rn

(6.1)

iD1

and inf.f / D infff .z/ W z 2 Rn g: We say that the vectors ai ; : : : ; am are collinear if there exist y; b 2 Rn and t1 ; : : : ; tm 2 R1 such that ai D y C ti b, i D 1; : : : ; m. We suppose that the vectors ai ; : : : ; am are not collinear. In this chapter we study the Fermat–Weber location problem f .x/ ! min; x 2 Rn by using Weiszfeld’s method [110] which was recently revisited in [18]. This problem is often called the Fermat–Torricelli problem named after the mathematicians originally formulated (Fermat) and solved (Torricelli) it in the case of three points; Weber (as well as Steiner) considered its extension for finitely many points. For a full treatment of this problem with a modified proof of the Weiszfeld algorithm by using the subdifferential theory of convex analysis (as well for generalized versions of the Fermat–Torricelli and related problems) with no presence of computational errors see [86]. Since the function f is continuous and satisfies a growth condition, this problem has a solution which is denoted by x 2 Rn . Thus f .x / D inf.f /:

(6.2)

6.2 Preliminaries

87

In view of Theorem 2.1 of [18] this solution is unique but in our study we do not use this fact. If x 62 A, then f 0 .x / D

m X

!i kx  ai k1 .x  ai / D 0:

(6.3)

iD1

If x D ai with some i 2 f1; : : : ; mg, then     X   m  1   !j kx  aj k .x  aj /   !i :   jD1;j6Di

(6.4)

6.2 Preliminaries For each x 2 Rn n A set T.x/ D

m X

!1 1

!i kx  ai k

iD1

m X .kx  ai k1 !i ai /:

(6.5)

iD1

Let y 2 Rn n A satisfy T.y/ D y: This equality is equivalent to the relation yD

m X

!1 1

!i ky  ai k

iD1

m X .ky  ai k1 !i ai / iD1

which in its turn is equivalent to the equality m X

.!i ky  ai k1 .y  ai // D 0:

iD1

It is easy to see that the last equality is equivalent to the relation f 0 .y/ D 0: Thus for every y 2 Rn n A, T.y/ D y if and only if f 0 .y/ D 0:

(6.6)

88

6 Weiszfeld’s Method

For each x 2 Rn and every y 2 Rn n A set h.x; y/ D

m X

!i ky  ai k1 kx  ai k2 :

(6.7)

iD1

Let y 2 Rn n A and consider the function s D h.; y/ W Rn ! R1 which is strictly convex and possesses a unique minimizer x satisfying the relation 0 D s0 .x/ D 2

m X

!i ky  ai k1 .x  ai /

iD1

which is equivalent to the equality T.y/ D x: This implies that h.T.y/; y/  h.x; y/ for all x 2 Rn :

(6.8)

Lemma 6.1 ([18]). (i) For every y 2 Rn n A, h.T.y/; y/  h.x; y/ for all x 2 Rn : (ii) For every y 2 Rn n A, h.y; y/ D f .y/: (iii) For every x 2 Rn and every y 2 Rn n A, h.x; y/  2f .x/  f .y/: Proof. Assertion (i) was already proved [see (6.8)]. Assertion (ii) is evident. Let us prove assertion (iii). Let x 2 Rn and y 2 Rn n A. Clearly, for each a 2 R1 and each b > 0, a2 b1  2a  b:

6.2 Preliminaries

89

Therefore for all i D 1; : : : ; m, kx  ai k2 ky  ai k1  2kx  ai k  ky  ai k: This implies that h.x; y/ D

m X

!i ky  ai k1 kx  ai k2

iD1

2

m X

!i kx  ai k 

iD1

m X

!i ky  ai k D 2f .x/  f .y/:

iD1

Lemma 6.1 is proved. Lemma 6.2 ([18]). For every y 2 Rn n A, f .T.y//  f .y/ and the equality holds if and only if T.y/ D y. Proof. Let y 2 Rn n A. In view of Lemma 6.1 (i), (6.8) holds. By the strict convexity of the function x ! h.x; y/, x 2 Rn , T.y/ is its unique minimizer, by Lemma 6.2 (ii), h.T.y/; y/  h.y; y/ D f .y/ and if T.y/ 6D y, then this implies that h.T.y/; y/ < h.y; y/ D f .y/: Together with Lemma 6.1 (iii) this implies that 2f .T.y//  f .y/  h.T.y/; y/  f .y/ and if T.y/ 6D y, then 2f .T.y//  f .y/  h.T.y/; y/ < h.y; y/ D f .y/: This completes the proof of Lemma 6.2. For every x 2 Rn n A set L.x/ D

m X iD1

!i kx  ai k1 :

(6.9)

90

6 Weiszfeld’s Method

For j D 1; : : : ; m set L.aj / D

m X

!i kaj  ai k1 :

(6.10)

iD1;i6Dj

Clearly, for each x 2 Rn n A, T.x/ D x  L.x/1 f 0 .x/:

(6.11)

Lemma 6.3. Let y 2 Rn n A. Then f .T.y//  f .y/ C hf 0 .y/; T.y/  yi C 21 L.y/kT.y/  yk2 : Proof. Clearly, the function x ! h.x; y/, x 2 Rn is quadratic. Therefore its secondorder Taylor expansion around y is exact and can be written as h.x; y/ D h.y; y/ C h@x h.y; y/; x  yi C L.y/kx  yk2 : Combined with the relations h.y; y/ D f .y/ and @x h.y; y/ D 2f 0 .y/ this implies that h.x; y/ D f .y/ C 2hf 0 .y/; x  yi C L.y/kx  yk2 : For x D T.y/ the relation above implies that h.T.y/; y/ D f .y/ C 2hf 0 .y/; T.y/  yi C L.y/kT.y/  yk2 : Together with Lemma 6.1 (iii) this implies that 2f .T.y//  f .y/  h.T.y/; y/ D f .y/ C 2hf 0 .y/; T.y/  yi C L.y/kT.y/  yk2 ; 2f .T.y//  2f .y/ C 2hf 0 .y/; T.y/  yi C L.y/kT.y/  yk2 and f .T.y//  f .y/ C hf 0 .y/; T.y/  yi C 21 L.y/kT.y/  yk2 : Lemma 6.3 is proved.

6.3 The Basic Lemma

91

6.3 The Basic Lemma Set Q D maxfkai k W i D 1; : : : ; mg: M

(6.12)

By (6.1) and (6.12), f .0/ 

m X

Q !i M:

(6.13)

iD1

Q Since x is the minimizer of f it follows from (6.1) We show that kx k  2M. and (6.13) that m X

Q  f .0/  f .x / D !i M

m X

iD1

!i kx  ai k:

(6.14)

iD1

There exists j0 2 f1; : : : ; mg such that kx  aj k  kx  ai k; i D 1; : : : ; m:

(6.15)

By (6.12), (6.14), and (6.15), m X iD1

Q  !i M

m X

!i kx  aj k;

iD1

Q kx  aj k  M 

and Q kx k  2M:

(6.16)

Q and y 2 Rn n A satisfy Lemma 6.4. Let M  M kyk  M:

(6.17)

Then kT.y/k  3M: Proof. In view of Lemma 6.2, f .T.y//  f .y/:

(6.18)

92

6 Weiszfeld’s Method

It follows from (6.1), (6.12), and (6.17) that f .y/ D

m X

!i ky  ai k  2M

iD1

m X

!i :

(6.19)

iD1

By (6.1), (6.18), and (6.19), m X

!i kT.y/  ai k D f .T.y//  f .y/  2M

m X

!i :

(6.20)

kT.y/  aj k  kT.y/  ai k; i D 1; : : : ; m:

(6.21)

iD1

iD1

There exists j 2 f1; : : : ; mg such that

Relations (6.20) and (6.21) imply that m X

!i kT.y/  aj k  2M

m X

iD1

!i ;

iD1

kT.y/  aj k  2M: Together with (6.12) this implies that kT.y/k  3M. Lemma 6.4 is proved. Q ı 2 .0; 1, y 2 Rn n A satisfy Lemma 6.5 (The Basic Lemma). Let M  M, kyk  M;

(6.22)

kT.y/  xk  ı

(6.23)

kzk  M:

(6.24)

x 2 Rn satisfy

and let z 2 Rn satisfy

Then 1

2

2

f .x/  f .z/  2 L.y/.kz  yk  kz  xk / C ı 8M C 1 C

m X

! !i :

iD1

Proof. Relations (6.1) and (6.23) imply that jf .x/  f .T.y//j  kx  T.y/k

m X iD1

!i  ı

m X iD1

!i :

(6.25)

6.3 The Basic Lemma

93

In view of Lemma 6.3, f .T.y//  f .y/ C hf 0 .y/; T.y/  yi C 21 L.y/kT.y/  yk2 :

(6.26)

Since the function f is convex we have f .y/  f .z/ C hf 0 .y/; y  zi:

(6.27)

By (6.26) and (6.27), f .T.y//  f .z/ C hf 0 .y/; y  zi C hf 0 .y/; T.y/  yi C 21 L.y/kT.y/  yk2 D f .z/ C hf 0 .y/; T.y/  zi C 21 L.y/kT.y/  yk2 :

(6.28)

It follows from (6.25) and (6.28) that 0

1

2

f .x/  f .z/ C hf .y/; T.y/  zi C 2 L.y/kT.y/  yk C ı

m X

!i :

(6.29)

iD1

In view of (6.11), f 0 .y/ D L.y/.y  T.y//:

(6.30)

By (6.29) and (6.30), f .x/  f .z/ C L.y/hy  T.y/; T.y/  zi C 21 L.y/kT.y/  yk2 C ı

m X

!i :

(6.31)

iD1

Lemma 2.1 implies that hy  T.y/; T.y/  zi D 21 Œkz  yk2  kz  T.y/k2  ky  T.y/k2 :

(6.32)

It follows from (6.31) and (6.32) that f .x/  f .z/ C 21 L.y/.kz  yk2  kz  T.y/k2 / C ı

m X

!i :

(6.33)

iD1

Lemma 6.4 and (6.22) imply that kT.y/k  3M:

(6.34)

94

6 Weiszfeld’s Method

By (6.23), (6.24), and (6.34), jkz  T.y/k2  kz  xk2 j D jkz  T.y/k  kz  xkj.kz  T.y/k C kz  xk/  kx  T.y/k.8M C 1/  ı.8M C 1/:

(6.35)

It follows from (6.33) and (6.35) that 1

2

2

f .x/  f .z/  2 L.y/.kz  yk  kz  xk / C ı 8M C 1 C

m X

! !i :

iD1

Lemma 6.5 is proved.

6.4 The Main Result Let ı 2 .0; 1 and a positive number 0  ı. Choose p 2 f1; : : : ; mg such that f .ap /  f .ai /; i D 1; : : : ; m:

(6.36)

For each j D 1; : : : ; m set rj D

m X

!i kai  aj k1 .aj  ai /:

(6.37)

iD1;i6Dj

In order to solve our minimization problem we need to calculate rp D

m X

!i kai  ap k1 .ap  ai /:

iD1;i6Dp

Since our computer system produces computational errors we can obtain only a vector rOp 2 Rn such that kOrp  rp k  0 . Proposition 6.6. Assume that rOp 2 Rn satisfies kOrp  rp k  0

(6.38)

kOrp k  !p C 20 :

(6.39)

and

6.4 The Main Result

95

Then Q 0: f .ap /  inf.f / C 9M Q was defined by (6.12).) (Note that M Proof. By (6.38) and (6.39), krp k  kOrp k C 0  !p C 30 :

(6.40)

It follows from (6.1), (6.37), and (6.40) that there exists l 2 @f .ap /

(6.41)

klk  30 :

(6.42)

such that

By the convexity of f , (6.2), (6.12), (6.16), (6.41), and (6.42), f .x /  f .ap / C hl; x  ap i Q 0  f .ap /  klk.kx k C kap k/  f .ap /  9M and Q 0: f .ap /  f .x / C 9M Proposition 6.6 is proved. In view of Proposition 6.6, if rOp satisfies (6.38) and (6.39), then ap is an approximate solution of our minimization problem. It is easy to see that the following proposition holds. Proposition 6.7. Assume that rOp 2 Rn satisfies kOrp  rp k  0 and kOrp k > !p C 20 : Then krp k > !p C 0 and ap is not a minimizer of f . Lemma 6.8 ([18]). Let krp k > !p . Then f .ap /  f .ap  .krp k  !p /L.ap /1 krp k1 rp /  .krp k  !p /2 .2L.ap //1 : Proposition 6.9. Assume that krp k > !p ;

(6.43)

96

6 Weiszfeld’s Method

dp 2 Rn satisfies kdp C krp k1 rp k  ı;

(6.44)

jtp  L.ap /1 .krp k  !p /j  ı

(6.45)

kx0  ap  tp dp k  ı:

(6.46)

tp  0 satisfies

and that x0 2 Rn satisfies

Then kdp k  1 C ı;

(6.47)

1

tp  L.ap / .krp k  !p / C ı;

(6.48)

Q C 2.2ı C L.ap /1 .krp k  !p //; kx0 k  M 1

(6.49) 1

kx0  ap C .krp k  !p /L.ap / krp k rp k  ı.3 C .krp k  !p /L.ap /1 /;

(6.50) 1

1

jf .x0 /  f .ap  .krp k  !p /L.ap / krp k rp /k  ı.3 C .krp k  !p /L.ap /1 /

m X

!i

(6.51)

iD1

and f .ap /  f .x0 /  .krp k  !p /2 .2L.ap //1  ı.3 C .krp k  !p /L.ap /1 /

m X

!i : (6.52)

iD1

Proof. In view of (6.44), (6.47) is true. Inequality (6.45) implies (6.48). By (6.12) and (6.46)–(6.48), kx0 k  ı C kap k C tp kdp k Q C 2tp  M Q C 2.2ı C L.ap /1 .krp k  !p //  ıCM and (6.49) is true. It follows from (6.44)–(6.47) that kx0  ap C .krp k  !p /L.ap /1 krp k1 rp k  ı C ktp dp C .krp k  !p /L.ap /1 krp k1 rp k

6.4 The Main Result

97

 ı C ktp dp  .krp k  !p /L.ap /1 dp k Ck.krp k  !p /L.ap /1 .dp C krp k1 rp /k  ı C ı.1 C ı/ C .krp k  !p /L.ap /1 ı  ı.3 C .krp k  !p /L.ap /1 / and (6.50) holds. Relations (6.1) and (6.50) imply (6.51). Relation (6.52) follows from (6.51) and Lemma 6.8. Proposition 6.9 is proved. The next theorem which is proved in Sect. 6.5, is our main result. Theorem 6.10. Let krp k > !p ;

(6.53)

Q C 4 C 2.krp k  !p /L.ap / ; M0 D 3M 1

(6.54)

a positive number ı satisfy 1

ı < 12 .krp k  !p /

m X

!1 !i

(6.55)

iD1

and 2ı 8M0 C 1 C

m X

! < .krp k  !p /2 .16L.ap //1 ;

!i

(6.56)

iD1

0 D 4ı 16M0 C 1 C

m X

! !i Œ144L.ap /2 .krp k  !p /4 M02

iD1 m X C1.. !i /2 C 1/

(6.57)

iD1

and n0 D bı

1

8M0 C 1 C

m X

!1 !i

.krp k  !p /2 .8L.ap //1 c C 1:

(6.58)

iD1 n Assume that tp  0, dp 2 Rn and x0 2 Rn satisfy (6.44)–(6.46), fxi g1 iD1  R and that for each integer i  0 satisfying xi 62 A,

kT.xi /  xiC1 k  ı:

(6.59)

98

6 Weiszfeld’s Method

Then x0 62 A and there exists j 2 f0; : : : ; n0 g such that xi 62 A; i 2 f0; : : : ; jg n fjg; f .xj /  inf.f / C 0 : Note that in the theorem above ı is the computational error produced by our computer system. In order to obtain a good approximate solution we needP bc1 ı 1 c iterations [see (6.58)], where c1 > 0 is a constant depending only on M0 , m iD1 !i , krp k  !p and L.ap /. As a result, we obtain a point x 2 Rn such that f .x/  inf.f / C c2 ı [see (6.57)], where the constant c2 > 0 depends only on M0 , L.ap /.

Pm iD1

!i , krp k  !p and

6.5 Proof of Theorem 6.10 Proposition 6.9, (6.44)–(6.46), (6.55), and (6.56) imply that f .x0 /  f .ap / 2

 .krp k  !p / .2L.ap //

1

1

C ı.3 C .krp k  !p /L.ap / /

m X

!i

iD1

 f .ap /  .krp k  !p /2 .4L.ap //1 :

(6.60)

By (6.36) and (6.60), x0 62 A:

(6.61)

If f .x0 /  inf.f / C 0 or f .x1 /  inf.f / C 0 ; then in view of (6.61) the assertion of the theorem holds with j D 0 or j D 1, respectively. Consider the case with f .x0 / > inf.f / C 0 and f .x1 / > inf.f / C 0 :

(6.62)

6.5

Proof of Theorem 6.10

99

Assume that k 2 Œ0; n0  is an integer, xi 62 A; i D 0; : : : ; k

(6.63)

f .xi / > inf.f / C 0 ; i D 0; : : : ; k C 1:

(6.64)

and

(Note that in view of (6.61) and (6.62), relations (6.63) and (6.64) hold for k D 0.) For all integers i  0, set 0 i D iı @8M0 C 1 C

m X

1 !j A :

(6.65)

jD1

By (6.56), (6.58), (6.60), and (6.64), for all i D 0; : : : ; n0 , i  n0 ı.8M0 C 1 C

m X

!j /

jD1

 ı.8M0 C 1 C

m X

!j / C .krp k  !p /2 .8L.ap //1

jD1

 .krp k  !p /2 .8L.ap //1 C .krp k  !p /2 .16L.ap //1  41  3.f .ap /  f .x0 //:

(6.66)

Remind (see Sect. 6.1) that x 2 Rn satisfies f .x / D inf.f /:

(6.67)

We show that for all j D 0; : : : ; k C 1, f .xj /  f .x0 / C j ; 

kxj  x k  M0 :

(6.68) (6.69)

In view of (6.16), Q kx k  2M:

(6.70)

Proposition 6.9, (6.44)–(6.46), and (6.49) imply that Q C 2.2 C L.ap /1 .krp k  !p //: kx0 k  M

(6.71)

100

6 Weiszfeld’s Method

By (6.54), (6.70), and (6.71), kx0  x k  M0 : Thus (6.68) and (6.69) hold for j D 0. Assume that an integer j 2 f0; : : : ; kg and (6.68) and (6.69) hold. By (6.36), (6.60), (6.66), (6.68), and the relation k  n0 , f .xj /  f .x0 / C n0  f .x0 / C 41  3.f .ap /  f .x0 // < f .ap / and xj 62 A:

(6.72)

vi 2 @f .ai /:

(6.73)

Let i 2 f1; : : : ; mg and

In view of (6.1), @f .ai / D

m X

!q kai  aq k1 .ai  aq / C !i BRn .0; 1/:

(6.74)

qD1;q6Di

Relations (6.73) and (6.74) imply that kvi k 

m X

!q C !i D

qD1;q6Di

m X

!q :

(6.75)

qD1

It follows from (6.68), (6.73), and (6.75) that f .ai /  f .x0 /  f .ai /  f .xj / C j  hvi ; ai  xj i C j  kvi kkai  xj k C j  kai  xj k

m X

!q C j :

(6.76)

!q C j ; i D 1; : : : ; m:

(6.77)

qD1

By (6.36) and (6.76), f .ap /  f .x0 /  kai  xj k

m X qD1

6.5

Proof of Theorem 6.10

101

In view of (6.66) and (6.77), for all i D 1; : : : ; m, 0 11 m X kai  xj k  41 .f .ap /  f .x0 // @ !q A qD1

and kai  xj k1  4.f .ap /  f .x0 //1

m X

!q :

(6.78)

qD1

It follows from (6.9), (6.72), and (6.78) that m m X X L.xj / D .!i kxj  ai k1 /  4 !i iD1

!2 .f .ap /  f .x0 //1 :

(6.79)

iD1

Lemma 6.2, (6.1), (6.59), (6.64), (6.68), and (6.72) imply that f .xjC1 /  f .T.xj // C kxjC1  T.xj /k

m X

!i

iD1

 f .xj / C ı

m X

!i  f .x0 / C jC1 :

(6.80)

iD1

It follows from (6.54), (6.59), (6.64), (6.67), (6.69), (6.70), (6.72), (6.79), and Lemma 6.5 applied with M D 2M0 , z D x , y D xj , and x D xjC1 that 0 < f .xjC1 /  f .x / !2 m X 2 !i .f .ap /  f .x0 //1 .kx  xj k2  kx  xjC1 k2 / iD1

C ı 16M0 C 1 C

m X

! !i :

(6.81)

iD1

By (6.57), (6.60), and (6.81), kx  xj k  kx  xjC1 k: Therefore in view of the relation above, (6.80) and (6.81), we showed by induction that (6.68) and (6.69) hold for j D 0; : : : ; kC1 and that (6.81) holds for j D 0; : : : ; k. It follows from (6.58), (6.60), (6.64), (6.68) and the relation k  n0 that

102

6 Weiszfeld’s Method

f .xkC1 /  f .x0 / C kC1  f .x0 / C .n0 C 1/ı 8M0 C 1 C

m X

! !i

iD1 2

< f .x0 / C .krp k  !p / .8L.ap //

1

C 2ı 8M0 C 1 C

m X

! !i

< f .ap /

iD1

and xkC1 62 A:

(6.82)

By (6.81) which holds for all j D 0; : : : ; k, .k C 1/0
0 set B.x; r/ D fy 2 X W kx  yk  rg: Let C be a nonempty closed convex subset of X. Assume that f W X ! R1 is a convex continuous function which is bounded from below on C. Recall that for each x 2 X, @f .x/ D fu 2 X W f .y/  f .x/  hu; y  xi for all y 2 Xg:

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_7

105

106

7 The Extragradient Method for Convex Optimization

We consider the minimization problem f .x/ ! min; x 2 C: Denote Cmin D fx 2 C W f .x/  f .y/ for all y 2 Cg:

(7.1)

We assume that Cmin 6D ;. We suppose that f is Gâteaux differential at any point x 2 X and for x 2 X denote by f 0 .x/ 2 X the Gâteaux derivative of f at x. This implies that for any x 2 X and any h 2 X hf 0 .x/; hi D lim t1 Œf .x C th/  f .x/: t!0

(7.2)

We suppose that the mapping f 0 W X ! X is Lipschitz on all the bounded subsets of X. Set inf.f I C/ D infff .z/ W z 2 Cg:

(7.3)

We study the minimization problem with the objective function f , over the set C, using the extragradient method introduced in Korpelevich [75]. By Lemma 2.2, for each nonempty closed convex set D  X and for each x 2 X, there is a unique point PD .x/ 2 D satisfying kx  PD .x/k D inffkx  yk W y 2 Dg; kPD .x/  PD .y/k  kx  yk for all x; y 2 X and hz  PD .x/; x  PD .x/i  0 for each x 2 X and each z 2 D. The following theorem is our first main result of this chapter. Theorem 7.1. Let M0 > 0, M1 > 0, L > 0,  2 .0; 1/, B.0; M0 / \ Cmin 6D ;;

(7.4)

f 0 .B.0; 3M0 //  B.0; M1 /;

(7.5)

kf 0 .z1 /  f 0 .z2 /k  Lkz1  z2 k for all z1 ; z1 2 B.0; 3M0 C M1 C 1/;

(7.6)

0 < ˛ < ˛   1; ˛  L  1;

(7.7)

7.1 Preliminaries and the Main Results

107

an integer k satisfy k > 4M02 ˛1  1

(7.8)

and let a positive number ı satisfy ı < 41 .2M0 C 1/1 ˛ :

(7.9)

Assume that  1 1 f˛i g1 iD0  Œ˛ ; ˛ ; fxi giD0  X; fyi giD0  X;

(7.10)

kx0 k  M0

(7.11)

and that for each integer i  0, kyi  PC .xi  ˛i f 0 .xi //k  ı; 0

kxiC1  PC .xi  ˛i f .yi //k  ı:

(7.12) (7.13)

Then there is an integer j 2 Œ0; k such that kxi k  3M0 ; i D 0; : : : ; j; f .PC .xj  ˛j f 0 .xj ///  inf.f I C/ C : In Theorem 7.1 our goal is to obtain a point  2 C such that f ./  inf.f I C/ C ; where  > 0 is given. In order to meet this goal, the computational errors, produced by our computer system, should not exceed c1 , where c1 > 0 is a constant depending only on M0 , ˛ [see (7.9)]. The number of iterations is bc2  1 c, where c2 > 0 is a constant depending only on M0 , ˛ . It is easy to see that the following proposition holds. Proposition 7.2. If limx2C; kxk!1 f .x/ D 1 and the space X is finitedimensional, then for each  > 0 there exists > 0 such that if x 2 C satisfies f .x/  inf.f I C/ C , then d.x; Cmin /  . The following theorem is our second main result of this chapter.

108

7 The Extragradient Method for Convex Optimization

Theorem 7.3. Let lim

x2C; kxk!1

f .x/ D 1

(7.14)

and let the following property hold: (C) for each  > 0 there exists > 0 such that if x 2 C satisfies f .x/  inf.f I C/ C , then d.x; Cmin /  =2. Let  2 .0; 1/, 2 .0; .=4/2 /

(7.15)

be as guaranteed by property (C), M0 > 1, M1 > 0, L > 0, Cmin  B.0; M0  1/;

(7.16)

f 0 .B.0; 3M0 //  B.0; M1 /;

(7.17)

0

0

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z1 2 B.0; 3M0 C M1 C 1/; 



0 < ˛ < ˛  1; ˛ L < 1;

(7.18) (7.19)

an integer k satisfy k > 8M02 1 minf˛ ; 1  .˛  /2 L2 g1

(7.20)

and let a positive number ı satisfy 16ı.8M0 C 8/  minf˛ ; 1  .˛  /2 L2 g:

(7.21)

 1 1 f˛i g1 iD0  Œ˛ ; ˛ ; fxi giD0  X; fyi giD0  X;

(7.22)

Assume that

kx0 k  M0

(7.23)

and that for each integer i  0, kyi  PC .xi  ˛i f 0 .xi //k  ı;

(7.24)

kxiC1  PC .xi  ˛i f 0 .yi //k  ı:

(7.25)

Then d.xi ; Cmin / <  for all integers i  k. The chapter is organized as follows. Section 7.2 contains auxiliary results. Theorem 7.1 is proved in Sect. 7.3 while Theorem 7.3 is proved in Sect. 7.4. The results of this chapter were obtained in [126].

7.2 Auxiliary Results

109

7.2 Auxiliary Results We use the assumptions, notation, and definitions introduced in Sect. 7.1. Lemma 7.4. Let u 2 Cmin ; u 2 X; ˛ > 0; 0

(7.26) 0

v D PC .u  ˛f .u//; uN D PC .u  ˛f .v//:

(7.27)

Then kNu  u k2  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2˛kNu  vkkf 0 .u/  f 0 .v/k: Proof. It is easy to see that hf 0 .v/; x  vi  f .x/  f .v/ for all x 2 X:

(7.28)

hu  uN ; f 0 .v/i  hv  uN ; f 0 .v/i  f .u /  f .v/:

(7.29)

In view of (7.28),

It follows from (7.27) and Lemma 2.2 that hNu  v; .u  ˛f 0 .u//  vi  0:

(7.30)

Relation (7.30) implies that hNu  v; .u  ˛f 0 .v//  vi  ˛hNu  v; f 0 .u/  f 0 .v/i:

(7.31)

Set z D u  ˛f 0 .v/:

(7.32)

It follows from (7.27) and (7.32) that kNu  u k2 D kz  u C PC .z/  zk2 D kz  u k2 C kz  PC .z/k2 C 2hPC .z/  z; z  u i:

(7.33)

Relation (7.26) and Lemma 2.2 imply that 2kz  PC .z/k2 C 2hPC .z/  z; z  u i D 2hz  PC .z/; u  PC .z/i  0:

(7.34)

110

7 The Extragradient Method for Convex Optimization

It follows from (7.33), (7.34), (7.32), (7.27), and (7.28) that kNu  u k2  kz  u k2  kz  PC .z/k2 D ku  ˛f 0 .v/  u k2  ku  ˛f 0 .v/  uN k2 D ku  u k2  ku  uN k2 C 2˛hu  uN ; f 0 .v/i  ku  u k2  ku  uN k2 C2˛hv  uN ; f 0 .v/i C 2˛Œf .u /  f .v/:

(7.35)

In view of (7.31) and (7.35), kNu  u k2  ku  u k2 C2˛hv  uN ; f 0 .v/i C 2˛Œf .u /  f .v/ hu  v C v  uN ; u  v C v  uN i D ku  u k2 C 2˛hv  uN ; f 0 .v/i C2˛Œf .u /  f .v/  ku  vk2 kv  uN k2  2hu  v; v  uN i D ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2hv  uN ; ˛f 0 .v/  u C vi  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2˛hNu  v; f 0 .u/  f 0 .v/i  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2˛kNu  vkkf 0 .u/  f 0 .v/k: This completes the proof of Lemma 7.4. Lemma 7.5. Let u 2 Cmin ; M0 > 0; M1 > 0; L > 0; ˛ 2 .0; 1;

(7.36)

f 0 .B.u ; M0 //  B.0; M1 /;

(7.37)

kf 0 .z1 /  f 0 .z2 /k  Lkz1  z2 k for all z1 ; z1 2 B.u ; M0 C M1 /;

(7.38)

˛L  1

(7.39)

and let u 2 B.u ; M0 /; v D PC .u  ˛f 0 .u//; uN D PC .u  ˛f 0 .v//:

(7.40)

7.2 Auxiliary Results

111

Then kNu  u k2  ku  u k2 C2˛Œf .u /  f .v/  ku  vk2 .1  ˛ 2 L2 /: Proof. Lemma 7.4 and (7.36) imply that kNu  u k2  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2˛kNu  vkkf 0 .u/  f 0 .v/k:

(7.41)

In view of (7.40) and (7.37), kf 0 .u/k  M1 :

(7.42)

It follows from (7.40), (7.36), Lemma 2.2, and (7.42) that kv  u k  ku  ˛f 0 .u/  u k  ku  u k C ˛kf 0 .u/k  M0 C ˛M1 :

(7.43)

In view of (7.40), (7.43), (7.36), and (7.39), kf 0 .u/  f 0 .v/k  Lku  vk:

(7.44)

By (7.41) and (7.44), kNu  u k2  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C 2˛kNu  vkku  vkL  ku  u k2  ku  vk2  kv  uN k2 C2˛Œf .u /  f .v/ C ˛ 2 L2 ku  vk2 C kNu  vk2  ku  u k2 C 2˛Œf .u /  f .v/  ku  vk2 .1  ˛ 2 L2 /: (7.45) By (7.45), kNu  u k2  ku  u k2 C 2˛Œf .u /  f .v/  ku  vk2 .1  ˛ 2 L2 /: This completes the proof of Lemma 7.5. Lemma 7.6. Let u 2 Cmin ; M0 > 0; M1 > 0; L > 0; ˛ 2 .0; 1; ı 2 .0; 1/;

(7.46)

f 0 .B.u ; M0 //  B.0; M1 /;

(7.47)

112

7 The Extragradient Method for Convex Optimization

kf 0 .z1 /  f 0 .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.u ; M0 C M1 C 1/;

(7.48)

˛L  1:

(7.49)

Assume that x 2 B.u ; M0 /; y 2 X;

(7.50)

ky  PC .x  ˛f 0 .x//k  ı; 0

xQ 2 X; kQx  PC .x  ˛f .y//k  ı:

(7.51) (7.52)

Then kQx  u k2  4ı.M0 C 1/ C kx  u k2 C2˛Œf .u /  f .PC .x  ˛f 0 .x/// jjx  PC .x  ˛f 0 .x//jj2 .1  ˛ 2 L2 /: Proof. Put v D PC .x  ˛f 0 .x//; z D PC .x  ˛f 0 .v//:

(7.53)

Lemma 7.5, (7.46), (7.47), (7.48), (7.49), (7.50), and (7.53) imply that kz  u k2  kx  u k2 C 2˛Œf .u /  f .v/  kx  vk2 .1  ˛ 2 L2 /:

(7.54)

It is clear that kQx  u k2 D kQx  z C z  u k2 D kQx  zk2 C 2hQx  z; z  u i C kz  u k2  kQx  zk2 C 2kQx  zkkz  u k C kz  u k2 :

(7.55)

In view of (7.51) and (7.53), kv  yk  ı:

(7.56)

zQ D PC .x  ˛f 0 .y//:

(7.57)

Put

By (7.53), (7.46), Lemma 2.2, (7.50), and (7.47), ku  vk  ku  xk C ˛kf 0 .x/k  M0 C ˛M1 :

(7.58)

7.3 Proof of Theorem 7.1

113

Relations (7.58), (7.56), and (7.46) imply that ku  yk  ku  vk C kv  yk  M0 C ˛M1 C 1:

(7.59)

It follows from (7.52), (7.57), (7.53), Lemma 2.2, (7.58), (7.59), (7.46), (7.48), (7.56), and (7.49) that kQx  zjj  kQx  zQk C kQz  zk  ı C kQz  zk  ı C ˛kf 0 .y/  f 0 .v/k  ı C ˛Lkv  yk  ı C ˛Lı D ı.1 C ˛L/  2ı:

(7.60)

In view of (7.55), (7.60), (7.54), (7.46), (7.50), and (7.53), kQx  u k2  4ı 2 C kx  u k2 C 2˛Œf .u /  f .v/ kx  vk2 .1  ˛ 2 L2 / C 4ıkx  u k  4ı.M0 C 1/ C kx  u k2 C2˛Œf .u /  f .PC .x  ˛f 0 .x///  kx  vk2 .1  ˛ 2 L2 / D 4ı.M0 C 1/ C kx  u k2 C 2˛Œf .u /  f .PC .x  ˛f 0 .x/// kx  PC .x  ˛f 0 .x//k2 .1  ˛ 2 L2 /: This completes the proof of Lemma 7.6.

7.3 Proof of Theorem 7.1 In view of (7.4), there exists a point u 2 Cmin \ B.0; M0 /:

(7.61)

It follows from (7.11) and (7.61) that kx0  u k  2M0 :

(7.62)

Assume that i  0 is an integer and that xi 2 B.u ; 2M0 /:

(7.63)

(It is clear that in view of (7.62), inclusion (7.63) is valid for i D 0.) It follows from (7.61), (7.63), (7.7), (7.11), (7.5), (7.6), (7.12), (7.13), and Lemma 7.6 applied

114

7 The Extragradient Method for Convex Optimization

with x D xi , y D yi xQ D xiC1 , ˛ D ˛i that kxiC1  u k2  4ı.2M0 C 1/ Ckxi  u k2 C 2˛i Œf .u /  f .PC .xi  ˛i f 0 .xi ///:

(7.64)

Thus we have shown that the following property holds: (P1) If an integer i  0 satisfies (7.63), then inequality (7.64) is valid. We claim that there exists an integer i 2 Œ0; k for which f .PC .xi  ˛i f 0 .xi ///  f .u / C :

(7.65)

Assume the contrary. Then relations (7.9) and (7.10) imply that for each integer i 2 Œ0; k, 2˛i Œf .u /  f .PC .xi  ˛i f 0 .xi /// C 4ı.2M0 C 1/  2˛i ./ C 4ı.2M0 C 1/  2˛  C 4ı.2M0 C 1/  ˛ :

(7.66)

It follows from (7.62), (7.66), and property (P1) that for each integer i 2 Œ0; k, kxiC1  u k2  kxi  u k2  ˛ :

(7.67)

In view of (7.63) and (7.67), 4M02  kx0  u k2  kxk  u k2 D

k1 X Œkxi  u k2  kxiC1  u k2   k˛  iD0

and k  4M02 ˛1  1 : This contradicts (7.8). The contradiction we have reached proves that there exists an integer j 2 Œ0; k such that f .PC .xj  ˛j f 0 .xj ///  f .u / C : We may assume without loss of generality that for each integer i  0 satisfying i < j, f .PC .xi  ˛i f 0 .xi /// > f .u / C :

(7.68)

7.4 Proof of Theorem 7.3

115

It follows from (7.68), (7.9), and (7.10) that for any integer i  0 satisfying i < j, inequality (7.66) is valid. Combined with (7.62), property (P1), and (7.61) this implies that for each integer i satisfying 0  i  j, we have kxi  u k  2M0 and kxi k  3M0 . This completes the proof of Theorem 7.1.

7.4 Proof of Theorem 7.3 Let u be an arbitrary element of Cmin :

(7.69)

In view of (7.69) and (7.16), ku k  M0  1:

(7.70)

Relations (7.70) and (7.23) imply that kxi  u k  2M0  1:

(7.71)

Assume that i  0 is an integer such that xi 2 B.u ; 2M0 /:

(7.72)

(It is clear that in view of (7.71) inclusion (7.72) is valid for i D 0.) It follows from (7.69), (7.9), (7.17)–(7.19), (7.72), (7.70), (7.24), and Lemma 7.6 applied with x D xi , y D yi , xQ D xiC1 , ˛ D ˛i that kxiC1  u k2  4ı.2M0 C 1/ Ckxi  u k2 C 2˛i Œf .u /  f .PC .xi  ˛i f 0 .xi /// kxi  PC .xi  ˛i f 0 .xi //k2 .1  ˛i2 L2 / and by (7.22), kxi  u k2  kxiC1  u k2  2˛ Œf .PC .xi  ˛i f 0 .xi ///  f .u / Ckxi  PC .xi  ˛i f 0 .xi //k2 .1  ˛i2 L2 /  4ı.2M0 C 1/:

(7.73)

116

7 The Extragradient Method for Convex Optimization

Thus we have shown that the following property holds: (P2)

If an integer i  0 satisfies (7.72), then inequality (7.73) is valid.

Assume that an integer i  0 satisfies (7.72) and that maxff .PC .xi  ˛i f 0 .xi ///  f .u /; kxi  PC .xi  ˛i f 0 .xi //k2 g  :

(7.74)

It follows from (7.74), property (C), and (7.15) that d.PC .xi  ˛i f 0 .xi /; Cmin /  =2; kxi  PC .xi  ˛i f 0 .xi //k  1=2 < =4 and d.xi ; Cmin /  3=4: Thus we have shown that the following property holds: (P3)

If an integer i  0 satisfies (7.72) and (7.74), then d.xi ; Cmin /  3=4:

Assume that an integer i  0 satisfies inclusion (7.72) and that maxff .PC .xi  ˛i f 0 .xi ///  f .u /; kxi  PC .xi  ˛i f 0 .xi //k2 g > :

(7.75)

It follows from (7.72), property (P2), (7.73), (7.75), (7.19), (7.21), and (7.22) that kxi  u k2  kxiC1  u k2  minf˛ ; 1  .˛  /2 L2 g  4ı.2M0 C 1/  21 minf˛ ; 1  .˛  /2 L2 g: Thus we have shown that the following property holds: (P4)

If an integer i  0 satisfies (7.72) and (7.75), then kxi  u k2  kxiC1  u k2  21 minf˛ ; 1  .˛  /2 L2 g

and since u is an arbitrary point of Cmin , we have d.xiC1 ; Cmin /2  d.xi ; Cmin /2  . =2/ minf˛ ; 1  .˛  /2 L2 /g: We claim that there exists an integer i 2 Œ0; k such that (7.74) is valid.

7.4 Proof of Theorem 7.3

117

Assume the contrary. Then (7.75) holds for each integer i 2 Œ0; k. Combined with (7.71) and property (P4) this implies that 4M02  kx0  u k2  kxk  u k2 D

k1 X Œkxi  u k2  kxiC1  u k2  iD0

 k. =2/ minf˛ ; 1  .˛  /2 L2 g and k  8M02 1 minf˛ ; 1  .˛  /2 L2 g1 : This contradicts (7.20). The contradiction we have reached proves that there exists an integer j 2 Œ0; k such that (7.74) is valid with i D j. We may assume that for all integers i  0 satisfying i < j Eq. (7.75) holds. It follows from property (P4) and (7.71) that xj 2 B.u ; 2M0 /:

(7.76)

Property (P3), (7.76), and (7.74) with i D j imply that d.xj ; Cmin /  3=4:

(7.77)

Assume that an integer i  j and that d.xi ; Cmin / < :

(7.78)

There are two cases: (7.74) is valid; (7.75) is valid. Assume that (7.74) is true. In view of property (P3), (7.78), and (7.16), d.xi ; Cmin / < 3=4: Since u is an arbitrary point of the set Cmin we may assume without loss of generality that kxi  u k < .4=5/: It follows from (7.79), (7.15), property (P2), (7.73), (7.19), and (7.21) that kxiC1  u k  kxi  u k C 2.8ı.M0 C 1//1=2 < .4=5/ C =5 and d.xiC1 ; Cmin / < :

(7.79)

118

7 The Extragradient Method for Convex Optimization

Assume that (7.75) holds. Property (P4), (7.78), and (7.16) imply that d.xiC1 ; Cmin /2  d.xi ; Cmin /2  . =2/ minf˛ ; 1  .˛  /2 L2 g and d.xiC1 ; Cmin / < :

(7.80)

Thus (7.80) holds in both cases. We have shown that if an integer i  j and (7.78) holds, then (7.80) is true. Therefore, d.xi ; Cmin / <  for all integers i  k: This completes the proof of Theorem 7.3.

Chapter 8

A Projected Subgradient Method for Nonsmooth Problems

In this chapter we study the convergence of the projected subgradient method for a class of constrained optimization problems in a Hilbert space. For this class of problems, an objective function is assumed to be convex but a set of admissible points is not necessarily convex. Our goal is to obtain an -approximate solution in the presence of computational errors, where  is a given positive number.

8.1 Preliminaries and Main Results Let .X; h; i/ be a Hilbert space with an inner product h; i which induces a complete norm k  k. For each x 2 X and each nonempty set A  X put d.x; A/ D inffkx  yk W y 2 Ag: For each x 2 X and each r > 0 set B.x; r/ D fy 2 X W kx  yk  rg: Assume that f W X ! R1 is a convex continuous function which is Lipschitz on all bounded subsets of X. For each point x 2 X and each positive number  let @f .x/ D fl 2 X W f .y/  f .x/  hl; y  xi for all y 2 Xg

(8.1)

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_8

119

120

8 A Projected Subgradient Method for Nonsmooth Problems

be the subdifferential of f at x and let @ f .x/ D fl 2 X W f .y/  f .x/  hl; y  xi   for all y 2 Xg

(8.2)

be the -subdifferential of f at x. Let C be a closed nonempty subset of the space X. Assume that lim f .x/ D 1:

kxk!1

(8.3)

It means that for each M0 > 0 there exists M1 > 0 such that if a point x 2 X satisfies the inequality kxk  M1 , then f .x/ > M0 . Define inf.f I C/ D infff .z/ W z 2 Cg:

(8.4)

Since the function f is Lipschitz on all bounded subsets of the space X it follows from (8.4) that inf.f I C/ is finite. Set Cmin D fx 2 C W f .x/ D inf.f I C/g:

(8.5)

It is well known that if the set C is convex, then the set Cmin is nonempty. Clearly, the set Cmin 6D ; if the space X is finite-dimensional. In this chapter we assume that Cmin 6D ;:

(8.6)

It is clear that Cmin is a closed subset of X. We suppose that the following assumption holds. (A1) For every positive number  there exists ı > 0 such that if a point x 2 C satisfies the inequality f .x/  inf.f I C/ C ı, then d.x; Cmin /  . (It is clear that (A1) holds if the space X is finite-dimensional.) We also suppose that the following assumption holds. (A2) There exists a continuous mapping PC W X ! X such that PC .X/ D C, PC .x/ D x for all x 2 C and kx  PC .y/k  kx  yk for all x 2 C and all y 2 X: For every number  2 .0; 1/ let ./ D supfı 2 .0; 1 W if x 2 C satisfies f .x/  inf.f I C/ C ı; then d.x; Cmin /  minf1; gg:

(8.7)

8.1 Preliminaries and Main Results

121

In view of (A1), ./ is well defined for every positive number . In this chapter we will prove the following two results obtained in [122]. Theorem 8.1. Let f˛i g1 iD0  .0; 1 satisfy lim ˛i D 0;

i!1

1 X

˛i D 1

iD1

and let M;  > 0. Then there exist a natural number n0 and ı > 0 such that the following assertion holds. Assume that an integer n  n0 , fxk gnkD0  X; kx0 k  M; vk 2 @ı f .xk / n f0g; k D 0; 1; : : : ; n  1; n1 f k gn1 kD0 ; fk gkD0  B.0; ı/;

and that for k D 0; : : : ; n  1, xkC1 D PC .xk  ˛k kvk k1 vk  ˛k k /  ˛k k : Then the inequality d.xk ; Cmin /   hods for all integers k satisfying n0  k  n. Theorem 8.2. Let M;  > 0. Then there exists ˇ0 2 .0; 1/ such that for each ˇ1 2 .0; ˇ0 / there exist a natural number n0 and ı > 0 such that the following assertion holds. Assume that an integer n  n0 , fxk gnkD0  X; kx0 k  M; vk 2 @ı f .xk / n f0g; k D 0; 1; : : : ; n  1; f˛k gn1 kD0  Œˇ1 ; ˇ0 ; n1 f k gn1 kD0 ; fk gkD0  B.0; ı/

and that for k D 0; : : : ; n  1, xkC1 D PC .xk  ˛k kvk k1 vk  ˛k k /  k : Then the inequality d.xk ; Cmin /   holds for all integers k satisfying n0  k  n. In this chapter we use the following definitions and notation. Define X0 D fx 2 X W f .x/  inf.f I C/ C 1g:

(8.8)

122

8 A Projected Subgradient Method for Nonsmooth Problems

In view of (8.3), there exists a number KN > 0 such that N X0  B.0; K/:

(8.9)

Since the function f is Lipschitz on all bounded subsets of the space X there exists a number LN > 1 such that N 1  z2 k for all z1 ; z2 2 B.0; KN C 4/: jf .z1 /  f .z2 /j  Lkz

(8.10)

8.2 Auxiliary Results We use the notation and definitions introduced in Sect. 8.1 and suppose that all the assumptions posed in Sect. 8.1 hold. Proposition 8.3. Let  2 .0; 1. Then for each x 2 X satisfying d.x; C/ < minfLN 1 21 .=2/; =2g; f .x/  inf.f I C/ C minf21 .=2/; =2g;

(8.11)

the inequality d.x; Cmin /   holds. Proof. In view of the definition of , .=2/ 2 .0; 1 and if x 2 C satisfies f .x/ < inf.f I C/ C .=2/; then d.x; Cmin /  minf1; =2g:

(8.12)

Assume that a point x 2 X satisfies (8.11). There exists a point y 2 C which satisfies kx  yk < 21 LN 1 .=2/ and kx  yk < =2:

(8.13)

Relations (8.11), (8.8), (8.9), and (8.13) imply that N y 2 B.0; KN C 1/: x 2 B.0; K/;

(8.14)

By (8.13), (8.14), and the definition of LN [see (8.10)], N  yk < .=2/21 : jf .x/  f .y/j  Lkx It follows from the choice of the point y, (8.11), and (8.15) that y2C

(8.15)

8.2 Auxiliary Results

123

and f .y/ < f .x/ C .=2/21  inf.f I C/ C .=2/: Combined with (8.12) this implies that d.y; Cmin /  =2: Together with (8.13) this implies that d.x; Cmin /  kx  yk C d.y; Cmin /  : This completes the proof of Proposition 8.3. Lemma 8.4. Assume that  > 0, x 2 X, y 2 X, f .x/ > inf.f I C/ C ; f .y/  inf.f I C/ C =4;

(8.16)

v 2 @=4 f .x/:

(8.17)

f .u/  f .x/  hv; u  xi  =4 for all u 2 X:

(8.18)

Then hv; y  xi  =2. Proof. In view of (8.2) and (8.17),

By (8.16) and (8.18), .3=4/  f .y/  f .x/  hv; y  xi  =4: The inequality above implies that hv; y  xi  =2: This completes the proof of Lemma 8.4. Lemma 8.5. Let xN 2 Cmin ;

(8.19)

K0 > 0,  2 .0; 1; ˛ 2 .0; 1; let a positive number ı satisfy N 1 ; ı.K0 C KN C 1/  .8L/

(8.20)

124

8 A Projected Subgradient Method for Nonsmooth Problems

let a point x 2 X satisfy kxk  K0 ; f .x/ > inf.f I C/ C ;

(8.21)

;  2 B.0; ı/; v 2 @=4 f .x/ n f0g

(8.22)

y D PC .x  ˛kvk1 v  ˛/  :

(8.23)

and let

Then N 1  C 2˛ 2 C k k2 C 2k k.K0 C KN C 2/: ky  xN k2  kx  xN k2  ˛.4L/ Proof. In view of (8.8)–(8.10) and (8.19), for every point z 2 B.Nx; 41  LN 1 /, we have N  xN k  f .Nx/ C =4 D inf.f I C/ C =4: f .z/  f .Nx/ C Lkz

(8.24)

Lemma 8.4, (8.21), (8.22), and (8.24) imply that for every point z 2 B.Nx; 41  LN 1 /; we have hv; z  xi  =2: Combined with (8.22) the inequality above implies that N 1 /: hkvk1 v; z  xi < 0 for all z 2 B.Nx; .4L/

(8.25)

zQ D xN C 41 LN 1 kvk1 v:

(8.26)

zQ 2 B.Nx; 41 LN 1 /:

(8.27)

Put

It is easy to see that

Relations (8.25), (8.26), and (8.27) imply that 0 > hkvk1 v; zQ  xi D hkvk1 v; xN C 41 LN 1 kvk1 v  xi:

(8.28)

8.2 Auxiliary Results

125

By (8.28), hkvk1 v; xN  xi < 41 LN 1 :

(8.29)

y0 D x  ˛kvk1 v  ˛:

(8.30)

Set

It follows from (8.30), (8.22), (8.21), (8.19), (8.8), (8.9), (8.29), and (8.20) that ky0  xN k2 D kx  ˛kvk1 v  ˛  xN k2 D kx  ˛kvk1 v  xN k2 C ˛ 2 kk2 2˛h; x  ˛kvk1 v  xN i  kx  ˛kvk1 v  xN k2 C˛ 2 ı 2 C 2˛ı.K0 C KN C 1/  kx  xN k2  2hx  xN ; ˛kvk1 vi C˛ 2 C ˛ 2 ı 2 C 2˛ı.K0 C KN C 1/ < kx  xN k2  2˛.41 LN 1 / C˛ 2 .1 C ı 2 / C 2˛ı.K0 C KN C 1/ N 1  C 2˛ 2 :  kx  xN k2  ˛.4L/

(8.31)

In view of (8.8), (8.9), (8.19), (8.21), and (8.31), N 2C2 ky0  xN k2  .K0 C K/ and ky0  xN k  K0 C KN C 2: By (8.23), (8.30), (8.19), (A2), (8.31), and (8.32), ky  xN k2 D kPC .y0 /   xN k2  kPC .y0 /  xN k2 C k k2 C 2k kkPC .y0 /  xN k  ky0  xN k2 C k k2 C 2k kky0  xN k N 1   kx  xN k2  ˛.4L/ C2˛ 2 C k k2 C 2k k.K0 C KN C 2/: This completes the proof of Lemma 8.5. Lemma 8.5 implies the following result.

(8.32)

126

8 A Projected Subgradient Method for Nonsmooth Problems

Lemma 8.6. Let K0 > 0,  2 .0; 1; ˛ 2 .0; 1, a positive number ı satisfy N 1 ; ı.K0 C KN C 1/  .8L/ let x 2 X satisfy kxk  K0 ; f .x/ > inf.f I C/ C ; let

;  2 B.0; ı/; v 2 @=4 f .x/ n f0g and let y D PC .x  ˛kvk1 v  ˛/  : Then N 1  d.y; Cmin /2  d.x; Cmin /2  ˛.4L/ C2˛ 2 C k k2 C 2k k.K0 C KN C 2/:

8.3 Proof of Theorem 8.1 We may assume without loss of generality that  < 1. In view of Proposition 8.3, there exists a number N 2 .0; =8/

(8.33)

such that if x 2 X; d.x; C/  2N and f .x/  inf.f I C/ C 2N ; then d.x; Cmin /  :

(8.34)

Fix xN 2 Cmin :

(8.35)

0 2 .0; 41 N /:

(8.36)

Fix

8.3 Proof of Theorem 8.1

127

Since limi!1 ˛i D 0 there is an integer p0 > 0 such that KN C 4 < p0

(8.37)

and that for all integers p  p0 , we have N 1 0 : ˛p < .32L/ Since

P1 iD0

(8.38)

˛i D 1 there exists a natural number n0 > p0 C 4 such that nX 0 1

N ˛i > .4p0 C M C kNxk/2 01 16L:

(8.39)

iDp0

Fix K > KN C 4 C M C 4n0 C 4kNxk

(8.40)

and a positive number ı such that N 1 0 : 6ı.K C 1/ < .16L/

(8.41)

Assume that an integer n  n0 and that fxk gnkD0  X; kx0 k  M;

(8.42)

n1 f k gn1 kD0 ; fk gkD0  B.0; ı/;

(8.43)

vk 2 @ı f .xk / n f0g; k D 0; 1; : : : ; n  1

(8.44)

and that for all integers k D 0; : : : ; n  1, we have xkC1 D PC .xk  ˛k kvk k1 vk  ˛k k /  ˛k k :

(8.45)

In order to prove the theorem it is sufficient to show that d.xk ; Cmin /   for all integers k satisfying n0  k  n:

(8.46)

Assume that an integer k 2 Œp0 ; n  1;

(8.47)

kxk k  K ; f .xk / > inf.f I C/ C 0 :

(8.48)

128

8 A Projected Subgradient Method for Nonsmooth Problems

In view of (8.35), (8.41), (8.48), (8.43), (8.44), and (8.45), the conditions of Lemma 8.5 hold with K0 D K ,  D 0 , ˛ D ˛k , x D xk ,  D k , v D vk , y D xkC1 . D ˛k k and combined with (8.43), (847), (8.38), (8.41), and (8.40) this lemma implies that N 1 0 kxkC1  xN k2  kxk  xN k2  ˛k .4L/ C2˛k2 C ˛k2 k k k2 C 2k k k˛k .K C KN C 2/ N 1 0 C 2˛k2 C ˛k2 ı 2 C 2ı˛k .K C KN C 2/  kxk  xN k2  ˛k .4L/ N 1 0 C 2ı˛k .K C KN C 3/  kxk  xN k2  ˛k .8L/ N 1 0 :  kxk  xN k2  ˛k .16L/ Thus we have shown that the following property holds: (P1)

If an integer k 2 Œp0 ; n  1 and (8.48) is valid, then we have N 1 ˛k 0 : kxkC1  xN k2  kxk  xN k2  .16L/

We claim that there exists an integer j 2 fp0 ; : : : ; n0 g such that f .xj /  inf.f I C/ C 0 Assume the contrary. Then f .xj / > inf.f I C/ C 0 ; i D p0 ; : : : ; n0 :

(8.49)

It follows from (8.45), (8.43), (8.41), (8.35), and (A2) that for all integers i D 0; : : : ; n  1, we have kxiC1  xN k  1 C kPC .xi  ˛i kvi k1 vi  ˛i /  xN k

(8.50)

1

 1 C kxi  ˛i kvi k vi  ˛i   xN k  1 C kxi  xN k C 2 D kxi  xN k C 3: By (8.40), (8.42), and (8.50), for all integers i D 0; : : : ; n0 , kxi k  kx0  xN k C 3i C kNxk  M C 3i C 2kNxk  M C 3n0 C 2kNxk < K :

(8.51)

Let i 2 fp0 ; : : : ; n0  1g:

(8.52)

It follows from (8.52), (8.51), (8.49), and property (P1) that N 1 ˛i 0 : kxiC1  xN k2  kxi  xN k2  .16L/

(8.53)

8.3 Proof of Theorem 8.1

129

Relations (8.42), (8.50), (8.52), and (8.53) imply that .M C 3p0 C kNxk/2  kxn0  xN k2  kxp0  xN k2 D

nX 0 1

nX 0 1

iDp0

iDp0

N 1 0 ŒkxiC1  xN k2  kxi  xN k2   .16L/

˛i

and nX 0 1

N 01 .M C 3p0 C kNxk//2 : ˛i  16L

iDp0

This contradicts (8.39). The contradiction we have reached proves that there exists an integer j 2 fp0 ; : : : ; n0 g such that f .xj /  inf.f I C/ C 0 :

(8.54)

By (8.45), (A2), (8.43), (8.41), and (8.36), we have d.xj ; C/  ˛j1 ı < N :

(8.55)

In view of (8.54), (8.55), (8.36), and (8.34), d.xj ; Cmin /  :

(8.56)

We claim that for all integers i satisfying j  i  n, d.xi ; Cmin /  : Assume the contrary. Then there exists an integer k 2 Œj; n for which d.xk ; Cmin / > :

(8.57)

k > j  p0 :

(8.58)

By (8.56) and (8.57), we have

By (8.56) we may assume without loss of generality that d.xi ; Cmin /   for all integers i satisfying j  i < k:

(8.59)

130

8 A Projected Subgradient Method for Nonsmooth Problems

Thus d.xk1 ; Cmin /  :

(8.60)

f .xk1 /  inf.f I C/ C 0 I

(8.61)

f .xk1 / > inf.f I C/ C 0 :

(8.62)

There are two cases:

Assume that (8.61) is valid. It follows from (8.61), (8.36), (8.33), (8.8), and (8.9) that N xk1 2 X0  B.0; K/:

(8.63)

By (8.45) and (8.43) there exists a point z 2 C such that kxk1  zk  ı:

(8.64)

By (8.45), (8.43), (8.64), and (A2), kxk  zk  ˛k1 ı C kz  PC .xk1  ˛k1 kvk1 k1 vk1  ˛k1 k1 /k  ı C kz  xk1 k C ˛k1 C ı D 3ı C ˛k1 :

(8.65)

Combined with (8.41), (8.58), and (8.38) the relation above implies that d.xk ; C/  3ı C ˛k1 < 0 :

(8.66)

In view of (8.64) and (8.65), kxk  xk1 k  4ı C ˛k1 :

(8.67)

It follows from (8.60), (8.67), (8.41), (8.38), and (8.58) that d.xk ; Cmin /  2:

(8.68)

Relations (8.63), (8.68), (8.8), and (8.9) imply that xk1 ; xk 2 B.0; KN C 2/: Together with (8.10) and (8.67) the inclusion above implies that N k1  xk k  L.4ı N C ˛k1 /: jf .xk1 /  f .xk /j  Lkx

(8.69)

8.4 Proof of Theorem 8.2

131

In view of (8.69), (8.51), (8.41), (8.38), and (8.58), we have N C ˛k1 / f .xk /  f .xk1 / C L.4ı N  inf.f I C/ C 0 C L.4ı C ˛k1 /  inf.f I C/ C 20 :

(8.70)

It follows from (8.70), (8.66), (8.36), and (8.34) that d.xk ; Cmin /  : This inequality contradicts (8.57). The contradiction we have reached proves (8.62). By (8.60), (8.8), and (8.9), we have kxk1 k  KN C 1:

(8.71)

It follows from (8.40), (8.41), (8.43), (8.44), (8.71), and (8.62) that Lemma 8.6 holds with x D xk1 ; y D xk ;  D k1 ; v D vk1 ; ˛ D ˛k1 ; K0 D KN C 1;  D 0 ; D ˛k1 k1 : Combined with (8.38), (8.58), (8.43), (8.41), and (8.60) this implies that 2 2 N 1 0 C 2˛k1 C 2˛k1 k k1 k2 d.xk ; Cmin /2  d.xk1 ; Cmin /2  ˛k1 .4L/

C2˛k1 k k1 k.2KN C 3/ 2 N 1 ˛k1 0 C 2ı 2 ˛k1  d.xk1 ; Cmin /2  .8L/ C 2˛k1 ı.2KN C 3/

N 1 ˛k1 0 C 2ı˛k1 .2KN C 4/  d.xk1 ; Cmin /2  .8L/ N 1 ˛k1 0  d.xk1 ; Cmin /2   2 :  d.xk1 ; Cmin /2  .16L/ This contradicts (8.57). The contradiction we have reached proves that d.xi ; Cmin /   for all integers i satisfying j  i  n. Since j  n0 this completes the proof of Theorem 8.1.

8.4 Proof of Theorem 8.2 We may assume that without loss of generality  < 1; M > KN C 4:

(8.72)

Proposition 8.3 implies that there exists N 2 .0; =8/

(8.73)

132

8 A Projected Subgradient Method for Nonsmooth Problems

such that if x 2 X; d.x; C/  2N and f .x/  inf.f I C/ C 2N ; then d.x; Cmin /  =4:

(8.74)

Put N 1 N : ˇ0 D .64L/

(8.75)

ˇ1 2 .0; ˇ0 /:

(8.76)

Let

There exists an integer n0  4 such that N ˇ1 n0 > 162 .3 C 2M/2 N 1 L:

(8.77)

K > 2M C 4 C 4n0 C 2KN C 2M

(8.78)

Fix

and a positive number ı such that N 1 N ˇ1 : 6ıK < .64L/

(8.79)

xN 2 Cmin :

(8.80)

Fix a point

Assume that an integer n  n0 , n1 n1 fxk gnkD0  X; f k gn1 kD0  X; fk gkD0  X; f˛k gkD0  Œˇ1 ; ˇ0 ;

(8.81)

kx0 k  M; k k k  ı; kk k  ı; k D 0; : : : ; n  1;

(8.82)

vk 2 @ı f .xk / n f0g; k D 0; 1; : : : ; n  1

(8.83)

and that for all integers k D 0; : : : ; n  1, xkC1 D PC .xk  ˛k kvk k1 vk  ˛k k /  k : We claim that d.xk ; Cmin /   for all integers k satisfying n0  k  n.

(8.84)

8.4 Proof of Theorem 8.2

133

Assume that an integer k 2 Œ0; n  1; kxk k  K f .xk / > inf.f I C/ C N =4:

(8.85)

It follows from (8.75), (8.78)–(8.81), (8.83), (8.85), (8.82), and (8.74) that Lemma 8.5 holds with  D N =4, K0 D K , ˛ D ˛k , x D xk , D k ,  D k , v D vk , y D xkC1 and combining with (8.79) this implies that N 1 N kxkC1  xN k2  kxk  xN k2  ˛k .16L/ C2˛k2 C ı 2 C 2ı.K C KN C 2/ N 1 N C 2˛k2 C 2ı.K C KN C 3/:  kxk  xN k2  ˛k .16L/ Together with (8.81), (8.75), (8.78), and (8.79) this implies that N 1 N C 2ı.KN C 3 C K / kxkC1  xN k2  kxk  xN k2  ˛k .32L/ N 1 N ˇ1 C 2ı.KN C 3 C K /  kxk  xN k2  .32L/ N 1 N :  kxk  xN k2  ˇ1 .64L/ Thus we have shown that the following property holds: (P2) if an integer k 2 Œ0; n  1 and (8.85) is valid, then we have N 1 ˇ1 N : kxkC1  xN k2  kxk  xN k2  .64L/ We claim that there exists an integer j 2 f1; : : : ; n0 g for which f .xj /  inf.f I C/ C N =4: Assume the contrary. Then we have f .xj / > inf.f I C/ C N =4; j D 1; : : : ; n0 :

(8.86)

It follows from (8.84), (8.82), (8.79), (A2), (8.80), (8.81), and (8.75) that for all integers i D 0; : : : ; n  1, we have kxiC1  xN k  1 C kxi  ˛i kvi k1 vi  ˛i i  xN k  kxi  xN k C 3:

(8.87)

By (8.80)–(8.82), (8.72), (8.87), and (8.78) for i D 0; : : : ; n0 , kxi  xN k  kx0  xN k C 3i; kxi k  2kNxk C M C 3n0 < K :

(8.88) (8.89)

134

8 A Projected Subgradient Method for Nonsmooth Problems

Let k 2 f1; : : : ; n0  1g. It follows from (8.89), (8.86), and property (P2) that N 1 ˇ1 N : kxkC1  xN k2  kxk  xN k2  .64L/

(8.90)

Relations (8.72), (8.80), (8.88), (8.82), and (8.90) imply that .M C kNxk C 3/2  kxn0  xN k2  kx1  xN k2 D

nX 0 1

ŒkxiC1  xN k2  kxi  xN k2 

iD1

N 1 N ˇ1  ˇ1 n0 =2.64L/ N 1 N ;  .n0  1/.64L/ N 1 N ˇ1  .M C kNxk C 3/2  .2M C 3/2 : .n0 =2/.64L/ This contradicts (8.77). The contradiction we have reached proves that there exists an integer j 2 f1; : : : ; n0 g for which f .xj /  inf.f I C/ C N =4:

(8.91)

By (8.84), (A2), and (8.82), we have d.xj ; C/  ı:

(8.92)

Relations (8.91), (8.92), (8.79), and (8.74) imply that d.xj ; Cmin /  :

(8.93)

We claim that for all integers i satisfying j  i  n, we have d.xi ; Cmin /  : Assume the contrary. Then there exists an integer k 2 Œj; n for which d.xk ; Cmin / > :

(8.94)

k > j:

(8.95)

It is easy to see that

8.4 Proof of Theorem 8.2

135

We may assume without loss of generality that d.xi ; Cmin /   for all integers i satisfying j  i < k:

(8.96)

Then d.xk1 ; Cmin /  :

(8.97)

f .xk1 /  inf.f I C/ C N =4I

(8.98)

f .xk1 / > inf.f I C/ C N =4:

(8.99)

There are two cases:

Assume that (8.98) is valid. In view of (8.98), (8.73), (8.8), and (8.9), N xk1 2 X0  B.0; K/:

(8.100)

By (8.82), (8.84), and (A2), there exists a point z 2 C such that kxk1  zk  ı:

(8.101)

It follows from (8.82), (8.84), (8.101), and (A2) that kxk  zk  ı C kz  PC .xk1  ˛k1 kvk1 k1 vk1  ˛k1 k1 /k  ı C kz  xk1 k C ˛k1 C ı < 3ı C ˛k1 :

(8.102)

Relations (8.101), (8.98), (8.79), and (8.74) imply that d.xk1 ; Cmin /  =4:

(8.103)

By (8.101), (8.102), (8.79), (8.81), (8.75), and (8.73), kxk  xk1 k  4ı C ˛k1 < N < =8:

(8.104)

In view of (8.103) and (8.104), d.xk ; Cmin / < : This inequality contradicts (8.94). The contradiction we have reached proves (8.99). In view of (8.97), (8.8), and (8.9), kxk1 k  KN C 1:

(8.105)

136

8 A Projected Subgradient Method for Nonsmooth Problems

It follows from (8.78), (8.79), (8.105), (8.99), and (8.82)–(8.84) that Lemma 8.6 holds with x D xk1 ; y D xk ;  D k1 ; D k1 ; v D vk1 ; ˛ D ˛k1 ; K0 D KN C 1  D 41 N and combining with (8.81), (8.75), (8.79), and (8.97) this implies that d.xk ; Cmin /2 2 N 1 N C 2˛k1  d.xk1 ; Cmin /2  ˛k1 .16L/ C ı 2 C 2ı.KN C 4/ 2 N 1 ˛k1 N C 2˛k1  d.xk1 ; Cmin /2  .16L/ C 2ı.KN C 5/

N 1 ˛k1 N C 2ı.KN C 5/  d.xk1 ; Cmin /2  .32L/ N 1 ˇ1 N  2ı.2KN C 5/  d.xk1 ; Cmin /2  .32L/ < d.xk1 ; Cmin /2   2 : This contradicts (8.94). The contradiction we have reached proves that d.xi ; Cmin /   for all integers i satisfying j  i  n. In view of inequality n0  j, Theorem 8.2 is proved.

Chapter 9

Proximal Point Method in Hilbert Spaces

In this chapter we study the convergence of a proximal point method under the presence of computational errors. Most results known in the literature show the convergence of proximal point methods when computational errors are summable. In this chapter the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by some constant.

9.1 Preliminaries and the Main Results We analyze the behavior of the proximal point method in a Hilbert space which is an important tool in the optimization theory. See, for example, [15, 16, 31, 34, 36, 53, 55, 69, 77, 81, 87, 103, 104, 106, 107, 111, 113] and the references mentioned therein. Let X be a Hilbert space equipped with an inner product h; i which induces the norm k  k. For each function g W X ! R1 [ f1g set inf.g/ D inffg.y/ W y 2 Xg: Suppose that f W X ! R1 [ f1g is a convex lower semicontinuous function and a is a positive constant such that dom.f / WD fx 2 X W f .x/ < 1g 6D ;; f .x/  a for all x 2 X

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_9

(9.1)

137

138

9 Proximal Point Method in Hilbert Spaces

and that lim f .x/ D 1:

kxk!1

(9.2)

In view of (9.1) and (9.2), the set argmin.f / WD fz 2 X W f .z/ D inf.f /g 6D ;:

(9.3)

x 2 argmin.f /

(9.4)

Let a point

and let M be any positive number such that M > inf.f / C 4:

(9.5)

In view of (9.2), there exists a number M0 > 1 such that f .z/ > M C 4 for all z 2 X satisfying kzk  M0  1:

(9.6)

kx k < M0  1:

(9.7)

0 < 1 < 2  M02 =2:

(9.8)

Clearly,

Assume that

The following theorem is the main result of this chapter. Theorem 9.1. Let k 2 Œ1 ; 2 ; k D 0; 1; : : : ;

(9.9)

 2 .0; 1, a natural number L satisfy L > 2.4M02 C 1/2 1

(9.10)

and let a positive number  satisfy 1=2

 1=2 .L C 1/.21 1 C 8M0 1

/  1 and .L C 1/  =4:

(9.11)

Assume that a sequence fxk g1 kD0  X satisfies f .x0 /  M

(9.12)

9.1 Preliminaries and the Main Results

139

and f .xkC1 / C 21 k kxkC1  xk k2  inf.f C 21 k k  xk k2 / C 

(9.13)

for all integers k  0. Then for all integers k > L, f .xk /  inf.f / C : By Theorem 9.1, for a given  > 0, we obtain  2 X satisfying f ./  inf.f / C  doing bc1 1 c iterations [see (9.10)] with the computational error  D c2 2 [see (9.11)], where the constant c1 > 0 depends only on M0 ; 2 and the constant c2 > 0 depends only on M0 ; L; 1 ; 2 . Theorem 9.1 implies the following result. Theorem 9.2. Let k 2 Œ1 ; 2 ; k D 0; 1; : : : ; a natural number L satisfy L > .4M02 C 1/22

(9.14)

and let a positive number N satisfy 1=2

N 1=2 .L C 1/.21 1 C 8M0 1

/  1 and N .L C 1/  1=4:

(9.15)

Assume that fi g1 iD0  .0; N /; lim i D 0 i!1

(9.16)

and that > 0. Then there exists a natural number T0 such that for each sequence fxk g1 kD0  X satisfying f .x0 /  M

(9.17)

and f .xkC1 / C 21 k kxkC1  xk k2  inf.f C 21 k k  xk k2 / C k for all integers k  0, the inequality f .xk /  inf.f / C holds for all integers k > T0 .

(9.18)

140

9 Proximal Point Method in Hilbert Spaces

Since the function f is convex and lower semicontinuous and satisfies (9.2), Theorem 9.2 easily implies the following result. Corollary 9.3. Suppose that all the assumptions of Theorem 9.2 hold and that the sequence fxk g1 kD0  X satisfies (9.17) and (9.18) for all integers k  0. Then limk!1 f .xk / D inf.f / and the sequence fxk g1 kD0 is bounded. Moreover, it possesses a weakly convergent subsequence and the limit of any weakly convergent subsequence of fxk g1 kD0 is a minimizer of f . Problem (P) is called well posed if the function f possesses a unique minimizer which is a limit in the norm topology of any minimizing sequence of f (see [60, 121] and the references mentioned therein). Corollary 9.3 easily implies the following result. Corollary 9.4. Suppose that problem (P) is well posed, all the assumptions of Theorem 9.2 hold and that the sequence fxk g1 kD0  X satisfies (9.17) and (9.18) for all integers k  0. Then fxk g1 converges in the norm topology to a unique kD0 minimizer of f . Note that in [60] it was shown that most problems of type (P) (in the sense of Baire category) are well posed. The results of the chapter were obtained in [120]. The chapter is organized as follows. Section 9.2 contains auxiliary results. Theorem 9.1 is proved in Sect. 9.3 and Theorem 9.2 is proved in Sect. 9.4.

9.2 Auxiliary Results We use the notation and definitions introduced in Sect. 9.1 and suppose that all the assumptions made in the introduction holds. Lemma 9.5. Assume that k 2 Œ1 ; 2 ; k D 0; 1; : : :

(9.19)

and that a sequence fxk g1 kD0 satisfies f .x0 /  M; f .xkC1 / C 21 k kxkC1  xk k2  inf.f C 21 k k  xk k2 / C 1 for all integers k  0. Then kxk k  M0 for all integers k  0. Proof. Relations (9.20) and (9.6) imply that kx0 k  M0 :

(9.20) (9.21)

9.2 Auxiliary Results

141

Assume that an integer k  0 and that kxk k  M0 :

(9.22)

It follows from (9.21), (9.19), (9.7), (9.8), (9.3), (9.4), (9.5), and (9.22) that f .xkC1 /  f .x / C 21 k kx  xk k2 C 1  f .x / C 21 2 .2M0 /2 C 1  f .x / C 2 D inf.f / C 2 < M: Together with (9.6) the inequality above implies that kxkC1 k  M0 . Thus we showed by induction that (9.22) holds for all integers k  0. This completes the proof of Lemma 9.5. Lemma 9.6. Assume that k 2 Œ1 ; 2 ; k D 0; 1; : : : ;

(9.23)

k 2 .0; 1, k D 0; 1; : : : , a sequence fxk g1 kD0  X satisfies f .x0 /  M

(9.24)

and that for all integers k  0, f .xkC1 / C 21 k kxkC1  xk k2  inf.f C 21 k k  xk k2 / C k :

(9.25)

Then the following assertions hold. 1. For every integer k  0, .2=k /.f .xkC1 /  f .x // C kxkC1  xk k2  2  2 1 1=2  2k 1 : 1 C kxk  x k  kxkC1  x k C 8M0 .k 1 /

2. For every pair of natural numbers m > n, m X

 21 2 .f .xi /  f .x // C

iDn

m X

kxi1  xi k2

iDn

 4M02 C

m1 X

1 1=2 Œ21 : 1 i C 8M0 .i 1 /

iDn1

Proof. It follows from (9.24), (9.25), and Lemma 9.5 that kxk k  M0 for all integers k  0:

(9.26)

142

9 Proximal Point Method in Hilbert Spaces

In view of (9.25), for every integer k  0, f .xkC1 /  f .xk / C k :

(9.27)

We will prove Assertion 1. Let k  0 be an integer. There exists a point ykC1 2 X such that f .ykC1 / C 21 k kykC1  xk k2  f .x/ C 21 k kx  xk k2 for all x 2 X:

(9.28)

We estimate kxkC1  ykC1 k. Set z D 21 .xkC1 C ykC1 /:

(9.29)

It is easy to see that 21 kykC1  xk k2 C 21 kxkC1  xk k2  k21 .xkC1 C ykC1 /  xk k2 D 21 kykC1 k2 C 21 kxk k2  hykC1 ; xk i C 21 kxkC1 k2 C 21 kxk k2 hxkC1 ; xk i  kxk k2 C hxk ; xkC1 C ykC1 i  k21 .ykC1 C xkC1 /k2 D 21 kykC1 k2 C 21 kxkC1 k2  k21 .ykC1 C xkC1 /k2 D k21 .ykC1  xkC1 /k2 :

(9.30)

In view of (9.29), convexity of the function f , (9.30), (9.28), and (9.25), f .z/ C 21 k kz  xk k2  21 f .xkC1 / C 21 f .ykC1 / C21 k .21 kykC1  xk k2 C 21 kxkC1  xk k2  k21 .ykC1  xkC1 /k2 /  infff .x/ C 21 k kx  xk k W x 2 Xg C21 k  21 k k21 .ykC1  xkC1 /k2 : Combining with (9.19) the inequality above implies that k21 .ykC1  xkC1 /k2  k 1 1 and that 1=2 : kykC1  xkC1 k  2.k 1 1 /

Now we estimate f .x /  f .xkC1 /. In view of (9.28), 0 2 @f .ykC1 / C k .ykC1  xk /

(9.31)

9.2 Auxiliary Results

143

and for every point u 2 X, f .u/  f .ykC1 /  k hxk  ykC1 ; u  ykC1 i:

(9.32)

f .x /  f .ykC1 /  k hxk  ykC1 ; x  ykC1 i:

(9.33)

By (9.32), we have

Relation (9.25) implies that f .xkC1 / C 21 k kxkC1  xk k2  f .ykC1 / C 21 k kykC1  xk k2 C k and f .ykC1 /  f .xkC1 /  21 k .kxkC1  xk k2  kykC1  xk k2 /  k : Together with (9.33) the relation above implies that f .x /  f .xkC1 / D f .x /  f .ykC1 / C f .ykC1 /  f .xkC1 /  k hxk  ykC1 ; x  ykC1 i C f .ykC1 /  f .xkC1 / D 21 k ŒkykC1  x k2  kxk  x k2 C kxk  ykC1 k2  C f .ykC1 /  f .xkC1 /  21 k ŒkykC1  x k2  kxk  x k2 C kxk  xkC1 k2   k :

(9.34)

It follows from (9.28), (9.23), (9.26), (9.7), (9.8), and (9.5) that for all integers q  1, f .yq /  f .x / C 21 2 kx  xq k2  f .x / C 1 < M: Combined with (9.6) the inequality above implies that kyq k  M0 ; q D 1; 2; : : :

(9.35)

Now we use (9.34) and (9.35) and obtain an estimation of f .x /f .xkC1 / without terms which contain ykC1 . In view of (9.26) and (9.31), kxk  ykC1 k2 D k.xk  xkC1 /  .ykC1  xkC1 /k2 D kxk  xkC1 k2 C kykC1  xkC1 k2  2hxk  xkC1 ; ykC1  xkC1 i  kxk  xkC1 k2  2kxk  xkC1 kkykC1  xkC1 k 1=2  kxk  xkC1 k2  8M0 .k 1 : 1 /

(9.36)

144

9 Proximal Point Method in Hilbert Spaces

By (9.26), (9.7), and (9.31), we have kykC1  x k2 D k.xkC1  x / C .ykC1  xkC1 /k2 D kxkC1  x k2 C kykC1  xkC1 k2 C 2hxkC1  x ; ykC1  xkC1 i 1=2  kxkC1  x k2  8M0 .k 1 : 1 /

(9.37)

Relations (9.34) and (9.37) imply that f .x /  f .xkC1 / 1=2  k C 21 k ŒkxkC1  x k2  kx  xk k2 C kxk  xkC1 k2  8M0 .k 1 ; 1 /

f .xkC1 /  f .x / C 21 k kxk  xkC1 k2 1=2  k C 21 k Œkxk  x k2  kxkC1  x k2  C 21 k 8M0 .k 1 : 1 /

and by (9.23), .2=k /.f .xkC1 /  f .x // C kxk  xkC1 k2  2  2 1 1=2  2k 1 : 1 C kxk  x k  kxkC1  x k C 8M0 .k 1 /

Thus Assertion 1 is proved. Let us prove Assertion 2. It follows from Assertion 1, (9.23), (9.7), and (9.26) that for all pairs of natural numbers m > n, m m X X .2=2 /.f .xi /  f .x // C kxi1  xi k2 iDn

iDn

 kxn1  x k2 C

m1 X

1 1=2 Œ2i 1  1 C 8M0 .i 1 /

iDn1

 4M02 C

m1 X

1 1=2 Œ21 : 1 i C 8M0 .i 1 /

iDn1

Assertion 2 is proved. This completes the proof of Lemma 9.6.

9.3 Proof of Theorem 9.1 It follows from (9.9), (9.10), (9.11), (9.12), (9.13), and Lemma 9.6 applied for a natural number n and m D n C L that

9.3 Proof of Theorem 9.1 nCL X

145

.2=2 /.f .xi /  f .x //

iDn

 4M02 C

n1CL X

1 1=2 Œ21  1  C 8M0 .1 /

iDn1 1=2

 4M02 C .L C 1/ 1=2 Œ21 1 C 8M0 1

  4M02 C 1:

(9.38)

Let n  1 be an integer. In view of (9.38),  2 .L C 1/21 2 minff .xi /  f .x / W i D n; : : : ; n C Lg  4M0 C 1

and by (9.10), minff .xi /  f .x / W i D n; : : : ; n C Lg  .4M02 C 1/.L C 1/1 21 2 < =4:

(9.39)

Since (9.39) holds for any natural number n there exists a strictly increasing sequence of natural numbers fSi g1 iD1 such that S1 2 f1; : : : ; 1 C Lg; SiC1  Si 2 Œ1; 1 C L; i D 1; 2; : : : ; f .xSi /  f .x /  =4; i D 1; 2; : : : :

(9.40) (9.41)

Let an integer j  L C 1. In view of (9.40), there is an integer i  1 such that Si  j < SiC1 and j  Si  L C 1: By (9.13) for every integer k  0, we have f .xkC1 /  f .xk / C : Combined with (9.42), (9.41) and (9.11) the inequality above implies that f .xj /  f .xSi / C .L C 1/  f .x / C =4 C =4  f .x / C : Theorem 9.1 is proved.

(9.42)

146

9 Proximal Point Method in Hilbert Spaces

9.4 Proof of Theorem 9.2 By Theorem 9.1 the following property holds: (P1) Let a sequence fxk g1 kD0  X satisfy f .x0 /  M; 1

2

f .xkC1 / C 2 k kxkC1  xk k  inf.f C 21 k k  xk k2 / C N ; k D 0; 1; : : : : Then f .xk /  inf.f / C 1 for all integers k > L: By Theorem 9.1 there exist ı 2 .0; N / and an integer L0  1 such that the following property holds: (P2) For every sequence fyi g1 iD0  X which satisfies f .y0 /  inf.f / C 1; f .ykC1 / C 21 k kykC1  yk k2  inf.f C 21 k k  yk k2 / C ı for all integers k  0 we have f .yk /  inf.f / C for all integers k  L0 :

(9.43)

(Here is as in the statement of the theorem.) In view of (9.16), there exists an integer L1  1 such that k < ı for all natural numbers k  L1 :

(9.44)

T0 > L0 C L1 C L:

(9.45)

Fix a natural number

Assume that a sequence fxi g1 iD0  X satisfies (9.17) and (9.18). It follows from property (P1), (9.17), (9.18), and (9.16) that f .xk /  inf.f / C 1 for all integers k > L:

(9.46)

For every nonnegative integer k set yk D xkCLCL1 :

(9.47)

9.4 Proof of Theorem 9.2

147

It follows from (9.47) and (9.46) that f .y0 /  inf.f / C 1:

(9.48)

By (9.18), (9.47), and (9.44), for all nonnegative integers k, f .ykC1 / C 21 k kykC1  yk k2  inf.f C 21 k k  yk k2 / C ı:

(9.49)

In view of (9.48), (9.49), and property (P2), f .yk /  inf.f / C for all integers k  L0 : Combined with (9.47) and (9.45) the inequality above implies that f .xk /  inf.f / C for all integers k > T0 : This completes the proof of Theorem 9.2.

(9.50)

Chapter 10

Proximal Point Methods in Metric Spaces

In this chapter we study the local convergence of a proximal point method in a metric space under the presence of computational errors. We show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by some constant. The principle assumption is a local error bound condition which relates the growth of an objective function to the distance to the set of minimizers, introduced by Hager and Zhang [55].

10.1 Preliminaries and the Main Results Let X be a metric space equipped with a metric d. For each x 2 X and each r > 0 set B.x; r/ D fy 2 X W d.x; y/  rg: For each x 2 X and each nonempty set A  X set D.x; A/ D inffd.x; y/ W y 2 Ag: For each g W X ! R1 [ f1g put inf.g/ D inffg.x/ W x 2 Xg: Let f W X ! R1 [ f1g be a lower semicontinuous bounded from below function which is not identically infinity. In this chapter we continue our study of proximal point methods which was began in the previous chapter. Literature connected with the analysis and development of proximal point methods and based on tools and methods of convex and variational © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_10

149

150

10 Proximal Point Methods in Metric Spaces

analysis includes [15, 16, 31, 34, 35, 42, 55, 56, 66, 67, 69, 77, 81, 87, 103, 104, 111, 113]. In the proximal point method iterates xk , k  0 are generated by the following rule: xkC1 2 argminff .x/ C 21 k d.x; xk /2 g; where f k g1 kD0 is a sequence of positive numbers and x0 2 X is an initial point. Most results, known in the literature, establish the convergence of proximal point methods when X is a Hilbert space and the function f is convex. Convergence of proximal point methods for a nonconvex objective function f was established in [55, 56]. In [56] convergence results were obtained for finite dimensional optimization problems. In a Hilbert space setting, convergence results for a nonconvex objective function were established in [55]. The principle assumption of [55, 56] is a local error bound condition [see (10.4)] which relates the growth of an objective function to the distance to the set of minimizers. Convergence results in Banach spaces were obtained in [7, 40, 61, 62, 100]. Variable-metric methods were used in order to obtain convergence results in [4, 26, 78]. Let xN 2 X satisfy f .Nx/ D inf.f /:

(10.1)

˝ D fx 2 X W f .x/ D inf.f /g

(10.2)

Set

and let a set ˝0  X satisfy xN 2 ˝0  ˝ \ B.Nx;  /

(10.3)

with a positive constant  . Suppose that ˛ > 0; 0 >  and that f .x/  f .Nx/  ˛D.x; ˝0 /2 for all x 2 B.Nx; 0 /:

(10.4)

 2 .0; 0 /; ˇ0 2 .0; 2˛=3/; ˇ1 2 .0; ˇ0 /;

(10.5)

D 2ˇ0 .2˛  ˇ0 /1 :

(10.6)

< 1:

(10.7)

Let

By (10.5) and (10.6),

10.1 Preliminaries and the Main Results

151

For each number  which satisfies  2 .0; 1/;  < .1 C .1  /1 /1 ;  < .0   /=3;

(10.8)

choose a natural number k0 ./ which satisfies k0 ./  < =8

(10.9)

and a positive number ./ such that 2.k0 .//2 .2./ˇ11 /1=2 C 4.k0 .//2 ./1 .2˛  ˇ0 /1 < 0  ; .2./ˇ11 /1=2 < 161 .k0 .//1 .1  /; ./ < .1  /321 .k0 .//1 2 .2˛  ˇ0 /:

(10.10) (10.11) (10.12)

The following theorem obtained in [124] is the main result of this chapter. Theorem 10.1. Let a number  satisfy (10.8), k0 D k0 ./ and  D ./. Assume that f k g1 kD0  Œˇ1 ; ˇ0 ;

(10.13)

a sequence fxk g1 kD0  X satisfies d.x0 ; xN /.1 C .1  /1 / < 

(10.14)

and that for all integers k  0, f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C  for all z 2 X:

(10.15)

Then xj 2 B.Nx; 0 / for all integers j  0; D.xj ; ˝0 /   for all integers j  k0 : It is easy to see that k0 ./ in (10.9) and ./ in (10.10)–(10.12) depend on  monotonically. Namely, k0 ./ (./, respectively) is an decreasing (increasing, respectively) function of . Clearly, k0 ./ and ./ also depend on the parameters and . The influence of the choice of these parameters on the convergence is not simple. For example, according to (10.14) it is desirable to choose  which is sufficiently close to 0 . But in view of (10.10), if  is close to 0 , the value ./ becomes very small. We can choose a small parameter ˇ0 [see (10.5)] and in this case in view of (10.5) and (10.6) and ˇ1 are small, by (10.9) the number of iterations k0 ./ is not large, but in view of (10.11) ./ is small. The following theorem is the second main result of this chapter.

152

10 Proximal Point Methods in Metric Spaces

Theorem 10.2. There exists N > 0 such that for each sequence fi g1 iD0  .0; N  satisfying lim i D 0

i!1

(10.16)

the following assertion holds. Let  > 0. Then there exists a natural number k1 such that for each sequence 1 f k g1 kD0  Œˇ1 ; ˇ0  and for each sequence fxk gkD0  X which satisfies d.x0 ; xN /.1 C .1  /1 /1 <  and for all integers k  0 f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C k for all z 2 X the following relations hold: xj 2 B.Nx; 0 / for all integers j  0; D.xj ; ˝0 /   for all integers j  k1 : Theorem 10.2 establishes the convergence of the proximal algorithm in the presence of computational errors fi g1 iD0 such that limi!1 i D 0 without assuming their summability. The local error bound condition (10.4) with parameters ˛; 0 ;  , introduced in [55, 56], hold for many functions and can be verified in principle. In the three examples below we show that the parameters ˛; 0 ;  can be calculated by investigating the function f in some cases. On the hand these parameters can be obtained as a result of numerical experiments. Example 10.3. Let .X; h; i/ be a Hilbert space with an inner product h; i which induces a complete norm k  k. Assume that f 2 C2 is a real-valued convex function on X such that lim f .x/ D 1

kxk!1

and inffhf 00 .x/h; hikhk2 W x 2 X and h 2 X n f0gg > 0; where f 00 .x/ is the second order Frechet derivative of f at a point x. It is not difficult to see that the function f possesses a unique minimizer xN and (10.4) holds with  D 1, ˝ D ˝0 D fNxg, 0 > 1 and

10.1 Preliminaries and the Main Results

153

˛ D 21 inffhf 00 .x/h; hikhk2 W x 2 X and h 2 X n f0gg > 0: Example 10.4. Let .X; h; i/ be a Hilbert space with an inner product h; i which induces a complete norm k  k. Assume that f W X ! R1 [ f1g is a lower semicontinuous function, 0 < a < b, the restriction of f to the set fz 2 X W kzk < bg is a convex function which possesses a continuous second order Frechet derivative f 00 ./, inffhf 00 .x/h; hikhk2 W x 2 X; kxk < b and h 2 X n f0gg > 0 and that f .z/ > infff .x/ W x 2 Xg for all z 2 X n B.0; a/: It is easy to see that there exists a unique minimizer xN of f , kNxk  a and that (10.4) holds with ˝ D ˝0 D fNxg, 0 D .b  a/=2, any positive constant  < 0 and ˛ D 21 inff< f 00 .x/h; h > khk2 W x 2 X; kxk < b and h 2 X n f0gg: Example 10.5. Assume that .X; d/ is a metric space, f W X ! R1 [ f1g is a lower semicontinuous function which is not identically infinity, xN 2 X satisfies (10.1), equations (10.2) and (10.3) hold with a positive constant  and (10.4) holds with constants ˛ > 0 and 0 >  . Assume that T W X ! X satisfies c1 d.x; y/  d.T.x/; T.y//  c2 d.x; y/ for all x; y 2 X;

(10.17)

where c1 ; c2 are positive constants such that c2 c1 1  < 0 :

(10.18)

g.x/ D f .T.x//; x 2 X:

(10.19)

Set

It is easy to see that g.T 1 .Nx// D inf.g/ D inf.f /; T 1 .˝/ D fz 2 X W g.z/ D inf.g/g:

(10.20)

Let x 2 B.T 1 .Nx/; c1 2 0 /: By (10.17) and (10.21)

(10.21)

154

10 Proximal Point Methods in Metric Spaces

d.T.x/; xN /  c2 d.x; T 1 .Nx//  0 :

(10.22)

It follows from (10.4), (10.17), (10.19) and (10.22) that g.x/  g.T 1 .Nx// D f .T.x//  f .Nx/  ˛D.T.x/; ˝0 /2 D ˛ inffd.T.x/; z/ W z 2 ˝0 g2  ˛ inffc1 d.x; T 1 .z// W z 2 ˝0 g2  ˛c21 inffd.x; v/ W v 2 T 1 .˝0 /g2 : Thus g.x/  g.T 1 .Nx//  ˛c21 D.x; T 1 .˝0 //2 for all x 2 B.T 1 .Nx/; c1 2 0 /: By (10.3), (10.17), and (10.18) for each z 2 T 1 .˝0 /, N /  c1 d.z; T 1 .Nx//  c1 1 d.T.z/; x 1  ; T 1 .˝0 /  T 1 .˝/ \ B.T 1 .Nx/; c1 1  /: 1 Note that by (10.18), c1 1  < c2 0 . Thus the local error bound condition also holds for the function g.

Example 10.6. Consider the following constrained minimization problem Z

2

jx.t/  sin.t/jdt ! min;

0

x W Œ0; 2 ! R1 is an absolutely continuous (a. c.) function such that x.0/ D x.2/ D 0; jx0 .t/j  1; t 2 Œ0; 2 almost everywhere (a.e.):

(10.23)

Clearly, this problem possesses a unique solution xN .t/ D sin.t/, t 2 Œ0; 2. Let us show that this constrained problem is a particular case of the problem considered in this session. Denote by X the set of all a.c. functions x W Œ0; 2 ! R1 such that (10.23) holds. For all x1 ; x2 2 X set d.x1 ; x2 / D maxfjx1 .t/  x2 .t/j W t 2 Œ0; 2g: Clearly, .X; d/ is a metric space. For x 2 X put

10.1 Preliminaries and the Main Results

155

Z f .x/ D

2

jx.t/  sin.t/jdt:

(10.24)

0

Clearly, the functional f W X ! R1 is continuous. Let x 2 X n fNxg; d.x; xN /  1:

(10.25)

jx.t/j  2; t 2 Œ0; 2:

(10.26)

By (10.23),

Clearly, there is t0 2 Œ0; 2 such that jx.t0 /  sin.t0 /j D d.x; xN /:

(10.27)

By (10.23) and (10.27) for each t 2 Œ0; 2 satisfying jt0  tj  d.x; xN /=4; jx.t/  x.t0 /j  jt  t0 j  d.x; xN /=4; jNx.t/  xN .t0 /j  jt  t0 j  d.x; xN /=4 and jx.t/  xN .t/j  jx.t0 /  xN .t0 /j  jx.t/  x.t0 /j  jNx.t/  xN .t0 /j  d.x; xN /  d.x; xN /=2: Together with (10.24) and (10.25) this implies that Z f .x/ D

0

2

jx.t/  xN .t/jdt  .d.x; xN /=2/d.x; xN /=4; f .x/  f .Nx/  81 d.x; xN /2

and (10.4) holds with 0 D 1, ˛ D 81 ,  D 1=2, ˝ D ˝0 D fNxg: The results of the chapter were obtained in [124]. The chapter is organized as follows. Section 10.2 contains auxiliary results. Section 10.3 contains the main lemma. Theorem 10.1 is proved in Sect. 10.4. Section 10.5 contains an auxiliary result for Theorem 10.2 which is proved in Sect. 10.6. In Sect. 10.7 we obtain extensions of the main results of the chapter for well-posed minimization problems. In Sect. 10.8 we construct an example of a function f for which under the conditions of Theorem 10.2 the sequence fxk g1 kD0 does not converge to a point.

156

10 Proximal Point Methods in Metric Spaces

10.2 Auxiliary Results Lemma 10.7. Let  > 0, z0 ; z1 2 X and 2 Œˇ1 ; ˇ0  satisfy f .z1 / C 21 d.z1 ; z0 /2  f .z/ C 21 d.z; z0 /2 C  for all z 2 X

(10.28)

and let x 2 ˝:

(10.29)

Then d.z1 ; z0 /  d.x; z0 / C .2 1 /1=2 : Proof. By (10.28), (10.29) and (10.2), f .z1 / C 21 d.z1 ; z0 /2  f .x/ C 21 d.x; z0 /2 C   f .z1 / C 21 d.x; z0 /2 C : This implies that d.z1 ; z0 /2  d.x; z0 /2 C 2 1 and d.z0 ; z1 /  d.x; z0 / C .2 1 /1=2 : Lemma 10.7 is proved. Lemma 10.7 implies the following result. Lemma 10.8. Let  > 0, z0 ; z1 2 X and 2 Œˇ1 ; ˇ0  satisfy f .z1 / C 21 d.z1 ; z0 /2  f .z/ C 21 d.z; z0 /2 C  for all z 2 X: Then d.z1 ; z0 /  D.z0 ; ˝0 / C .2 1 /1=2 : Lemma 10.9. Let

2 Œˇ1 ; ˇ0 

(10.30)

10.2 Auxiliary Results

157

and let  > 0; z0 ; z1 2 X satisfy z1 2 B.Nx; 0 /; D.z1 ; ˝0 / > 0; f .z1 / C 21 d.z1 ; z0 /2  f .z/ C 21 d.z; z0 /2 C  for all z 2 X:

(10.31) (10.32)

Then D.z1 ; ˝0 /  2.D.z1 ; ˝0 //1 .2˛  ˇ0 /1 C D.z0 ; ˝0 / C .2 1 /1=2 : Proof. Let x 2 ˝0 :

(10.33)

By (10.32), f .z1 / C 21 d.z1 ; z0 /2  f .x/ C 21 d.x; z0 /2 C : Together with (10.33), (10.3), and (10.2) this implies that f .z1 /  f .Nx/  21 .d.x; z0 /2  d.z1 ; z0 /2 / C    C 21 d.x; z1 /.2d.z1 ; z0 / C d.x; z1 //: Since the inequality above holds for any x 2 ˝0 we obtain that f .z1 / f .Nx/   C 21 D.z1 ; ˝0 /.2d.z1 ; z0 / C D.z1 ; ˝0 //: By (10.31) and (10.4), f .z1 /  f .Nx/  ˛D.z1 ; ˝0 /2 : Together with (10.34) this implies that ˛D.z1 ; ˝0 /2   C 21 D.z1 ; ˝0 /.2d.z1 ; z0 / C D.z1 ; ˝0 //: The inequality above implies that .˛  21 /.D.z1 ; ˝0 //2   C D.z1 ; ˝0 /d.z1 ; z0 /: Combined with (10.30) and (10.31) this implies that .˛  21 ˇ0 /D.z1 ; ˝0 /  ˇ0 d.z1 ; z0 / C .D.z1 ; ˝0 //1 :

(10.34)

158

10 Proximal Point Methods in Metric Spaces

By the inequality above and (10.5) D.z1 ; ˝0 /  ˇ0 .˛  21 ˇ0 /1 d.z1 ; z0 / C .D.z1 ; ˝0 //1 .˛  21 ˇ0 /1 :

(10.35)

By Lemma 10.8, (10.30), and (10.32), d.z1 ; z0 /  D.z0 ; ˝0 / C .2 1 /1=2 :

(10.36)

In view of (10.35), (10.36), (10.6), and (10.7), D.z1 ; ˝0 /  2.D.z1 ; ˝0 //1 .2˛  ˇ0 /1 C D.z0 ; ˝0 / C .2 1 /1=2  2.D.z1 ; ˝0 //1 .2˛  ˇ0 /1 C D.z0 ; ˝0 / C .2 1 /1=2 : Lemma 10.9 is proved.

10.3 The Main Lemma Lemma 10.10. Let a number  satisfy (10.8), k0 D k0 ./,  D ./, a sequence 1 f k g1 kD0 satisfy (10.13) and let a sequence fyk gkD0  X satisfy d.y0 ; xN /.1 C .1  /1 / < 

(10.37)

and for all integers k  0, f .ykC1 / C 21 k d.ykC1 ; yk /2  f .z/ C 21 k d.z; yk /2 C  for all z 2 X:

(10.38)

Then there exists an integer k 2 Œ0; k0  such that d.yj ; xN /  0 ; j D 0; : : : ; k;

(10.39)

D.yk ; ˝0 /  =2:

(10.40)

j D 4j1 .2˛  ˇ0 /1 C j.2ˇ11 /1=2 :

(10.41)

Proof. For all integers j  0 set

10.3 The Main Lemma

159

Assume that an integer k satisfies 0  k < k0

(10.42)

and that for all integers j D 0; : : : ; k, yj 2 B.Nx; 0 /; D.yj ; ˝0 /  j D.y0 ; ˝0 / C j :

(10.43)

(Clearly for k D 0 this assumption holds.) If there is an integer k1 2 Œ0; kC1 such that D.yk1 ; ˝0 /  =2, then the assertion of the lemma holds. Therefore we may assume without loss of generality that D.yj ; ˝0 / > =2 for al integers j 2 Œ0; k C 1:

(10.44)

For all integers j D 0; : : : ; k, it follows from (10.13), (10.38), and Lemma 10.8 applied with z0 D yj , z1 D yjC1 , D j that 1=2 : d.yj ; yjC1 /  D.yj ; ˝0 / C .2 1 j /

(10.45)

By (10.45) and (10.43), d.ykC1 ; y0 / 

k X jD0



d.yjC1 ; yj / 

k X 1=2 ŒD.yj ; ˝0 / C .2 1  j / jD0

k X 1=2 Œ j D.y0 ; ˝0 / C j C .2 1  j / jD0

 .1  /1 D.y0 ; ˝0 / C

k X

j C .k C 1/.2ˇ11 /1=2 : (10.46)

jD0

By (10.46), (10.37), (10.3), (10.41), (10.42), and (10.10), d.ykC1 ; xN /  d.ykC1 ; y0 / C d.y0 ; xN / 1

 d.y0 ; xN /.1 C .1  / / C

k X

j C .k C 1/.2ˇ11 /1=2

jD0

  C .k C 1/k C .k C 1/.2ˇ11 /1=2   C 4k02 1 .2˛  ˇ0 /1 Ck02 .2ˇ11 /1=2 C k0 .2ˇ11 /1=2 < 0 :

(10.47)

160

10 Proximal Point Methods in Metric Spaces

By (10.13), (10.47), (10.44), (10.38), (10.43), (10.7), (10.41), and Lemma 10.9 applied with z0 D yk , z1 D ykC1 , D k , D.ykC1 ; ˝0 /  2.D.ykC1 ; ˝0 //1 .2˛  ˇ0 /1 1=2 C D.yk ; ˝0 / C .2 1 k /

 41 .2˛  ˇ0 /1 C D.yk ; ˝0 / C .2ˇ11 /1=2  41 .2˛  ˇ0 /1 C . k D.y0 ; ˝0 / C k / C .2ˇ11 /1=2  kC1 D.y0 ; ˝0 / C kC1 :

(10.48)

In view of (10.47) and (10.48) we conclude that (10.43) holds for all j D 0; : : : ; kC1. Thus by induction we have shown that at least one of the following cases holds: (i) there is an integer k 2 Œ0; k0  such that (10.39) and (10.40) hold. (ii) (10.43) holds for all j D 0; : : : ; k0 . In the case (i) the assertion of the lemma holds. Assume that the case (ii) holds. By (10.43) with j D k0 , (10.3), (10.37), (10.41), (10.9), (10.11), and (10.12), D.yk0 ; ˝0 /  k0 D.y0 ; ˝0 / C k0  k0  C 4k0 1 .2˛  ˇ0 /1 C k0 .2ˇ11 /1=2 < =2 and the assertion of the lemma holds. This completes the proof of Lemma 10.10.

10.4 Proof of Theorem 10.1 By Lemma 10.10 there is an integer k 2 Œ0; k0  such that d.xj ; xN /  0 ; j D 0; : : : ; k;

(10.49)

D.xk ; ˝0 /  =2:

(10.50)

We show that for all integers j  k D.xj ; ˝0 /  :

(10.51)

This will complete the proof of the theorem. Assume that an integer j  k satisfies (10.51). In view of (10.3) and (10.8), in order to complete the proof it is sufficient to show that

10.5

An Auxiliary Result for Theorem 10.2

161

D.xjC1 ; ˝0 /  : We may assume without loss of generality that D.xjC1 ; ˝0 / > =2:

(10.52)

By Lemma 10.8 applied with z0 D xj , z1 D xjC1 , D j , (10.13), (10.15), and (10.51), 1=2   C .2ˇ11 /1=2 : d.xj ; xjC1 /  D.xj ; ˝0 / C .2 1 j /

Together with (10.51) and (10.11) this implies that D.xjC1 ; ˝0 /  d.xjC1 ; xj / C D.xj ; ˝0 /  2 C .2ˇ11 /1=2 < 3: Together with (10.3) and (10.8) this implies that d.xjC1 ; xN / < 0 :

(10.53)

By (10.13), (10.53), (10.31), (10.15), (10.52), (10.51), (10.11), (10.12), and Lemma 10.9 applied to z0 D xj , z1 D xjC1 , D j , 1=2 D.xjC1 ; ˝0 /  2.D.xjC1 ; ˝0 //1 .2˛  ˇ0 /1 C D.xj ; ˝0 / C .2 1 j /

 41 .2˛  ˇ0 /1 C  C .2ˇ11 /1=2 < : This completes the proof of Theorem 10.1.

10.5 An Auxiliary Result for Theorem 10.2 Let  2 .0; 1/;

(10.54)

k0 0 < =8

(10.55)

4k0 1 .2˛  ˇ0 /1 < .1  /=8;

(10.56)

k0 .2ˇ11 /1=2 < .1  /=8:

(10.57)

a natural number k0 satisfy

and let a positive number  satisfy

162

10 Proximal Point Methods in Metric Spaces

Proposition 10.11. Assume that f k g1 kD0  Œˇ1 ; ˇ0 ;

(10.58)

a sequence fxk g1 kD0  X satisfies xk 2 B.Nx; 0 / for all integers k  0

(10.59)

and that for all integers k  0 f .xkC1 / C 21 k d.xkC1 ; xk /  f .z/ C 21 k d.z; xk / C  for all z 2 X:

(10.60)

D.xj ; ˝0 /   for all integers j  k0 :

(10.61)

Then

Proof. For all integers j D 0; 1; : : : put j D j.21 .2˛  ˇ0 /1 C .2ˇ11 /1=2 /:

(10.62)

Assume that an integer j  0 and that D.xj ; ˝0 /  :

(10.63)

We show that D.xjC1 ; ˝0 /  . We may assume without loss of generality that D.xjC1 ; ˝0 / > =2:

(10.64)

By (10.58), (10.60), (10.64), (10.59), and Lemma 10.9 applied with z0 D xj , z1 D xjC1 , D j , (10.11), (10.12), (10.63), (10.56), and (10.57), 1=2 D.xjC1 ; ˝0 /  2.D.xjC1 ; ˝0 //1 .2˛  ˇ0 /1 C D.xj ; ˝0 / C .2 1 j /

 41 .2˛  ˇ0 /1 C  C .2ˇ11 /1=2 < : Thus we have shown that if an integer j  0 satisfies (10.63), then D.xjC1 ; ˝0 /  : Therefore in order to prove the proposition it is sufficient to show that (10.63) holds with some integer j 2 Œ0; k0 . Assume the contrary. Thus D.xj ; ˝0 / >  for all integers j 2 Œ0; k0 :

(10.65)

10.6

Proof of Theorem 10.2

163

Assume that an integer k satisfies 0  k < k0 ;

(10.66)

D.xj ; ˝0 /  j D.x0 ; ˝0 / C j :

(10.67)

(Clearly, for k D 0 this assumption holds.) By (10.58), (10.59), (10.65), (10.60), (10.67), (10.7), (10.62), (10.66), and Lemma 10.9 applied with z0 D xk , z1 D xkC1 , D k , 1=2 D.xkC1 ; ˝0 /  2.D.xkC1 ; ˝0 //1 .2˛  ˇ0 /1 C D.xk ; ˝0 / C .2 1 k /

 21 .2˛  ˇ0 /1 C kC1 D.x0 ; ˝0 / C k C .2ˇ11 /1=2 D kC1 D.x0 ; ˝0 / C kC1 : Thus (10.67) holds for j D k C 1. By induction we have shown that (10.67) holds for all j D 0; : : : ; k0 . Together with (10.59), (10.55), (10.57), and (10.62) this implies that D.xk0 ; ˝0 /  k0 D.x0 ; ˝0 / C k0  k0 0 C k0 .21 .2˛  ˇ0 /1 C .2ˇ11 /1=2 / < : This contradicts (10.65). The contradiction we have reached proves Proposition 10.11.

10.6 Proof of Theorem 10.2 By Theorem 10.1 there exists N > 0 such that the following property holds: (P1) For each sequence f k g1 kD0  Œˇ1 ; ˇ0 

(10.68)

and each sequence fxk g1 kD0  X which satisfies d.x0 ; xN /.1 C .1  /1 / <  and for all integers k D 0; 1; : : : ; f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C N for all z 2 X

(10.69)

164

10 Proximal Point Methods in Metric Spaces

we have xj 2 B.Nx; 0 / for all integers j  0:

(10.70)

Assume that fi g1 iD0  .0; N  and lim i D 0 i!1

(10.71)

and  > 0. By Proposition 10.11 there are O > 0 and a natural number q1 such that the following property holds: (P2) Assume that (10.68) holds, a sequence fxk g1 kD0  X satisfies (10.70) and that for all integers k  0, f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C O for all z 2 X: Then D.xj ; ˝0 /   for all integers j  q1 . By (10.71) there exists a natural number q2 such that j < O for all integers j  q2 :

(10.72)

k1 D q1 C q2 :

(10.73)

Set

Assume that (10.68) holds, a sequence fxk g1 kD0  X satisfies (10.69) and that for all integers k  0 f .xkC1 / C 21 k d.xk ; xkC1 /2  f .z/ C 21 k d.z; xk /2 C k for all z 2 X:

(10.74)

Equations (10.68), (10.69), (10.74), (10.71), and (P1) imply (10.70). By (10.68), (10.70), (10.72), and (10.74) we can apply (P2) to the sequence fxq2 Cj g1 jD0 and obtain that D.xj ; ˝0 /   for all integers j  q1 C q2 D k1 . Theorem 10.2 is proved.

10.7 Well-Posed Minimization Problems We use the notation and definitions from Sect. 10.1. Suppose that ˝ D fNxg

10.7 Well-Posed Minimization Problems

165

and that for each sequence fzi g1 iD0  X satisfying limi!1 f .zi / D inf.f / we have lim .zi ; xN / D 0:

i!1

In other words the problem f .x/ ! min, x 2 X is well posed in the sense of [121]. Fix M > 1 C 0 . Proposition 10.12. There exist  ;  > 0 such that for each 2 .0;  , each z0 2 B.Nx; M/ and each z1 2 X satisfying f .z1 / C 21 d.z1 ; z0 /2  f .z/ C 21 d.z; z0 /2 C  for all z 2 X

(10.75)

the inequality d.z1 ; xN /  21 .1 C .1  /1 /1

(10.76)

holds. Proof. Since the problem f .z/ ! min, z 2 X is well posed there is ı > 0 such that if z 2 X satisfies f .z/  inf.f / C ı; then d.z; xN /  21 .1 C .1  /1 /1 :

(10.77)

Choose positive numbers

 < .M 2 C 1/1 ı;  2 .0; ı=2/:

(10.78)

2 .0;  ; z0 2 B.Nx; M/

(10.79)

Let

and let z1 2 X satisfy (10.75). By (10.79), (10.78), and (10.75), f .z1 /  f .z1 / C 21 d.z1 ; z0 /2  f .Nx/ C 21  d.Nx; z0 /2 C   inf.f / C 21  M 2 C   inf.f / C ı: Together with (10.77) this implies that (10.76). Proposition 10.12 is proved. Let  ;  > 0 be as guaranteed by Proposition 10.12. We suppose that ˇ 0   :

(10.80)

166

10 Proximal Point Methods in Metric Spaces

We may assume without loss of generality that ./ 2 .0;   for all  2 .0; 1/:

(10.81)

Theorem 10.13. Let a number  satisfy  2 .0; 1/;  < 21 .1 C .1  /1 /1 ;  < 0 =3; k0 D k0 ./ and let a positive number  D ./. Assume that f k g1 kD0  Œˇ1 ; ˇ0 ;

(10.82)

a sequence fxk g1 kD0  X satisfies d.x0 ; xN /  M

(10.83)

and that for all integers k  0, f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C  for all z 2 X:

(10.84)

Then xj 2 B.Nx; 0 / for all integers j  1; xj 2 B.Nx; / for all integers j  k0 C 1: Proof. By the choice of  and  , Proposition 10.12, (10.84), (10.82), (10.80), (10.83), and (10.81) d.x1 ; xN /  21 .1 C .1  /1 /1 : Since  can be arbitrarily small positive number the assertion of the theorem now follows from Theorem 10.1. Theorem 10.14. Let N > 0 be as guaranteed by Theorem 10.2, O D minfN ;  g;

(10.85)

fi g1 iD0  .0; O ; lim i D 0;

(10.86)

i!1

 > 0 and let a natural number k1 be as guaranteed by Theorem 10.2 with the sequence fiC1 g1 iD0 . Assume that f k g1 kD0  Œˇ1 ; ˇ0 ;

(10.87)

10.8 An Example

167

a sequence fxk g1 kD0  X satisfies d.x0 ; xN /  M

(10.88)

and that for all integers k  0, f .xkC1 / C 21 k d.xkC1 ; xk /2  f .z/ C 21 k d.z; xk /2 C k for all z 2 X:

(10.89)

Then xj 2 B.Nx; 0 / for all integers j  1; xj 2 B.Nx; / for all integers j  k1 C 1: Proof. By the choice of  and  , Proposition 10.12, (10.88), (10.86), (10.85), (10.87), and (10.80), d.x1 ; xN /  21 .1 C .1  /1 /1 : The assertion of the theorem now follows from Theorem 10.2.

10.8 An Example Let X D Rn be equipped with the Euclidean norm k  k which induces the metric d.x; y/ D kx  yk, x; y 2 Rn . Set ˝ D B.0; 1/; f .x/ D D.x; B.0; 1//2 ; x 2 Rn : Clearly, all the assumptions made in Sect. 10.1 hold with xN D 0, ˝ D ˝0 D B.0; 1/,  D 1, ˛ D 1 and any positive constant 0 > 1. Thus Theorems 10.1 and 10.2 hold for the function f . We prove the following result. Proposition 10.15. Assume that > 0, a sequence fi g1 iD0  .0; 1 satisfies 1 X iD0

1=2

i

D1

(10.90)

168

10 Proximal Point Methods in Metric Spaces

and that x0 2 B.0; 1/. Then there exists a sequence fxk g1 kD0  B.0; 1/ such that for all integers k  0, f .xkC1 / C 21 kxkC1  xk k2  f .z/ C 21 kz  xk k2 C k for all z 2 Rn

(10.91)

and that for all z 2 B.0; 1/, lim inf kxk  zk D 0: k!1

Proposition 10.15 easily follows from the following auxiliary result. Lemma 10.16. Assume that > 0, a sequence fi g1 iD0  .0; 1 satisfies (10.90) and that y0 ; y1 2 B.0; 1/. Then there exist a natural number q and a sequence q fxk gkD0  B.0; 1/ such that x0 D y0 , xq D y1 ant that for all integers k 2 Œ0; q  1, Eq. (10.91) holds. Proof. Set F D fty1 C .1  t/y0 W t 2 Œ0; 1g: Set t0 D 0 and for all integers i  0 tiC1 D minfti C 21 .2i 1 /1=2 ; 1g:

(10.92)

By (10.90) and (10.92) there exists a natural number q such that tq D 1; ti < 1 for all nonnegative integers i < q:

(10.93)

For any integer k 2 Œ0; q set xk D tk y1 C .1  tk /y0 :

(10.94)

Clearly, q

fxk gkD0  F  B.0; 1/: Let an integer k satisfy 0  k < q. By (10.92), (10.94) and (10.95) f .xkC1 / C 21 kxkC1  xk k2 D 21 ktkC1 y1 C .1  tkC1 /y0  .tk y1 C .1  tk /y0 /k2 D 21 k.tkC1  tk /.y1  y0 /k2  2 .tkC1  tk /2  k : This implies (10.91) and completes the proof of Lemma 10.16.

(10.95)

Chapter 11

Maximal Monotone Operators and the Proximal Point Algorithm

In a finite-dimensional Euclidean space, we study the convergence of a proximal point method to a solution of the inclusion induced by a maximal monotone operator, under the presence of computational errors. The convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.

11.1 Preliminaries and the Main Results Let Rn be the n-dimensional Euclidean space equipped with an inner product h; i which induces the norm k  k. n A multifunction T W Rn ! 2R is called a monotone operator if hz  z0 ; w  w0 i  0 8z; z0 ; w; w0 2 Rn such that w 2 T.z/ and w0 2 T.z0 /:

(11.1)

It is called maximal monotone if, in addition, the graph f.z; w/ 2 Rn  Rn W w 2 T.z/g is not strictly contained in the graph of any other monotone operator T 0 W Rn ! Rn . A fundamental problem consists in determining an element z such that 0 2 T.z/. A proximal point algorithm is an important tool for solving this problem. This algorithm has been studied extensively because of its role in convex analysis and optimization. See, for example, [15–17, 31, 34, 36, 53, 55, 69, 81–83, 87, 103, 104, 106, 107, 111, 113] and the references mentioned therein. © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_11

169

170

11 Maximal Monotone Operators and the Proximal Point Algorithm

Let T W Rn ! Rn be a maximal monotone operator. The algorithm for solving the inclusion 0 2 T.z/ is based on the fact established by Minty [82], who showed that, for each z 2 Rn and each c > 0, there is a unique u 2 Rn such that z 2 .I C cT/.u/; where I W Rn ! Rn is the identity operator (Ix D x for all x 2 Rn ). The operator Pc WD .I C cT/1

(11.2)

is therefore single-valued from all of Rn onto Rn (where c is any positive number). It is also nonexpansive: kPc .z/  Pc .z0 /k  kz  z0 k for all z; z0 2 Rn

(11.3)

and Pc .z/ D z if and only if 0 2 T.z/. Following the terminology of Moreau [87] Pc is called the proximal mapping associated with cT. The proximal point algorithm generates, for any given sequence fck g1 kD0 of n positive real numbers and any starting point z0 2 Rn , a sequence fzk g1 kD0  R , where zkC1 WD Pck .zk /; k D 0; 1; : : :

(11.4)

We study the convergence of a proximal point method to the set of solutions of the inclusion 0 2 T.x/ under the presence of computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by some constant. More precisely, we show (Theorem 11.2) that, for given positive numbers M; , there exist a natural number n0 and ı > 0 such that, if the computational errors do not exceed ı for any iteration and if kx0 k  M, then the algorithm generates a 0 sequence fxk gnkD0 such that kxn0  xN k  , where xN 2 Rn satisfies 0 2 T.Nx/. The results of this chapter were obtained in [125]. Let T W Rn ! Rn be a maximal monotone operator. It is not difficult to see that its graph graphT WD f.x; w/ 2 Rn  Rn W w 2 T.x/g is closed. Assume that F WD fz 2 Rn W 0 2 T.z/g 6D ;: For each x 2 Rn and each nonempty set A  Rn put

11.1 Preliminaries and the Main Results

171

d.x; A/ D inffkx  yk W y 2 Ag: Fix N > 0:

(11.5)

For each x 2 Rn and each r > 0 set B.x; r/ D fy 2 Rn W kx  yk  rg: We prove the following result, which establishes the convergence of the proximal point algorithm without computational errors. Theorem 11.1. Let M;  > 0. Then there exists a natural number n0 such that, for 1 n N each sequence fk g1 kD0  Œ; 1/ and each sequence fxk gkD0  R such that kx0 k  M; xkC1 D Pk .xk / for all integers k  0;

(11.6)

the inequality d.xk ; F/   holds for all integers k  n0 . Since n0 depends only on M;  we can say that Theorem 11.1 establishes the uniform convergence of the proximal point algorithm without computational errors on bounded sets. Theorem 11.1 is proved in Sect. 11.3. The following theorem is one of the main results of this chapter. Theorem 11.2. Let M;  > 0. Then there exist a natural number n0 and a positive 0 1 N 1/ and each sequence number ı such that, for each sequence fk gnkD0  Œ; n0 n fxk gkD0  R such that kx0 k  M; kxkC1  Pk .xk /k  ı; k D 0; 1; : : : ; n0  1; the following inequality holds: d.xn0 ; F/  : Theorem 11.2 easily follows from the following result, which is proved in Sect. 11.4. Theorem 11.3. Let M; 0 > 0, let a natural number n0 be as guaranteed by Theorem 11.1 with  D 0 =2, and let ı D 0 .2n0 /1 . 0 1 N 1/ and each sequence fxk gn0  Rn Then, for each sequence fk gnkD0  Œ; kD0 such that

172

11 Maximal Monotone Operators and the Proximal Point Algorithm

kx0 k  M; kxkC1  Pk .xk /k  ı; k D 0; 1; : : : :n0  1; the following inequality holds: d.xn0 ; F/  : Theorem 11.2 easily implies the following result. Theorem 11.4. Let M;  > 0 and let a natural number n0 and ı > 0 be as N guaranteed by Theorem 11.2. Assume that fk g1 kD0  Œ; 1/ and that a sequence fxk g1  B.0; M/ satisfies kD0 kxkC1  Pk .xk /k  ı; k D 0; 1; : : : Then d.xk ; F/   for all integers k  n0 . Next result is proved in Sect. 11.4. It establishes a convergence of the proximal point algorithm with computational errors, which converge to zero under an assumption that all the iterates are bounded by the same prescribed bound. This convergence is uniform since n depends only on M, , and fık g1 kD0 . Theorem 11.5. Let M > 0, fık g1 kD0 be a sequence of positive numbers such that limk!1 ık D 0 and let  > 0. Then there exists a natural number n such that, for 1 N each sequence fk g1 kD0  Œ; 1/ and each sequence fxk gkD0  B.0; M/ satisfying kxkC1  Pk .xk /k  ık for all integers k  0; the inequality d.xk ; F/   holds for all integers k  n . In the last two theorems, which are proved in Sect. 11.4, we consider the case when the set F is bounded. In Theorem 11.6 it is assumed that computational errors do not exceed a certain positive constant, while in Theorem 11.7 computational errors tend to zero. Theorem 11.6. Suppose that the set F be bounded. Let M;  > 0. Then there exists N ı > 0 and a natural number n0 such that, for each fk g1 kD0  Œ; 1/ and each n sequence fxk g1  R satisfying kD0 kx0 k  M; kxkC1  Pk .xk /k  ı ; k D 0; 1; : : : the inequality d.xk ; F/   holds for all integers k  n0 .

11.2 Auxiliary Results

173

Theorem 11.7. Suppose that the set F be bounded and let M > 0. Then there exists ı > 0 such that the following assertion holds. Assume that fık g1 kD0  .0; ı satisfies lim ık D 0

k!1

and that  > 0. Then there exists a natural number n such that, for each sequence 1 n N fk g1 kD0  Œ; 1/ and each sequence fxk gkD0  R satisfying kx0 k  M; kxkC1  Pk .xk /k  ık ; k D 0; 1; : : : ; the inequality d.xk ; F/   holds for all integers k  n . Note that in Theorem 11.6 ı depends on , while in Theorem 11.7 ı does not depend on .

11.2 Auxiliary Results It is easy to see that the following lemma holds. Lemma 11.8. Let z; x0 ; x1 2 Rn . Then 21 kz  x0 k2  21 kz  x1 k2  21 kx0  x1 k2 D hx0  x1 ; x1  zi: 1 n Lemma 11.9. Let fk g1 kD0  .0; 1/, fxk gkD0  R satisfy for all integers k  0,

xkC1 D Pk .xk / D .I C k T/1 .xk /

(11.7)

0 2 T.z/:

(11.8)

and let z 2 Rn satisfies

Then, for all integers k  0, kz  xk k2  kz  xkC1 k2  kxk  xkC1 k2  0: Proof. Let k  0 be an integer. By Lemma 11.8, 21 kz  xk k2  21 kz  xkC1 k2  21 kxk  xkC1 k2 D hxk  xkC1 ; xkC1  zi:

(11.9)

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11 Maximal Monotone Operators and the Proximal Point Algorithm

By (11.7), xk  xkC1 2 k T.xkC1 /: Together with (11.9) and (11.8) this completes the proof of Lemma 11.8. Using (11.3) we can easily deduce the following lemma. Lemma 11.10. Assume that z 2 Rn satisfies (11.8), M > 0, 1 n fk g1 kD0  .0; 1/; fxk gkD0  R ;

kx0  zk  M and that (11.7) holds for all integers k  0. Then kxk  zk  M for all integers k  0. Lemma 11.11. Let M;  > 0. Then there exists ı > 0 such that, for each x 2 B.0; M/, each   N and each z 2 B.0; ı/ satisfying z 2 T.x/ the inequality d.x; F/   holds. Proof. Assume the contrary. Then, for each natural number k there exist xk 2 B.0; M/; zk 2 B.0; k1 /; k  N

(11.10)

d.xk ; F/ > ; zk 2 k T.xk /:

(11.11)

such that

By (11.11) and (11.10), for all integers k  1, 1 k zk 2 T.xk /

(11.12)

and N 1 N 1 1 ! 0 as k ! 1: k1 k zk k   kzk k   k

(11.13)

By (11.10), extracting a subsequence and re-indexing, we can assume that these exists x WD lim xk : k!1

(11.14)

Since graph T is closed, then (11.12), (11.13) and (11.14) imply that 0 2 T.x/ and that x 2 F. Together with (11.14) this implies that d.xk ; F/  =2 for all sufficiently large natural numbers k. This contradicts (11.11) and proves Lemma 11.11. Lemma 11.12. Assume that the integers p; q, with 0  p < q, are such that q1

q1

fk gkDp  .0; 1/; fk gkDp  .0; 1/; q

q

fxk gkDp  Rn ; fyk gkDp  Rn ; yp D xp ;

(11.15)

11.3

Proof of Theorem 11.1

175

and that for all integers k 2 fp; : : : ; q  1g, ykC1 D Pk .yk /; kxkC1  Pk .xk /k  k :

(11.16)

Then, for any integer k 2 fp C 1 : : : ; qg, kyk  xk k 

k1 X

i :

(11.17)

iDp

Proof. We prove the lemma by induction. In view of (11.16) and (11.15), equation (11.17) holds for k D p C 1. Assume that an integer j satisfies p C 1  j  q, (11.17) holds for all k D p C 1; : : : ; j and that j < q. By (11.16), (11.3) and (11.17) with k D j kyjC1  xjC1 k  kPj yj  xjC1 k  kPj yj  Pj xj k C kPj xj  xjC1 k  kyj  xj k C j 

j1 X

i C j D

iDp

j X

i

iDp

and (11.17) holds for all k D pC1; : : : ; jC1. Therefore we showed by induction that (11.17) holds for all k D p C 1; : : : ; q. This completes the proof of Lemma 11.12.

11.3 Proof of Theorem 11.1 Fix z 2 F:

(11.18)

By Lemma 11.11, there exists ı 2 .0; 1/ such that the following property holds: (P1) For each x 2 B.0; M C 2kzk/, each   N and each z 2 B.0; ı/ satisfying z 2 T.x/ we have d.x; F/  =2. Choose a natural number n0 such that 2 .kzk C M/2 n1 0 1 C supfkzk W z 2 Fg;  < 1:

(11.33)

By Theorem 11.2 there exist ı > 0 and a natural number n0 such that the following property holds: (P2) 0 1 N 1/ and each sequence fxk gn0  Rn which For each sequence fk gnkD0  Œ; kD0 satisfies kx0 k  M; kxkC1  Pk .xk /k  ı; k D 0; : : : ; n0  1 we have d.xn0 ; F/  =4: Put ı D minfı; .=4/n1 0 g:

(11.34)

1 n N fk g1 kD0  Œ; 1/; fxk gkD0  R

(11.35)

Assume that

and kx0 k  M; kxkC1  Pk .xk /k  ı ; k D 0; 1; : : :

(11.36)

By (11.35), (11.36), (11.34) and (P2), d.xn0 ; F/  =4:

(11.37)

kxn0 k  M:

(11.38)

In view of (11.37) and (11.33),

We show by induction that for any natural number j, d.xjn0 ; F/  =4; kxjn0 k  M: Equations (11.37) and (11.38) imply that (11.39) is valid for j D 1.

(11.39)

11.4

Proofs of Theorems 11.3,11.5,11.6, and 11.7

179

Assume that j is a natural number and (11.39) holds. By (11.39), (11.35), (11.36), (11.34), and (P2), d.x.jC1/n0 ; F/  =4: Together with (11.33) this implies that kx.jC1/n0 k  M. Thus (11.39) holds for all natural numbers j. Let j be a natural number. Put yjn0 D xjn0 ; ykC1 D Pk .yk /; k D jn0 ; : : : ; 2jn0  1:

(11.40)

By Lemma 11.12, (11.35), (11.36), (11.40), and (11.34) for all k D jn0 C 1; : : : ; 2.j C 1/n0 , kyk  xk k  n0 ı  =4:

(11.41)

Since the set F is closed and bounded there is z 2 F such that d.xjn0 ; F/ D kxjn0  zk:

(11.42)

It follows from (11.39) and (11.42) that kxjn0  zk  =4:

(11.43)

By (11.35), (11.40), the inclusion z 2 F, Lemma 11.9 and (11.43), kyk  zk  kyjn0  zk D kxjn0  zk  =4 for all integers k D jn0 C 1; : : : ; 2jn0 :

(11.44)

By (11.41), (11.44) and the inclusion z 2 F for all integers k D jn0 C 1; : : : ; 2.j C 1/n0 , d.xk ; F/  kxk  zk  kxk  yk k C kyk  zk  =4 C =4: Since j is any natural number we conclude that d.xk ; F/  =2 for all integers j  n0 : Theorem 11.6 is proved. Proof of Theorem 11.7. We may assume without loss of generality that M > 2 C supfkzk W z 2 Fg:

(11.45)

180

11 Maximal Monotone Operators and the Proximal Point Algorithm

By Theorem 11.6 there are ı > 0 and a natural number n0 such that the following property holds: 1 n N (P3) for each fk g1 kD0  Œ; 1/ and each fxk gkD0  R satisfying kx0 k  M; kxkC1  Pk .xk /k  ı; k D 0; 1; : : : the inequality d.xk F/  1 holds for all integers k  n0 . Assume that fık g1 kD0  .0; ı; lim ık D 0;  > 0: k!1

(11.46)

We may assume without loss of generality that  < 1:

(11.47)

By Theorem 11.6 there are ı 2 .0; ı/ and a natural number n such that the following property holds: 1 n N (P4) for each fk g1 kD0  Œ; 1/ and each fxk gkD0  R satisfying kx0 k  M, kxkC1  Pk .xk /k  ı ; k D 0; 1; : : : we have d.xk ; F/   for all integers k  n . By (11.46) there is a natural number p such that ık < ı for all integers k  p:

(11.48)

n D n0 C p C n :

(11.49)

1 n N fk g1 kD0  Œ; 1/; fxk gkD0  R ;

(11.50)

Put

Assume that

kx0 k  M; kxkC1  Pk .xk /k  ık ; k D 0; 1; : : :

(11.51)

By (11.50), (11.51), (11.46) and (P3), d.xk ; F/  1 for all integers k  n0 :

(11.52)

11.4

Proofs of Theorems 11.3,11.5,11.6, and 11.7

181

It follows from (11.52) and (11.45) that kxk k  M for all integers k  n0 :

(11.53)

By (11.48) and (11.51) for all integers k  n0 C p, kxkC1  Pk .xk /k  ı :

(11.54)

It follows from (11.50), (11.53), (11.54), (11.49), and property (P4) applied to the sequence fxk g1 kDn0 Cp that d.xk ; F/   for all integers k  n0 C p C n D n : This completes the proof of Theorem 11.7.

Chapter 12

The Extragradient Method for Solving Variational Inequalities

In a Hilbert space, we study the convergence of the subgradient method to a solution of a variational inequality, under the presence of computational errors. The convergence of the subgradient method for solving variational inequalities is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.

12.1 Preliminaries and the Main Results The studies of gradient-type methods and variational inequalities are important topics in optimization theory. See, for example, [3, 12, 30, 31, 37, 44, 52, 54, 68, 71– 74] and the references mentioned therein. In this chapter we study convergence of the subgradient method, introduced in [75] and known in the literature as the extragradient method, to a solution of a variational inequality in a Hilbert space, under the presence of computational errors. Let .X; h; i/ be a Hilbert space with an inner product h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set B.x; r/ D fy 2 X W kx  yk  rg: Let C be a nonempty closed convex subset of X. By Lemma 2.2, for each x 2 X there is a unique point PC .x/ 2 C satisfying kx  PC .x/k D inffkx  yk W y 2 Cg: Moreover, kPC .x/  PC .y/k  kx  yk for all x; y 2 X © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_12

183

184

12 The Extragradient Method for Solving Variational Inequalities

and hz  PC .x/; x  PC .x/i  0 for each x 2 X and each z 2 C. Consider a mapping f W X ! X. We say that the mapping f is monotone on C if hf .x/  f .y/; x  yi  0 for all x; y 2 C: We say that f is pseudo-monotone on C if for each x; y 2 C the inequality hf .y/; x  yi  0 implies that hf .x/; x  yi  0: Clearly, if f is monotone on C, then f is pseudo-monotone on C. Denote by S the set of all x 2 C such that hf .x/; y  xi  0 for all y 2 C:

(12.1)

S 6D ;:

(12.2)

We suppose that

For each  > 0 denote by S the set of all x 2 C such that hf .x/; y  xi  ky  xk   for all y 2 C:

(12.3)

In the sequel, we present examples which provide simple and clear estimations for the sets S in some important cases. These examples show that elements of S can be considered as -approximate solutions of the variational inequality. In this chapter, in order to solve the variational inequality (to find x 2 S), we use the algorithm known in the literature as the extragradient method [75]. In each iteration of this algorithm, in order to get the next iterate xkC1 , two orthogonal projections onto C are calculated, according to the following iterative step. Given the current iterate xk calculate yk D PC .xk  k f .xk // and then xkC1 D PC .xk  k f .yk //; where k is some positive number. It is known that this algorithm generates sequences which converge to an element of S. In this chapter, we study the behavior of the sequences generated by the algorithm taking into account computational errors which are always present in practice. Namely, in practice the algorithm 1 generates sequences fxk g1 kD0 and fyk gkD0 such that for each integer k  0, kyk  PC .xk  k f .xk //k  ı

12.1 Preliminaries and the Main Results

185

and kxkC1  PC .xk  k f .yk //k  ı; with a constant ı > 0 which depends only on our computer system. Surely, in this situation one cannot expect that the sequence fxk g1 kD0 converges to the set S. The goal is to understand what subset of C attracts all sequences fxk g1 kD0 generated by the algorithm. The main result of this chapter (Theorem 12.2) shows that this subset of C is the set S with some  > 0 depending on ı [see (12.9) and (12.10)]. The examples considered in this section show that one cannot expect to find an attracting set smaller than S , whose elements can be considered as approximate solutions of the variational inequality. The results of this chapter were obtained [127]. We suppose that the mapping f is Lipschitz on all bounded subsets of X and that hf .y/; y  xi  0 for all y 2 C and all x 2 S:

(12.4)

Remark 12.1. Note that (12.4) holds if f is pseudo-monotone on C. Usually algorithms, studied in the literature, generate sequences which converge weakly to an element of S. In this chapter, for a given  > 0, we are interested to find a point x 2 X such that inffkx  yk W y 2 S g  : This point x is considered as an -approximate solution of the variational inequality. We will prove the following result, which shows that an -approximate solution can be obtained after k iterations of the subgradient method and under the presence of computational errors bounded from above by a constant ı, where ı and k are constants depending on . Theorem 12.2. Let  2 .0; 1/, M0 > 0; M1 > 0; L > 0 be such that B.0; M0 / \ S 6D ;;

(12.5)

f .B.0; 3M0 C 1//  B.0; M1 /;

(12.6)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.0; 3M0 C M1 C 1/;

(12.7)

0 < Q <   1;  L < 1;

(12.8)

0 > 0 satisfy 30 .M1 C Q 1 .1 C M0 C M1 / C .L C Q 1 // < ;

(12.9)

let ı 2 .0; 1/ satisfy 4ı.1 C 2M0 / < 02 .1  2 L2 /=2

(12.10)

186

12 The Extragradient Method for Solving Variational Inequalities

and let an integer k > 8M02 02 .1  2 L2 /1 :

(12.11)

Assume that 1 1 fxi g1 iD0  X; fyi giD0  X; fi giD0  ŒQ ;  ;

(12.12)

kx0 k  M0 ;

(12.13)

kyi  PC .xi  i f .xi //k  ı;

(12.14)

kxiC1  PC .xi  i f .yi //k  ı:

(12.15)

and that for each integer i  0,

Then there is an integer j 2 Œ0; k such that xi 2 B.0; 3M0 /; i D 0; : : : ; j;

(12.16)

kxj  yj k  20 ; kxi  yi k > 20 for all integers i satisfying 0  i < j:

(12.17)

Moreover, if an integer j 2 Œ0; k satisfies (12.17), then hf .xj /;   xj i    k  xj k for all  2 C

(12.18)

and there is y 2 S such that kxj  yk  . Note that Theorem 12.2 provides the estimations for the constants ı and k, which follow from (12.9)–(12.11). Namely, ı D c1  2 and k D c2  2 , where c1 and c2 are positive constants depending on M0 . Let us consider the following particular example. Example 12.3. Assume that f .x/ D x for all x 2 X and C D X. Then S D f0g. Let  2 .0; 1=2/ and M0 D 102 . Clearly, in this case S  fy 2 X W kyk  2g and the assertion of Theorem 12.2 holds with M1 D 400 [see (12.6)], L D 1 [see (12.7)], Q D 21 ,  D 3=4 [see (12.8)], 0 D 51  103 

12.1 Preliminaries and the Main Results

187

[see (12.9)], ı D 21  1011  2 [see (12.10)] and with k, which is the smallest integer larger than 1012  16 2 [see (12.11)]. The following example demonstrates that the set S can be easily calculated if the mapping f is strongly monotone. Example 12.4. Let rN 2 .0; 1/. Set CrN D fx 2 X W kx  PC .x/k  rN g: We say that f is strongly monotone with a constant ˛ > 0 on CrN if hf .x/  f .y/; x  yi  ˛kx  yk2 for all x; y 2 CrN : Fix u 2 S. We suppose that there is ˛ 2 .0; 1/ such that hf .x/; x  u i  ˛kx  u k2 for all x 2 CrN :

(12.19)

Remark 12.5. Note that inequality (12.19) holds, if f is strongly monotone with a constant ˛ on CrN . Let  > 0 and x 2 S . Then for all y 2 C, hf .x/; y  xi  ky  xk   and in particular ku  xk    hf .x/; u  xi  ˛kx  u k2 : This implies that ˛kx  u k2  2 maxf; kx  u kg; kx  u k  maxf2˛ 1 ; .2˛ 1 /1=2 g and if   21 ˛, then kx  u k  .2˛ 1 /1=2 and S  fx 2 X W kx  u k  .2˛ 1 /1=2 g:

188

12 The Extragradient Method for Solving Variational Inequalities

According to Theorem 12.2, under its assumptions, there is an integer j 2 Œ0; k such that kxj  u k   C .2˛ 1 /1=2 : Note that the constant ˛ can be obtained by analyzing an explicit form of the mapping f . In next example we show what is the set S when C D X. Example 12.6. Assume that C D X. It is easy to see that S D fx 2 X W f .x/ D 0g: Let  > 0 and x 2 S . Then for all z 2 B.0; 1/, hf .x/; zi  kzk    2 and kf .x/k  2. Thus S  fx 2 X W kf .x/k  2g: In the following example, we demonstrate that, if computational errors made by our computer system are ı > 0, then in principle any element of the set S , where  is a positive constant depending on ı, can be a limit point of the sequence fxi g1 iD0 generated by the subgradient method. This means that Theorem 12.2 cannot be improved. Example 12.7. Assume that f .x/ D x for all x 2 X and C D X. Clearly, f is strongly monotone on X with a constant 1 and S D f0g. According to Example 12.6 for any  2 .0; 21 /, S  fy 2 X W kyk  2g: Let  2 .0; 1/, ı 2 .0; 1/, v 2 B.0; ı/ 1 and let sequences fxi g1 iD0 ; fyi giD0  X satisfy for all integers i  0,

yi D xi  f .xi / D .1   /xi ; xiC1 D xi  f .yi / C v D .1   C  2 /xi C v:

(12.20)

By induction it follows from (12.20) that for all integers n  1, n1 X .1   C  2 /i v ! .   2 /1 v as n ! 1: xn D .1   C  / x0 C 2 n

iD0

12.2 Auxiliary Results

189

Thus any  2 B.0; ı.   2 /1 / can be a limit of a sequence fxn g1 nD0 generated by the subgradient method under the present of computational errors ı. In next example we obtain an estimation for the sets S when f ./ is the Gâteaux derivative of a convex function. Example 12.8. Assume that F W X ! R1 is a convex Gâteaux differentiable function and f .x/ D F 0 .x/ for all x 2 X, where F 0 .x/ is the Gâteaux derivative of F at the point x 2 X. We suppose that F 0 D f be Lipschitz on all bounded subsets of X and that x 2 C satisfies F.x /  F.z/ for all z 2 C: Q can be known a priori or Q > kx k be given. (Note that M Assume that a constant M obtained by analyzing an explicit form of the function F.) Let  2 .0; 1/ and Q x 2 S \ B.0; M/: Then for each y 2 C, F.y/  F.x/  hF 0 .x/; y  xi D hf .x/; y  xi  ky  xk   and, in particular, F.x /  F.x/ Q  :  kx  x k    2 M Thus Q  fx 2 C W F.x/  F.x / C .2M Q C 1/g: S \ B.0; M/ The chapter is organized as follows. Section 12.2 contains auxiliary results. Theorem 12.2 is proved in Sect. 12.3. Convergence results for the finite-dimensional space X are obtained in Sects. 12.4 and 12.5.

12.2 Auxiliary Results We use the assumptions, definitions, and the notation introduced in Sect. 12.1.

190

12 The Extragradient Method for Solving Variational Inequalities

Lemma 12.9. Assume that  > 0; u 2 S; M0 > 0; M1 > 0; L > 0;

(12.21)

f .B.u ; M0 //  B.0; M1 /;

(12.22)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.u ; M0 C  M1 /:

(12.23)

u 2 B.u ; M0 /; v D PC .u   f .u//;

(12.24)

T WD fw 2 X W hu  f .u/  v; w  vi  0g;

(12.25)

Let

D be a convex and closed subset of X such that CDT

(12.26)

uQ D PD .u  f .v//:

(12.27)

(by Lemma 2.2, C  T) and let

Then kQu  u k2  ku  u k2  .1   2 L2 /ku  vk2 :

(12.28)

Proof. By (12.22) and (12.24), kf .u/k  M1 : Together with (12.21), (12.24), and Lemma 2.2, this implies that ku  vk  ku  .u  f .u//k  M0 C  M1 :

(12.29)

By (12.4), (12.21), and (12.24), hf .v/; uQ  u i  hf .v/; uQ  vi:

(12.30)

In view of (12.25), (12.26), and (12.27), hQu  v; .u  f .u//  vi  0: This implies that hQu  v; .u  f .v//  vi   hQu  v; f .u/  f .v/i:

(12.31)

12.2 Auxiliary Results

191

Set z D u   f .v/:

(12.32)

By (12.27) and (12.32), jjQu  u jj2 D jjz  u jj2 C jjz  PD .z/jj2 C 2hPD .z/  z; z  u i:

(12.33)

By (12.21), (12.26) and Lemma 2.2, 2kz  PD .z/k2 C 2hPD .z/  z; z  u i D 2hz  PD .z/; u  PD .z/i  0: Together with (12.27), (12.30), (12.32), and (12.33) this implies that kQu  u k2  kz  u k2  kz  PD .z/k2 D ku  f .v/  u k2  ku   f .v/  uQ k2 D ku  u k2  ku  uQ k2 C2hu  uQ ; f .v/i  ku  u k2  ku  uQ k2 C 2 hv  uQ ; f .v/i: Together with (12.31) this implies that kQu  u k2  ku  u k2 C2hv  uQ ; f .v/i  hu  v C v  uQ ; u  v C v  uQ i D ku  u k2  ku  vk2  kv  uQ k2 C2hQu  v; u  v   f .v/i  ku  u k2  ku  vk2  kv  uQ k2 C2hQu  v; f .u/  f .v/i:

(12.34)

By Cauchy-Schwarz inequality, (12.23), (12.24), (12.29), and (12.34), kQu  u k2  ku  u k2  kv  uk2  kv  uQ k2 C 2 LkQu  vkku  vk  ku  u k2  ku  vk2  kv  uQ k2 C  2 L2 ku  vk2 C kQu  vk2  ku  u k2  .1   2 L2 /ku  vk2 : Lemma 12.9 is proved.

192

12 The Extragradient Method for Solving Variational Inequalities

Lemma 12.10. Let u 2 S; M0 > 0; M1 > 0; L > 0; ı 2 .0; 1/; f .B.u ; M0 //  B.0; M1 /;

(12.35) (12.36)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.u ; M0 C M1 C 1/;

(12.37)

 2 .0; 1; L < 1:

(12.38)

Assume that x 2 B.u ; M0 /; y 2 X; ky  PC .x  f .x//k  ı; xQ 2 X; kQx  PC .x  f .y//k  ı:

(12.39) (12.40)

Then kQx  u k2  4ı.1 C M0 / C kx  u k2  .1   2 L2 /kx  PC .x   f .x//k2 : Proof. Set v D PC .x   f .x//; z D PC .x  f .v//; zQ D PC .x   f .y//:

(12.41)

By Lemma 12.9, (12.41), (12.35), (12.36), (12.37), (12.38), and (12.39), kz  u k2  kx  u k2  .1   2 L2 /kx  vk2 :

(12.42)

Clearly, kQx  u k2 D kQx  z C z  u k2 D kQx  zk2 C 2hQx  z; z  u i C kz  u k2  kQx  zk2 C 2kQx  zkkz  u k C kz  u k2 :

(12.43)

By (12.39) and (12.41), kv  yk  ı:

(12.44)

It follows from (12.41), (12.35), Lemma 2.2, (12.39), and (12.36) that ku  vk  ku  xk C kf .x/k  M0 C  M1 :

(12.45)

12.3 Proof of Theorem 12.2

193

By (12.45), (12.44), and (12.35), ku  yk  ku  vk C kv  yk  M0 C  M1 C 1:

(12.46)

By (12.46), (12.41), Lemma 2.2, (12.45), (12.40), (12.37), (12.38), and (12.44), kQx  zk  kQx  zQk C kQz  zk  ı C kQz  zk  ı C  kf .y/  f .v/k  ı C  Lı D ı.1 C L/:

(12.47)

By (12.43), (12.47), (12.42), (12.41), (12.38), and (12.39), kQx  u k2  .ı.1 C L//2 C 2ı.1 C  L/kz  u k C kz  u k2  .ı.1 C L//2 C 2ı.1 C  L/kx  u k C kz  u k2  ı 2 .1 C  L/2 C 2ı.1 C L/kx  u k Ckx  u k2  .1   2 L2 /kx  PC .x   f .x//k2  4ı 2 C 4ıM0 C kx  u k2  .1   2 L2 /kx  PC .x   f .x//k2 : This completes the proof of Lemma 12.10.

12.3 Proof of Theorem 12.2 By (12.5) there is u 2 S \ B.0; M0 /:

(12.48)

kx0  u k  2M0 :

(12.49)

By (12.13) and (12.48),

Assume that i  0 is an integer and that xi 2 B.u ; 2M0 /:

(12.50)

(Note that for i D 0 (12.50) holds.) By Lemma 12.10 applied with x D xi , y D yi , xQ D xiC1 and  D i , (12.48), (12.50), (12.6), (12.7), (12.12), (12.14), and (12.15), kxiC1  u k2  4ı.1 C 2M0 / C kxi  u k2  .1  i2 L2 /kxi  PC .xi  i f .xi //k2 :

(12.51)

194

12 The Extragradient Method for Solving Variational Inequalities

There are two cases: kxi  yi k  20 I

(12.52)

kxi  yi k > 20 :

(12.53)

Assume that (12.53) holds. Then by (12.53), (12.14), (12.10), and the inequality 0 < 1, kxi  PC .xi  i f .xi //k  kxi  yi k  k  yi C PC .xi  i f .xi //k > 20  ı > 0 > 02 :

(12.54)

Then in view of (12.54), (12.51), (12.12), and (12.10), kxiC1  u k2  4ı.1 C 2M0 / C kxi  u k2  02 .1  2 L2 /  kxi  u k2  02 .1  2 L2 /21 :

(12.55)

Thus we have shown that the following property holds: (P) if an integer i  0 satisfies (12.50) and (12.53), then (12.55) holds. Property (P), (12.52), (12.53), and (12.49) imply that at least one of the following cases holds: (a) for all integers i D 0; : : : ; k the relations (12.50), (12.53), and (12.55) are true; (b) there is an integer j 2 f0; : : : ; kg such that for all integers i D 0; : : : ; j, (12.50) is valid, for all integers i satisfying 0  i < j (12.53) holds and that kxj  yj k  20 : Assume that case (a) holds. Then by (12.49) and (12.55), 4M02  ku  x0 k2  ku  xk k2 D

k1 X Œku  xi k2  ku  xiC1 k2   21 k02 .1  2 L2 / iD0

and k  8M02 02 .1  2 L2 /1 :

(12.56)

12.3 Proof of Theorem 12.2

195

This contradicts (12.11). The contradiction we have reached proves that case (a) does not hold. Then case (b) holds and there is an integer j 2 f0; : : : ; kg guaranteed by (b). Then (12.16) and (12.17) hold. Assume that an integer j 2 Œ0; k satisfies (12.17). (Clearly, in view of (b) such integer j is unique.) Then kxj  u k  2M0 ; kxj  yj k  20 :

(12.57)

By (12.57), (12.10), and (12.14), kxj  PC .xj  j f .xj //k  kxj  yj k C kyj  PC .xj  j f .xj //k  20 C ı  30 :

(12.58)

By Lemma 2.2, for each  2 C, 0  hxj  j f .xj /  PC .xj  j f .xj //;   PC .xj  j f .xj //i:

(12.59)

By (12.58) and (12.59) for each  2 C, 0  hxj  PC .xj  j f .xj //;   PC .xj  j f .xj //i j hf .xj /;   PC .xj  j f .xj //i  kxj  PC .xj  j f .xj //k.k  xj k C kxj  PC .xj  j f .xj //k/ j hf .xj /;   xj i  j hf .xj /; xj  PC .xj  j f .xj //i  30 k  xj k  902  j hf .xj /;   xj i  3j kf .xj /k0 :

(12.60)

By (12.60), (12.12), (12.6), (12.9), (12.57), (12.48) for each  2 C, j hf .xj /;   xj i  30 .1 C  M1 /  30 k  xj k and in view of (12.8) and (12.12) hf .xj /;   xj i  30 j1 .1 C  M1 /  30 j1 k  xj k  30 Q 1 .1 C M1 /  30 Q 1 k  xj k:

(12.61)

By (12.9) and (12.61), hf .xj /;   xj i    k  xj k for all  2 C:

(12.62)

Clearly, (12.62) is the claimed (12.18). Set yN D PC .xj  j f .xj //:

(12.63)

196

12 The Extragradient Method for Solving Variational Inequalities

By (12.63) and (12.58), yN 2 C; kxj  yN k  30   < 1:

(12.64)

By (12.57), (12.48), (12.64), and (12.7), kf .xj /  f .Ny/k  Lkxj  yN k  30 L:

(12.65)

By (12.64), (12.57), (12.48), (12.6), (12.9), (12.61), (12.65), and (12.4) for each  2 C, hf .Ny/;   yN i  hf .Ny/;   xj i  kf .Ny/kkxj  yN k  hf .Ny/;   xj i  3M1 0  hf .xj /;   xj i  kf .Ny/  f .xj /kk  xj k  3M1 0  30 Q 1 .1 C M1 /  30 Q 1 k  xj k  30 Lk  xj k  3M1 0  30 .M1 C Q 1 .1 C M1 //  30 .L C Q 1 /.k  yN k C kNy  xj k/  30 .L C Q 1 /.k  yN k/  30 .M1 C Q 1 .1 C M1 / C .L C Q 1 //  k  yN k   for all  2 C. Thus yN 2 S . This completes the proof of Theorem 12.2.

12.4 The Finite-Dimensional Case We use the assumptions, definitions, and notation introduced in Sect. 12.1 and we prove the following result. Theorem 12.11. Let X D Rn ,  2 .0; 1/, M0 > 0 be such that B.0; M0 / \ S 6D ;; M1 > 0 be such that f .B.0; 3M0 C 1//  B.0; M1 /; L > 0 be such that kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.0; 3M0 C M1 C 1/; 0 <   1;  L < 1:

12.4 The Finite-Dimensional Case

197

n Then there exist ı 2 .0; / and an integer k  1 such that for each fxi g1 iD0  R and 1 n each fyi giD0  R which satisfy kx0 k  M0 and for each integer i  0,

kyi  PC .xi  f .xi //k  ı; kxiC1  PC .xi  i f .yi //k  ı there is an integer j 2 Œ0; k such that kxj k  3M0 and inffkxj  zk W z 2 Sg  : Theorem 12.11 follows immediately from Theorem 12.2 and the following result. Lemma 12.12. Let M0 > 0, > 0. Then there exists  2 .0; / such that for each z 2 S \ B.0; M0 / the following relation holds: inffkz  uk W u 2 Sg  : Proof. Assume the contrary. Then there exists a sequence fk g1 kD1  .0; / which converges to zero and a sequence z.k/ 2 B.0; M0 / \ Sk ; k D 1; 2; : : : such that for each integer k  1, inf.fkz.k/  uk W u 2 Sg/ > : We may assume without loss of generality that there is z D lim z.k/ : k!1

By definition, for each integer k  1 and each  2 C, hf .z.k/ /;   z.k/ i  k k  z.k/ k  k : This implies that for each  2 C, hf .z/;   zi D lim hf .z.k/ /;   z.k/ i k!1

 lim .k k  z.k/ k  k / D 0: k!1

(12.66)

198

12 The Extragradient Method for Solving Variational Inequalities

Thus z 2 S and for all natural numbers k large enough kz  z.k/ k < =2: This contradicts (12.66). The contradiction we have reached proves Lemma 12.12.

12.5 A Convergence Result We use the assumptions, definitions, and notation introduced in Sect. 12.1. Let X D Rn . For each x 2 Rn and each A  Rn set d.x; A/ D inffkx  yk W y 2 Ag: N 1; M N 2 > 0 such that N 0 > 2, M Suppose that the set S is bounded and choose M N 0  2/; S  B.0; M N 1 /; f .B.0; 3M N 0 C 3M N 1 C 1//  B.0; M N 2 /: N 0 C 1//  B.0; M f .B.0; 3M

(12.67) (12.68)

Assume that N0CM N1CM N 2 ; M1 > 0; L > 0; M0 > M

(12.69)

f .B.0; 3M0 C 1//  B.0; M1 /;

(12.70)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.0; 3M0 C M1 C 1/;

(12.71)

0 <   1 and L < 1:

(12.72)

2 .0; 41 /

(12.73)

By Theorem 12.11 there exist

and a natural number kN such that the following property holds: 1 n n (P2) for each pair of sequences fxi g1 iD0  R and fyi giD0  R with kx0 k  M0 and for each integer i  0 with kyi  PC .xi  f .xi //k  ; kxiC1  PC .xi  i f .yi //k  N such that d.xj ; S/  1=4. there is an integer j 2 Œ0; k We prove the following result.

(12.74)

12.5 A Convergence Result

199

Theorem 12.13. Let fıi g1 iD0  .0; ; lim ıi D 0 i!1

(12.75)

and let  2 .0; 1/. Then there exists a natural number k0 such that for each pair of 1 n n sequences fxi g1 iD0  R and fyi giD0  R which satisfies kx0 k  M0

(12.76)

and for each integer i  0 with kyi  PC .xi  f .xi //k  ıi ; kxiC1  PC .xi  i f .yi //k  ıi

(12.77)

the inequality d.xi ; S/   holds for all integers i  k0 . Proof. By Theorem 12.11, there exist a positive number ıN <  2 .1 C M0 /1 641

(12.78)

and an integer k1  1 such that the following property holds: 1 n n (P3) for each pair of sequences fui g1 iD0  R and fvi giD0  R such that ku0 k  M0

(12.79)

and that for each integer i  0, N kuiC1  PC .ui  i f .vi //k  ıN kvi  PC .ui  f .ui //k  ı;

(12.80)

there is an integer j 2 Œ0; k1  such that d.uj ; S/  =8:

(12.81)

By (12.75) there is an integer k2  1 such that ıi < k12 ıN for all integers i  k2 :

(12.82)

k0 D 2 C kN C k1 C k2 :

(12.83)

Set

200

12 The Extragradient Method for Solving Variational Inequalities

1 n n Assume that sequences fxi g1 iD0  R and fyi giD0  R satisfy (12.76) and that for each integer i  0 equation (12.77) holds. Assume that an integer j  0 satisfies

kxj k  M0 :

(12.84)

N such that kxi k  M0 . We show that there exists an integer i 2 Œ1 C j; 1 C j C k N such that By (12.75), (12.77), (12.84), and (P2) there is an integer p 2 Œj; j C k d.xp ; S/  1=4:

(12.85)

In view of (12.67), (12.69), and (12.85), kxp k  M0 : If p > j we put i D p and obtain N kxi k  M0 : i 2 Œj C 1; j C k;

(12.86)

Assume that p D j. Then in view of (12.67) and (12.85), N 0  2 C 1=4: kxj k  M

(12.87)

By Lemma 2.2, (12.73), (12.85), (12.87), (12.68), (12.72), (12.77), and (12.75), kyj k  C kPC .xj  f .xj //k  C kPC .xj /k C kf .xj /k  kxj k C 1=2 C  kf .xj /k N 0  2 C 3=4 C M N 1: M

(12.88)

N 2: kf .yj /k  M

(12.89)

By (12.88) and (12.68),

By (12.69), (12.77), (12.75), (12.73), (12.85), (12.72), (12.87), (12.89), and Lemma 2.2, kxjC1 k  1=4 C kPC .xj  f .yj //k  kPC .xj /k C kf .yj /k C 1=4 N0CM N 2 < M0 :  kxj k C 1=2 C kf .yj k  M

(12.90)

12.5 A Convergence Result

201

N By (12.90), (12.77), (12.75), and (P2) there exists an integer i 2 Œj C 1; j C 1 C k such that d.xi ; S/  1=4 and together with (12.67) and (12.69) the equation above implies that kxi k < M0 . Thus we have shown that the following property holds: (P4) if an integer j  0 satisfies kxj k  M0 , then there is an integer i 2 Œj C 1; j C N such that kxi k  M0 . 1 C k Set j0 D supfi W i is an integer; i  k2 and kxi k  M0 g:

(12.91)

By (12.76) the number j0 is well defined and satisfies 0  j0  k2 :

(12.92)

j0 C 1 C kN  k2 :

(12.93)

In view of (P4) and (12.91),

By (P4) and (12.91) there is an integer j1 such that N and kxj1 k  M0 : j1 2 Œj0 C 1; j0 C 1 C k

(12.94)

By (12.91) and (12.94), N j1 > k2 ; j1  k2  j1  j0  1 C k:

(12.95)

Assume that an integer j  j1 satisfies kxj k  M0 :

(12.96)

We show that there is an integer i 2 Œj C 1; j C 1 C k1  such that d.xi ; S/  =8. By (P3), (12.96), (12.77), (12.95), and (12.82) there is an integer p 2 Œj; j C k1  such that d.xp ; S/  =8:

(12.97)

If p > j, then we set i D p. Assume that p D j. Clearly, (12.87)–(12.90) hold and kxjC1 k  M0 :

(12.98)

By (P3), (12.98), (12.77), (12.95), and (12.82) there is an integer i 2 ŒjC1; jCk1 C1 for which d.xi ; S/  =8. Thus we have shown that the following property holds: (P5) If an integer j  j1 and kxj k  M0 , then there is an integer i 2 ŒjC1; jC1Ck1  such that d.xi ; S/  =8:

202

12 The Extragradient Method for Solving Variational Inequalities

(P5), (12.94), (12.67), and (12.69) imply that there exists a sequence of natural numbers fjp g1 pD1 such that for each integer p  1, 1  jpC1  jp  1 C k1

(12.99)

d.xjp ; S/  =8:

(12.100)

and for each integer p  2,

We show that d.xi ; S/   for all integers i  j2 : Set 0 D 41 k11 :

(12.101)

Let p  2 be an integer. We show that for each integer l satisfying 0  l < jpC1  jp , d.xjp Cl ; S/  .=8/ C l0 :

(12.102)

By (12.100) estimate (12.102) holds for l D 0. Assume that an integer l satisfies 0  l < jpC1  jp

(12.103)

and that (12.102) holds. By (12.102), (12.103), (12.99), and (12.101), d.xjp Cl ; S/  .=8/ C k1 0 < =2:

(12.104)

By (12.102) there is u 2 S such that d.xjp Cl ; S/ D kxjp Cl  u k  .=8/ C l0 :

(12.105)

It follows from (12.67), (12.69)–(12.72), (12.105), (12.99), (12.101)–(12.103), (12.82), (12.78), (12.77), (12.95), and Lemma 12.10 applied with u , M0 , M1 , L, ı D ıjp Cl ; x D xjp Cl ; y D yjp Cl ; xQ D xjp ClC1 that kxjp ClC1  u k  kxjp Cl  u k C .4ıjp Cl .1 C M0 //1=2  =8 C l0 N 2 .1 C M0 //1=2  =8 C l0 C k1 =4 D =8 C .l C 1/0 : C.4ık 1 1

12.5 A Convergence Result

203

This implies that d.xjp ClC1 ; S/  .=8/ C .l C 1/0 : Thus by induction we have shown that for all l D 0; : : : ; jpC1  jp relation (12.102) holds and it follows from (12.102), (12.99), and (12.101) that for all integers l D 0; : : : ; jpC1  jp  1, d.xjp Cl ; S/  .=8/ C l0  =8 C k1 0  =2: Since the inequality above holds for all integers p  2, we conclude that d.xi ; S/  =2 for all integers i  j2 : By (12.99), (12.95), and (12.83), j2  k1 C j1 C 1  k1 C 2 C kN C k2 D k0 : Together with (12.106) this implies that d.xi ; S/  =2 for all integers i  k0 . Theorem 12.13 is proved.

(12.106)

Chapter 13

A Common Solution of a Family of Variational Inequalities

In a Hilbert space, we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors. The convergence of the subgradient method is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.

13.1 Preliminaries and the Main Result Let .X; h; i/ be a Hilbert space equipped with an inner product h; i which induces a complete norm k  k. We denote by Card.A/ the cardinality of the set A. For every point x 2 X and every nonempty set A  X define d.x; A/ WD inffkx  yk W y 2 Ag: For every point x 2 X and every positive number r put B.x; r/ D fy 2 X W kx  yk  rg: Let cN 2 .0; 1/ and 0 <  <    1. Let C be a nonempty closed convex subset of X. In view of Lemma 2.2, for every point x 2 X there is a unique point PC .x/ 2 C satisfying kx  PC .x/k D inffkx  yk W y 2 Cg: Moreover, kPC .x/  PC .y/k  kx  yk © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_13

205

206

13 A Common Solution of a Family of Variational Inequalities

for all x; y 2 X and for each x 2 X and each z 2 C, hz  PC .x/; x  PC .x/i  0: Let L1 be a finite set of pairs .f ; C/ where C is a nonempty closed convex subset of X, and f W X ! X and L2 be a finite set of mappings T W X ! X. We suppose that the set L1 [ L2 is nonempty. (Note that one of the sets L1 or L2 may be empty.) We suppose that for each .f ; C/ 2 L1 the mapping f is Lipschitz on all bounded subsets of X, the set S.f ; C/ WD fx 2 C W hf .x/; y  xi  0 for all y 2 Cg

(13.1)

is nonempty and that hf .y/; y  xi  0 for all y 2 C and all x 2 S.f ; C/:

(13.2)

Evidently, every point x 2 S.f ; C/ is considered as a solution of the variational inequality associated with the pair .f ; C/ 2 L1 : Find x 2 C such that hf .x/; y  xi  0 for all y 2 C: For every pair .f ; C/ 2 L1 and every positive number  define S .f ; C/ D fx 2 C W hf .x/; y  xi  ky  xk   for all y 2 Cg:

(13.3)

Note that this set was introduced in Chap. 12 and that every point x 2 S .f ; C/ is an -approximate solution of the variational inequality associated with the pair .f ; C/ 2 L1 . The examples considered in Chap. 12, show that elements of S .f ; C/ can be considered as -approximate solutions of the corresponding variational inequality. We suppose that for every mapping T 2 L2 the set Fix.T/ WD fz 2 X W T.z/ D zg 6D ;;

(13.4)

kT.z1 /  T.z2 /k  kz1  z2 k for all z1 ; z2 2 X;

(13.5)

kz  xk2  kz  T.x/k2 C cN kx  T.x/k2 for all x 2 X and all z 2 Fix.T/:

(13.6)

For every mapping T 2 L2 and every positive number  define Fix .T/ WD fz 2 X W kT.z/  zk  g:

(13.7)

S WD Œ\.f ;C/2L1 S.f ; C/ \ Œ\T2L2 Fix.T/ 6D ;:

(13.8)

Suppose that the set

13.1 Preliminaries and the Main Result

207

Let  > 0 and .f ; C/ 2 L1 . For every point x 2 X define Q;f ;C .x/ D PC .x  f .x//; P;f ;C .x/ D PC .x  f .Q;f ;C .x///:

(13.9) (13.10)

Let a natural number l  Card.L1 [ L2 /: Denote by R the set of all mappings A W f0; 1; 2; : : : g ! L2 [ fP;f ;C W .f ; C/ 2 L1 ;  2 Π;   g such that the following properties hold: (P1) (P2)

for every nonnegative integer p and every mapping T 2 L2 there exists an integer i 2 fp; : : : ; p C l  1g such that A.i/ D T; for each integer p  0 and every pair .f ; C/ 2 L1 there exist an integer i 2 fp; : : : ; p C l  1g and  2 Π;    such that A.i/ D P;f ;C .

We are interested to find solutions of the inclusion x 2 S. In order to meet this goal we apply algorithms generated by A 2 R. More precisely, we associate with any A 2 R the algorithm which generates, for any starting point x0 2 X, a sequence fxk g1 kD0  X, where xkC1 WD ŒA.k/.xk /; k D 0; 1; : : : : According to the results known in the literature, this sequence should converge weakly to an element of S. In this chapter, we study the behavior of the sequences generated by A 2 R taking into account computational errors which are always present in practice. Namely, in practice the algorithm associated with A 2 R generates a sequence fxk g1 kD0 such that for each integer k  0, if A.k/ D T 2 L2 ; then kxkC1  A.k/.xk /k  ı and if A.k/ D P;f ;C with  2 Œ ;   ; .f ; C/ 2 L1 ; then there is vk 2 X such that kvk  Q;f ;C .xk /k  ı; kxkC1  PC .xk  f .vk //k  ı

208

13 A Common Solution of a Family of Variational Inequalities

with a constant ı > 0 which depends only on our computer system. Surely, in this situation one cannot expect that the sequence fxk g1 kD0 converges to the set S. Our goal is to understand what subset of X attracts all sequences fxk g1 kD0 generated by algorithms associated with A 2 R. In Chap. 12 we showed that in the case when L2 D ; and the set L1 is a singleton, this subset of X is the set of -approximate solutions of the corresponding variational inequality with some  > 0 depending on ı. In this chapter we generalize the main result of Chap. 12 and show that in the general case (see Theorem 13.1 stated below) this subset of X is the set S WD fx 2 X W x 2 Fix .T/ for each T 2 L2 and d.x; S .f ; C//   for all .f ; C/ 2 L1 g with some  > 0 depending on ı [see (13.15) and (13.17)]. Our goal is also, for a given  > 0, to find a point x 2 S . This point x is considered as an -approximate common solution of the problems associated with the family of operators L1 [ L2 . We will prove the following result (Theorem 13.1), which shows that an -approximate common solution can be obtained after l.n0  1/ iterations of the algorithm associated with A 2 R and under the presence of computational errors bounded from above by a constant ı, where ı and n0 are constants depending on  [see (13.15)–(13.17)]. This result was obtained in [128]. Theorem 13.1. Let  2 .0; 1, M0 > 0 be such that B.0; M0 / \ S 6D ;

(13.11)

and let M1 > 0; L > 0 be such that for each .f ; C/ 2 L1 , f .B.0; 3M0 C 2//  B.0; M1 /;

(13.12)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.0; 5M0 C M1 C 2/

(13.13)

and   L < 1:

(13.14)

4 1 .5l1 C 2  1 M1 C 4L C 2M1 /  ;

(13.15)

n0 > 16M02 cN 1 .1  .  /2 L2 /1 12

(13.16)

Let 1 2 .0; 21 / satisfy

an integer

13.1 Preliminaries and the Main Result

209

and a number ı 2 .0; 1/ satisfy 4ı.2 C 2M0 /l < 161 cN 12 .1  .  /2 L2 /:

(13.17)

A 2 R; fxk g1 kD0  X; kx0 k  M0

(13.18)

Assume that

and that for each integer k  0, if A.k/ D T 2 L2 ; then kxkC1  A.k/.xk /k  ı;

(13.19)

A.k/ D P;f ;C with  2 Π;   ; .f ; C/ 2 L1 ;

(13.20)

and if

then there is vk 2 X such that kvk  Q;f ;C .xk /k  ı; kxkC1  PC .xk   f .vk //k  ı:

(13.21)

Then there is an integer p 2 Œ0; n0  1 such that kxi k  3M0 C 1; i D 0; : : : ; .p C 1/l

(13.22)

and for each integer i 2 fpl; : : : ; .p C 1/l  1g, (P3) (P4)

If A.i/ D T 2 L2 , then kxiC1  xi k  1 ; If A.i/ D P;f ;C with  2 Π;   , .f ; C/ 2 L1 , then kxi  vi k  1 .

Moreover, if an integer p 2 Œ0; n0 1 be such that for each integer i 2 Œpl; .pC1/l1 properties (P3) and (P4) hold and kxi k  3M0 C 1, then for each pair i; j 2 fpl; : : : ; .p C 1/lg, kxi  xj k   and for each i 2 fpl; : : : ; .p C 1/lg, xi 2 Fix .T/ for all T 2 L2 ; d.xi ; S .f ; C//   for all .f ; C/ 2 L1 : Note that Theorem 13.1 provides the estimations for the constants ı and n0 , which follow from relations (13.15)–(13.17). Namely, ı D c1  2 and n0 D c2  2 , where c1 and c2 are positive constants depending only on M0 . Let  2 .0; 1, a positive number ı be defined by relations (13.15) and (13.17), and let an integer n0  1 satisfy inequality (13.16). Assume that we apply an algorithm associated with a mapping A 2 R under the presence of computational

210

13 A Common Solution of a Family of Variational Inequalities

errors bounded from above by a positive constant ı and that our goal is to find an approximate solution x 2 S : It is not difficult to see that Theorem 13.1 also answers an important question how we can find an iteration number i such that xi 2 S : According to Theorem 13.1, we should find the smallest integer q 2 Œ0; n0  1 such that for every integer i 2 Œql; .q C 1/l  1 properties (P3) and (P4) hold and that the relation kxi k  3M0 C 1 is true. Then the inclusion xi 2 S is valid for all integers i 2 Œql; .q C 1/l. Consider the following convex feasibility problem. Suppose that C1 ; : : : ; Cm are nonempty closed convex subsets of X, where m is a natural number, such that the set C D \m iD1 Ci is also nonempty. We are interested to find a solution of the feasibility problem x 2 C. For every point x 2 X and every integer i D 1; : : : ; m there exists a unique element Pi .x/ 2 Ci such that kx  Pi .x/k D inffkx  yk W y 2 Ci g: The feasibility problem is a particular case of the problem discussed above with L1 D ; and L2 D fPi W i D 1; : : : ; mg.

13.2 Auxiliary Results The next result follows from Lemma 12.10. Lemma 13.2. Let .f ; C/ 2 L1 , u 2 S; m0 > 0; M1 > 0; L > 0; ı 2 .0; 1/; f .B.u ; m0 //  B.0; M1 /; kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.u ; m0 C M1 C 1/ and let  2 .0; 1; L < 1: Assume that x 2 B.u ; m0 /; y 2 X; ky  PC .x   f .x//k  ı; xQ 2 X; kQx  PC .x  f .y//k  ı: Then kQx  u k2  4ı.1 C m0 / C kx  u k2  .1   2 L2 /kx  PC .x   f .x//k2 :

13.2 Auxiliary Results

211

Lemma 13.3. Let .f ; C/ 2 L1 , M0 > 0; M1 > 0; L > 0; ı 2 .0; 1/; B.0; M0 / \ S 6D ;;

(13.23)

f .B.0; 3M0 C 1//  B.0; M1 /;

(13.24)

kf .z1 /  f .z2 /k  Lkz1  z2 k for all z1 ; z2 2 B.0; 5M0 C M1 C 2/

(13.25)

and let  2 .0; 1; L < 1:

(13.26)

x 2 B.0; 3M0 C 1/; y 2 X; ky  PC .x   f .x//k  ı;

(13.27)

xQ 2 X; kQx  PC .x  f .y//k  ı:

(13.28)

Let

Then kQx  xk  2ı C .1 C  L/kx  PC .x   f .x//k: Proof. In view of (13.28) and Lemma 2.2, kQx  PC .x  f .x//k  ı C kPC .x  f .y//  PC .x   f .x//k  ı C  kf .y/  f .x/k:

(13.29)

It follows from (13.23) that there exists a point u 2 B.0; M0 / \ S:

(13.30)

Lemma 2.2, (13.27), and (13.30) imply that ky  u k  ı C kPC .x  f .x//  u k  ı C kx   f .x/  u k  ı C kxk C ku k C  kf .x/k: Combined with (13.24), (13.27), and (13.30) the relation above implies that kyk  1 C 3M0 C 1 C 2M0 C  M1 :

(13.31)

In view of (13.25), (13.26), (13.27), (13.29), and (13.31), kQx  PC .x  f .x//k  ı C  Lky  xk:

(13.32)

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13 A Common Solution of a Family of Variational Inequalities

It follows from (13.32) that kQx  xk  kQx  PC .x  f .x//k C kx  PC .x   f .x//k  ı C Lky  xk C kx  PC .x   f .x//k: Combined with (13.26) and (13.27) the relation above implies that kQx  xk  2ı C ky  xk.1 C  L/; kQx  xk  2ı C .1 C L/kx  PC .x   f .x//k: Lemma 13.3 is proved. Lemma 13.4. Suppose that all the assumptions of Theorem 13.1 hold. Then ı < 1 =4;

(13.33)

2ı.2M0 C 1/ < 161 cN 12 ;

(13.34)

4ı.2 C 2M0 /  161 12 .1  .  /2 L2 /:

(13.35)

Proof. It is not difficult to see that inequality (13.33) follows from (13.14), (13.17) and the relations l  1, cN < 1 and 1 < 1=2. Relation (13.34) follows from (13.17), (13.14) and the inequality l  1. Relation (13.35) follows from (13.17) and the inequalities l  1 and cN < 1.

13.3 Proof of Theorem 13.1 In view of (13.11), there exists u 2 B.0; M0 / \ S:

(13.36)

It follows from (13.18) and (13.36) that ku  x0 k  2M0 :

(13.37)

Assume that a nonnegative integer p satisfies ku  xpl k  2M0 :

(13.38)

(Clearly, in view of (13.37), inequality (13.38) holds with p D 0.) Assume that an integer i 2 fpl; : : : ; .p C 1/l  1g satisfies kxi  u k  2M0 C 1 .i  pl/:

(13.39)

13.3 Proof of Theorem 13.1

213

Then one of the following two cases hold: A.i/ D T 2 L2 I

(13.40)

A.i/ D P;f ;C with  2 Π;   ; .f ; C/ 2 L1 :

(13.41)

Assume that (13.40) is valid. Then it follows from (13.19), (13.36), (13.40), (13.5), (13.39), Lemma 13.4, and (13.33) that kxiC1  u k  kxiC1  A.i/.xi /k C kA.i/.xi /  u k  ı C kxi  u k  2M0 C .i C 1  pl/ 1 :

(13.42)

In view of (13.39), (13.42), the inclusions i 2 Œpl; .p C 1/l  1 and  2 .0; 1/ and (13.15), kxi  u k  2M0 C l 1  2M0 C 1; kxiC1  u k  2M0 C l 1  2M0 C 1: Combined with (13.42) these inequalities above imply that kxiC1  u k2  kxi  u k2  ı.kxiC1  u k C kxi  u k/  2ı.2M0 C 1/:

(13.43)

Assume that (13.41) is valid. In view of (13.36), (13.39), (13.15), (13.12), (13.13), (13.17), (13.41), kxi  u k  2M0 C 1 and all the assumptions of Lemma 13.2 hold with x D xi , y D vi , xQ D xiC1 , m0 D 2M0 C 1, and this implies that kxiC1  u k2  4ı.2 C 2M0 / C kxi  u k2

(13.44)

and kxiC1  u k  kxi  u k C 2.ı.2 C 2M0 //1=2  kxi  u k C 1  2M0 C .i C 1  pl/ 1 :

(13.45)

(Note that the first inequality of (13.45) follows from (13.44), the second inequality follows from (13.17) and the inequalities cN < 1 and l  1, and the third inequality follows from (13.39).)

214

13 A Common Solution of a Family of Variational Inequalities

It follows from (13.42)–(13.45) that in both cases kxiC1  u k  2M0 C .i C 1  pl/ 1 ; kxiC1  u k2  kxi  u k2 C 4ı.2 C 2M0 /: Thus by induction we proved that for all integers i D pl; : : : ; .p C 1/l the inequality kxi  u k  2M0 C .i  pl/ 1 holds and that for all integers i D pl; : : : ; .p C 1/l  1 the inequality kxiC1  u k2  kxi  u k2 C 4ı.2 C 2M0 / is valid. By (13.15), we have shown that the following property holds: (P5)

If a nonnegative integer p satisfies the inequality ku  xpl k  2M0 ;

then we have kxi  u k < 2M0 C 1 for all i D pl; : : : ; .p C 1/l

(13.46)

and kxi  u k2  kxiC1  u k2  4ı.2 C 2M0 / for all i D pl; : : : ; .p C 1/l  1:

(13.47)

Assume that an integer qQ 2 Œ0; n0  1 and that for every integer p 2 Œ0; qQ  the following property holds: (P6)

there exists i 2 fpl; : : : ; .p C 1/l  1g such that (P3) and (P4) do not hold.

Assume now that an integer p 2 Œ0; qQ  satisfies ku  xpl k  2M0 :

(13.48)

In view of property (P5) and relation (13.48), inequalities (13.46) and (13.47) are valid. Property (P6) implies that there exists an integer j 2 fpl; : : : ; .p C 1/l  1g such that properties (P3) and (P4) do not hold with i D j. Evidently, one of the following cases holds: A.j/ D T 2 L2 I

(13.49)

A.j/ D P;f ;C with  2 Π;   ; .f ; C/ 2 L1 :

(13.50)

13.3 Proof of Theorem 13.1

215

Assume that relation (13.49) holds. Since property (P3) does not hold with i D j we have kxjC1  xj k > 1 :

(13.51)

It follows from (13.49), (13.51), (13.19), and Lemma 13.4 that kxj  T.xj /k  kxjC1  xj k  kT.xj /  xjC1 k  1  ı  .3=4/ 1 :

(13.52)

Relations (13.49), (13.6), (13.36), (13.8), and (13.52) imply that ku  xj k2  ku  T.xj /k2 C cN kxj  T.xj /k2  ku  T.xj /k2 C cN .9=16/ 12 D ku  xjC1 k2 C kxjC1  T.xj /k2 C2hu  xjC1 ; xjC1  T.xj /i C cN .9=16/ 12  ku  xjC1 k2  2ku  xjC1 kkxjC1  T.xj /k C cN .9=16/ 12 : (13.53) In view of (13.53), (13.19), (13.46), and (13.34), ku  xj k2  ku  xjC1 k2  cN .9=16/ 12  2ı.2M0 C 1/  .Nc=2/ 12 : Hence if (13.49) holds, then ku  xj k2  ku  xjC1 k2  .Nc=2/ 12 :

(13.54)

Assume that relation (13.50) is valid. Then relations (13.50), (13.36), (13.46), (13.12), (13.13), (13.20), and (13.21) imply that all the assumptions of Lemma 13.2 hold with x D xj ; y D vj ; xQ D xjC1 ; m0 D 2M0 C 1 and this implies that ku  xjC1 k2  4ı.2 C 2M0 / C ku  xj k2  .1  . L/2 /kxj  PC .xj   f .xj //k2 : (13.55)

216

13 A Common Solution of a Family of Variational Inequalities

Since property (P4) does not hold with i D j we conclude that kxj  vj k > 1 :

(13.56)

In view of (13.46), (13.21), and (13.56), kxj  PC .xj  f .xj //k  kxj  vj k  kPC .xj  f .xj //  vj k > 1  ı > .3=4/ 1 :

(13.57)

It follows from (13.35), (13.55), and (13.57) that kxj  u k2  kxjC1  u k2  .1  .  L/2 /.9=16/ 12  4ı.2 C 2M0 /  .1  .  L/2 / 12 =2:

(13.58)

By relations (13.54) and (13.58), in both cases we have kxj  u k2  kxjC1  u k2  cN .1  .  L/2 / 12 =2:

(13.59)

It follows from (13.17), (13.47), and (13.59) that ku  xpl k2  ku  x.pC1/l k2 .pC1/l1

D

X

Œku  xi k2  ku  xiC1 k2 

iDpl

 cN .1  .  L/2 / 12 =2  4ı.2 C 2M0 /l > cN .1  .  L/2 / 12 =4:

(13.60)

By (13.48) and (13.60), ku  x.pC1/l k  ku  xpl k  2M0 :

(13.61)

Thus we have shown that the following property holds: (P7)

If an integer p 2 Œ0; qQ  satisfies inequality (13.48), then [see relations (13.61), (13.46), (13.60)] we have ku  x.pC1/l k  2M0 ; ku  xi k  2M0 C 1; i D pl; : : : ; .p C 1/l and ku  xpl k2  ku  x.pC1/l k2 > cN .1  .  L/2 / 12 =4:

(13.62)

13.3 Proof of Theorem 13.1

217

In view of (13.37), property (P7), (13.62), and (13.16), 4M02  ku  x0 k2  ku  x0 k2  ku  x.QqC1/l k2 D

qQ X

Œku  xpl k2  ku  x.pC1/l k2 

pD0

 .Qq C 1/Nc.1  .  L/2 / 12 =4 and qQ C 1  16M02 cN 1 .1  .  L/2 /1 12 < n0 :

(13.63)

We assumed that an integer qQ 2 Œ0; n0  1 and that for every integer p 2 Œ0; qQ  property (P6) holds and proved that qQ C 1 < n0 . This implies that there exists an integer q 2 Œ0; n0  1 such that for every integer p satisfying 0  p < q, property (P6) holds and that the following property holds: (P8)

For every integer i 2 fql; : : : ; .q C 1/l  1g properties (P3) and (P4) hold.

Property (P7) (with qQ D q  1) implies that ku  xjl k  2M0 ; j D 0; : : : ; q:

(13.64)

In view of (13.64) and property (P5), ku  xi k  2M0 C 1; i D 0; : : : ; .q C 1/l:

(13.65)

It follows from (13.64), (13.65), and (13.36) that kxi k  3M0 C 1; i D 0; : : : ; .q C 1/l: Assume that p is a nonnegative integer, kxi k  3M0 C 1; i D pl; : : : ; .p C 1/l  1

(13.66)

and that for all integers i D pl; : : : ; .p C 1/l  1 properties (P3) and (P4) hold. Let i 2 fpl; : : : ; .p C 1/l  1g:

(13.67)

A.i/ D T 2 L2 I

(13.68)

A.i/ D P;f ;C with  2 Π;   ; .f ; C/ 2 L1 :

(13.69)

There are two cases:

218

13 A Common Solution of a Family of Variational Inequalities

Assume that relation (13.68) is valid. Then in view of (13.68) and property (P3), kxiC1  xi k  1 :

(13.70)

It follows from (13.68), (13.19), (13.70), and (13.17) that kT.xi /  xi k  kT.xi /  xiC1 k C kxiC1  xi k  ı C 1  .5=4/ 1 ; xi 2 Fix.5=4/ 1 .T/:

(13.71)

Thus we have shown that the following property holds: (P9)

If (13.68) is true, then relations (13.70) and (13.71) hold.

Assume that (13.69) holds. In view of (13.69), (13.67), property (P4), (13.20), and (13.21), kxiC1  PC .xi  f .vi //k  ı; kvi  PC .xi  f .xi //k  ı; kxi  vi k  1 :

(13.72)

Relations (13.17), (13.21), and (13.72) imply that kxi  PC .xi  f .xi //k  kxi  vi k C kvi  PC .xi   f .xi //k  ı C 1  .5=4/ 1 :

(13.73)

It follows from (13.11), (13.69), (13.12), (13.13), (13.14), (13.67), (13.66), (13.72), (13.73), and (13.33) that all the assumptions of Lemma 13.3 hold with x D xi ; y D vi ; xQ D xiC1 (and with the constants M0 ; M1 ; L as in Theorem 13.1), and this implies that kxiC1  xi k  2ı C .1 C  L/kxi  PC .xi   f .xi //k  2ı C .5=4/.1 C L/ 1  .5=4/.1 C   L/ 1 C 2ı < 5 1 :

(13.74)

(Note that the second inequality in (13.74) follows from (13.73), the third one follows from (13.69) and the last inequality follows from (13.14) and (13.33).) In view of Lemma 2.2, (13.73), (13.69), (13.66), and (13.12), for every point  2 C,

13.3 Proof of Theorem 13.1

219

0  hxi  f .xi /  PC .xi  f .xi //;   PC .xi   f .xi //i D hxi  PC .xi  f .xi //;   PC .xi   f .xi //i hf .xi /;   PC .xi  f .xi //i  kxi  PC .xi  f .xi //k.k  xi k C kxi  PC .xi   f .xi //k/  hf .xi /;   xi i  hf .xi /; xi  PC .xi   f .xi //i  2 1 .k  xi k C 2 1 /  hf .xi /;   xi i  2  kf .xi /k 1 ;  hf .xi /;   xi i  2 1 k  xi k  4 12  2 1   M1 and hf .xi /;   xi i  2 1 1 k  xi k  41 12  2 1   1 M1 for each  2 C:

(13.75)

Set yN D PC .xi  f .xi //:

(13.76)

Relations (13.76) and (13.73) imply that kxi  yN k  .5=4/ 1 :

(13.77)

It follows from (13.77), (13.66), (13.13), (13.12), and (13.10) that kf .xi /  f .Ny/k  Lkxi  yN k  L.5=4/ 1 ; yN 2 B.xi ; 1/  B.0; 3M0 C 2/; kf .Ny/k  M1 :

(13.78)

(Note that the inclusion in (13.78) follows from (13.76), the inequality 1 < 1=2 and (13.66), and the last inequality in (13.78) follows from (13.12).) In view of (13.75), (13.77), (13.78), and (13.15), for every point  2 C, hf .Ny/;   yN i  hf .Ny/;   xi i  kf .Ny/kkxi  yN k  hf .Ny/;   xi i  2M1 1  hf .xi /;   xi i  kf .Ny/  f .xi /kk  xi k  2M1 1  2 1 1 k  xi k  4 1 1 2M1 1 1    2L 1 k  xi k  2M1 1

220

13 A Common Solution of a Family of Variational Inequalities

 2 1 1 k  yN k  2 1 1 kNy  xi k 41 1  2 1 M1   1 2L 1 k  yN k  2L 1 kxi  yN k  2M1 1  .2 1 1 C 2L 1 /k  yN k  6 1 1 2 1 M1   1  4L 1 2M1 1  kNy  k  : Combined with relation (13.76) the relation above implies that yN 2 S .f ; C/: It follows from the inclusion above and (13.77) that d.xi ; S .f ; C//  .5=4/ 1 : Thus we have shown that the following property holds: (P10)

if (13.69) is valid, then the inequalities kxi  xiC1 k  5 1 [see (13.74)] and d.xi ; S .f ; C//  .5=4/ 1 hold.

In view of properties (P9) and (P10), for every integer i 2 fpl; : : : ; .p C 1/l  1g, kxi  xiC1 k  5 1 : This implies that for every pair of integers i; j 2 fpl; : : : ; .p C 1/lg, we have kxi  xj k  5l 1 < =4:

(13.79)

[see (13.15)]. Let j 2 fpl; : : : ; .p C 1/lg. Assume that T 2 L2 . In view of (13.18) and property (P1) there exists an integer i 2 fpl; : : : ; .p C 1/l  1g such that A.i/ D T: It follows from (13.68) and property (P9) that xi 2 Fix.2 1 / .T/

13.4 Examples

221

and kxi  T.xi /k  2 1 : Combined with relations (13.79) and (13.15) this implies that kxj  T.xj /k  kxj  xi k C kxi  T.xi /k C kT.xj /  T.xi /k  =2 C 2 1 < ; xj 2 Fix .T/ for all T 2 L2 :

(13.80)

Assume that .f ; C/ 2 L1 . In view of (13.18) and property (P2), there exists an integer i 2 fpl; : : : ; .p C 1/l  1g such that A.i/ D P;f ;C with  2 Π;   : It follows from (13.69) and property (P10) that d.xi ; S .f ; C//  2 1 : Combined with relations (13.79) and (13.15) (the choice of 1 ) the inequality above implies that d.xj ; S .f ; C//  d.xi ; S .f ; C// C kxi  xj k  2 1 C =4 <  for all .f ; C/ 2 L1 . Theorem 13.1 is proved.

13.4 Examples In this section we present examples for which Theorem 13.1 can be used. Example 13.5. Let p  1 be an integer, Ci , i D 1; : : : ; p be nonempty closed convex subsets of the Hilbert space X and let for every integer i 2 f1; : : : ; pg, gi W X ! R1 be a convex Fréchet differentiable function and g0i .x/ 2 X be its Fréchet derivative at a point x 2 X. We assume that for all integers i D 1; : : : ; p the mapping g0i W X ! X is Lipschitzian on all bounded subsets of X. Consider the following multi-objective minimization problem p

Find x 2 \iD1 Ci such that gi .x/ D inffgi .z/ W z 2 Ci g for all i D 1; : : : ; p: It is clear that this problem is equivalent to the following problem which is a particular case of the problem discussed in this section with fi D g0i , i D 1; : : : ; p, and for which Theorem 13.1 was stated:

222

13 A Common Solution of a Family of Variational Inequalities p

Find x 2 \iD1 Ci such that for all i D 1; : : : ; p; hg0i .x/; y  xi  0 for all y 2 Ci : Let S be the set of solutions of these two problems. We assume that S 6D ;. Now it is not difficult to see that all the assumptions needed for Theorem 13.1 hold with fi D g0i , i D 1; : : : ; p, L1 D f.g0i ; Ci / W i D 1; : : : ; pg and L2 D ;. The constants M0 ; M1 ; L can be found, in principle, using the analytic description of the functions gi and the sets Ci , i D 1; : : : ; p which is usually given. In many cases p the set \iD1 Ci is contained in a ball or one of the sets Ci , i D 1; : : : ; p is contained in a ball and the radius of the ball can be found. Then we can found the constant M1 ; L, choose a positive constant   < L1 and apply Theorem 13.1 with A 2 R such that for each integer i  0 and each integer j 2 Œ0; p  1, A.ip C j/ D P  ;fjC1 ;CjC1 : Our next example is a particular case of Example 13.5. Example 13.6. Let X D R4 , p D 2, C1 D fx D .x1 ; x2 ; x3 ; x4 / 2 R4 W jx1 j  10; x3 D 2g; C2 D fx D .x1 ; x2 ; x3 ; x4 / 2 R4 W jx2 j  10; x4 D 2g; g1 .x/ D .2x1 C x2 C x3  x4  3/2 ; x D .x1 ; x2 ; x3 ; x4 / 2 R4 ; g2 .x/ D .x1 C 2x2 C x3  x4  3/2 ; x D .x1 ; x2 ; x3 ; x4 / 2 R4 : Evidently, the functions g1 ; g2 are convex and g1 ; g2 2 C2 . Consider the problem Find x 2 C1 \ C2 such that gi .x/ D inffgi .z/ W z 2 Ci g for i D 1; 2: As it was shown in Example 13.5, we can apply Theorem 13.1 for this problem with fi D g0i , i D 1; 2. Now we define the constants which appear in Theorem 13.1. Set l D 2. Clearly, C1 \ C2  fx D .x1 ; x2 ; x3 ; x4 / 2 R4 W jx1 j  10; jx2 j  10; x3 D 2; x4 D 2g  B.0; 16/: Thus we can set M0 D 16. It is easy to see that the set of solutions of our problem S D fx D .x1 ; x2 ; 2; 2/ W jx1 j  10; jx2 j  10; g1 .x/ D 0; g2 .x/ D 0g D fx D .x1 ; x2 ; 2; 2/ W jx1 j  10; jx2 j  10; 2x1 C x2  3 D 0; x1 C 2x2  3 D 0g D f.1; 1; 2; 2/g:

13.4 Examples

223

For all points x D .x1 ; x2 ; x3 ; x4 / 2 R4 we have f1 .x/ D g01 .x/ D 2.2x1 C x2 C x3  x4  3/.2; 1; 1; 1/; f2 .x/ D g02 .x/ D 2.x1 C 2x2 C x3  x4  3/.1; 2; 1; 1/: These equalities imply that for i D 1; 2, fi .B.0; 50//  B.0; 1530/ (thus M1 D 1530) and that the functions f1 ; f2 are Lipschitzian on R4 with the Lipschitz constant L D 12. Put   D 161 , cN D 1=2. We apply Theorem 13.1 with these constants and with  D 103 . Then (13.15) implies that we can set 1 D 41 107 . By (13.16), we have n0 > 2  163  1014 and in view of (13.17) the following inequality holds: ı < .16  34/1  163  1014 : Note that this example can also be considered as an example of a convex feasibility problem Find x 2 C1 \ C2 \ fz 2 Rn W g1 .z/  0g \ fz 2 Rn W g2 .z/  0g; or equivalently Find x 2 fz 2 C1 W g1 .z/  0g \ fz 2 C2 W g2 .z/  0g: Now we describe how the subgradient algorithm is applied for our example. First of all, note that for any y D .y1 ; y2 ; y3 ; y4 / 2 R4 , PC1 .y/ D .minfmaxfy1 ; 10g; 10g; y2 ; 2; y4 /; PC2 .y/ D .y1 ; minfmaxfy2 ; 10g; 10g; y3 ; 2/: We apply Theorem 13.1 with x0 D .0; 0; 0; 0/ and A 2 R such that for each integer i  0, A.2i/ D P161 ;f1 ;C1 ; A.2i C 1/ D P161 ;f2 ;C2 : 1 4 Then our algorithm generates two sequences fxi g1 iD0 ; fvi giD0  R such that for every nonnegative integer i,

kv2i  PC1 .x2i  161 f1 .x2i //k  ı; kx2iC1  PC1 .x2i  161 f1 .v2i //k  ı

224

13 A Common Solution of a Family of Variational Inequalities

and kv2iC1  PC2 .x2iC1  161 f2 .x2iC1 //k  ı; kx2iC2  PC2 .x2iC1  161 f2 .v2iC1 //k  ı: For every nonnegative integer p we calculate p D maxfkv2p  x2p k; kv2pC1  x2pC1 kg and find the smallest (first) integer p  0 such that p  1 . In view of Theorem 13.1, this nonnegative integer p exists and satisfies p < n0 . Then it follows from Theorem 13.1 that x2p ; x2pC1 ; x2pC2 2 S . Example 13.7. Let p; q  1 be integers, Ci , i D 1; : : : ; p, Di , i D 1; : : : ; q be nonempty closed convex subsets of the Hilbert space X and let for every integer i 2 f1; : : : ; pg, gi W X ! R1 be a convex Frechet differentiable function and g0i .x/ 2 X be its Frechet derivative at a point x 2 X. We assume that for every integer i D 1; : : : ; p, gi .z/  0 for all z 2 Ci and that the mapping g0i W X ! X is Lipschitzian on all bounded subsets of X. Consider the following convex feasibility problem p

q

Find a point belonging to .\iD1 fz 2 Ci W gi .z/  0g/ \ .\jD1 Dj /: It is easy to show that this problem is equivalent to the following problem which is a particular case of the problem discussed in this section, and for which Theorem 13.1 was stated: q

Find x 2 \jD1 Dj such that for all i D 1; : : : ; p; x 2 Ci and hg0i .x/; y  xi  0 for all y 2 Ci : Let S be the set of solutions of these two problems. We assume that S 6D ;. Now it is not difficult to see that all the assumptions needed for Theorem 13.1 hold with fi D g0i , i D 1; : : : ; p, L1 D f.g0i ; Ci / W i D 1; : : : ; pg and L2 D fPDi W i D 1; : : : ; qg, l D p C q. The constants M0 ; M1 ; L can be found as it was explained in Example 13.5, using the analytic description of the functions gi and the sets Ci ; Dj , i D 1; : : : ; p, j D 1; : : : ; q which is usually given. Then we choose a positive constant   < L1 and apply Theorem 13.1 with A 2 R such that A.i C .p C q// D A.i/ for all integers i  0; A.i/ D P  ;fiC1 ;CiC1 ; i D 0; : : : ; p  1; A.p  1 C j/ D PDj ; j D 1; : : : ; q:

Chapter 14

Continuous Subgradient Method

In this chapter we study the continuous subgradient algorithm for minimization of convex functions, under the presence of computational errors. We show that our algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how much time one needs for this.

14.1 Bochner Integrable Functions Let .Y; k  k/ be a Banach space and 1 < a < b < 1. A function x W Œa; b ! Y is strongly measurable on Œa; b if there exists a sequence of functions xn W Œa; b ! Y, n D 1; 2; : : : such that for any integer n  1 the set xn .Œa; b/ is countable and the set ft 2 Œa; b W xn .t/ D yg is Lebesgue measurable for any y 2 Y, and xn .t/ ! x.t/ as n ! 1 in .Y; k  k/ for almost every t 2 Œa; b. The function x W Œa; b ! Y is Bochner integrable if it is strongly measurable and Rb there exists a finite a kx.t/kdt. If x W Œa; b ! Y is a Bochner integrable function, then for almost every (a. e.) t 2 Œa; b, lim .t/1

Z

t!0

tCt

kx. /  x.t/kd D 0

t

and the function Z y.t/ D

t

x.s/ds; t 2 Œa; b

a

is continuous and a. e. differentiable on Œa; b. © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_14

225

226

14 Continuous Subgradient Method

Let 1 < 1 < 2 < 1. Denote by W 1;1 .1 ; 2 I Y/ the set of all functions x W Œ1 ; 2  ! Y for which there exists a Bochner integrable function u W Œ1 ; 2  ! Y such that Z t u.s/ds; t 2 .1 ; 2  x.t/ D x.1 / C 1

(see, e.g., [11, 27]). It is known that if x 2 W 1;1 .1 ; 2 I Y/, then this equation defines a unique Bochner integrable function u which is called the derivative of x and is denoted by x0 .

14.2 Convergence Analysis for Continuous Subgradient Method The study of continuous subgradient algorithms is an important topic in optimization theory. See, for example, [6, 10, 23, 27, 28] and the references mentioned therein. In this chapter we analyze its convergence under the presence of computational errors. We suppose that X is a Hilbert space equipped with an inner product denoted by h; i which induces a complete norm k  k. For each x 2 X and each r > 0 set B.x; r/ D fy 2 X W kx  yk  rg: For each x 2 X and each nonempty set E  X set d.x; E/ D inffkx  yk W y 2 Eg: Let D be a nonempty closed convex subset of X. Then for each x 2 X there is a unique point PD .x/ 2 D satisfying kx  PD .x/k D inffkx  yk W y 2 Dg (see Lemma 2.2). Suppose that f W X ! R1 [ f1g is a convex, lower semicontinuous and bounded from below function such that dom.f / WD fx 2 X W f .x/ < 1g 6D ;: Set inf.f / D inf.f .x/ W x 2 Xg

14.2 Convergence Analysis for Continuous Subgradient Method

227

and argmin.f / D fx 2 X W f .x/ D inf.f /g: For each set D  X put inf.f I D/ D infff .z/ W z 2 Dg; sup.f I D/ D supff .z/ W z 2 Dg: Recall that for each x 2 dom.f /, @f .x/ D fl 2 X W hl; y  xi  f .y/  f .x/ for all y 2 Xg: In Sect. 14.4 we will prove the following result. Theorem 14.1. Let ı 2 .0; 1, 0 < 1 < 2 , M > 0, 2 1 1 0 D 2ı.2M C 1/ 1 1 ; T0 > 4M 1 0

(14.1)

and let W Œ0; T0  ! R1 be a Lebesgue measurable function such that

1  .t/  2 for all t 2 Œ0; T0 :

(14.2)

Assume that x 2 W 1;1 .0; T0 I X/, x.0/ 2 dom.f / \ B.0; M/

(14.3)

and that for almost every t 2 Œ0; T0 , x.t/ 2 dom.f /

(14.4)

B.x0 .t/; ı/ \ . .t/@f .x.t/// 6D ;:

(14.5)

and

Then minff .x.t// W t 2 Œ0; T0 g  inf.f I B.0; M// C 0 : In Theorem 14.1 ı is the computational error. According to this result we can find a point  2 X such that f ./  inf.f I B.0; M// C c1 ı

228

14 Continuous Subgradient Method

during a period of time c2 ı 1 , where c1 ; c2 > 0 are constants depending only on 1 ; M.

14.3 An Auxiliary Result Let V  X be an open convex set and g W V ! R1 be a convex locally Lipschitzian function. Let T > 0, x0 2 X and let u W Œ0; T ! X be a Bochner integrable function. Set Z x.t/ D x0 C

t 0

u.s/ds; t 2 Œ0; T:

Then x W Œ0; T ! X is differentiable and x0 .t/ D u.t/ for almost every t 2 Œ0; T. Assume that x.t/ 2 V for all t 2 Œ0; T: We claim that the restriction of g to the set fx.t/ W t 2 Œ0; Tg is Lipschitzian. Indeed, since the set fx.t/ W t 2 Œ0; Tg is compact, the closure of its convex hull C is both compact and convex, and so the restriction of g to C is Lipschitzian. Hence the function .g  x/.t/ WD g.x.t//, t 2 Œ0; T, is absolutely continuous. It follows that for almost every t 2 Œ0; T, both the derivatives x0 .t/ and .g  x/0 .t/ exist: x0 .t/ D lim h1 Œx.t C h/  x.t/;

(14.6)

.g  x/0 .t/ D lim h1 Œg.x.t C h//  g.x.t//:

(14.7)

h!0 h!0

We continue with the following fact (see Proposition 8.3 of [101]). Proposition 14.2. Assume that t 2 Œ0; T and that both the derivatives x0 .t/ and .g  x/0 .t/ exist. Then .g  x/0 .t/ D lim h1 Œg.x.t/ C hx0 .t//  g.x.t//: h!0

(14.8)

Proof. There exist a neighborhood U of x.t/ in X and a constant L > 0 such that jg.z1 /  g.z2 /j  Ljjz1  z2 jj for all z1 ; z2 2 U :

(14.9)

Let  > 0 be given. In view of (14.6), there exists ı > 0 such that x.t C h/; x.t/ C hx0 .t/ 2 U for each h 2 Œı; ı \ Œt; T  t;

(14.10)

14.4 Proof of Theorem 14.1

229

and such that for each h 2 Œ.ı; ı/ n f0g \ Œt; T  t, kx.t C h/  x.t/  hx0 .t/k < jhj:

(14.11)

h 2 Œ.ı; ı/ n f0g \ Œt; T  t:

(14.12)

Let

It follows from (14.10), (14.9), (14.11), and (14.12) that jg.x.t C h//  g.x.t/ C hx0 .t//j  Lkx.t C h/  x.t/  hx0 .t/k < Ljhj:

(14.13)

Clearly, Œg.x.t C h//  g.x.t//h1 D Œg.x.t C h//  g.x.t/ C hx0 .t//h1 CŒg.x.t/ C hx0 .t//  g.x.t//h1 :

(14.14)

Relations (14.13) and (14.14) imply that jŒg.x.t C h//  g.x.t//h1  Œg.x.t/ C hx0 .t//  g.x.t//h1 j  jg.x.t C h//  g.x.t/ C hx0 .t//jjh1 j  L:

(14.15)

Since  is an arbitrary positive number, (14.7) and (14.15) imply (14.8). Corollary 14.3. Let z 2 X and g.y/ D kz  yk2 for all y 2 X. Then for almost every t 2 Œ0; T, the derivative .g  x/0 .t/ exists and .g  x/0 .t/ D 2hx0 .t/; x.t/  zi:

14.4 Proof of Theorem 14.1 Assume that the theorem does not hold. Then there exists z 2 B.0; M/

(14.16)

f .x.t// > f .z/ C 0 for all t 2 Œ0; T0 :

(14.17)

.t/ D kz  x.t/k2 ; t 2 Œ0; T0 :

(14.18)

such that

Set

In view of Corollary 14.3, for a. e. t 2 Œ0; T0 , there exist derivatives x0 .t/, 0 .t/ and

230

14 Continuous Subgradient Method

0 .t/ D 2hx0 .t/; x.t/  zi:

(14.19)

By (14.5), for a. e. t 2 Œ0; T0 , there exist .t/ 2 @f .x.t//

(14.20)

kx0 .t/ C .t/.t/k  ı:

(14.21)

such that

It follows from (14.19) that for almost every t 2 Œ0; T0 , 0 .t/ D 2hx.t/  z; x0 .t/i D 2hx.t/  z;  .t/.t/i C 2hx.t/  z; x0 .t/ C .t/.t/i:

(14.22)

In view of (14.21), for almost every t 2 Œ0; T0 , jhz  x.t/; x0 .t/ C .t/.t/ij  ıkz  x.t/k:

(14.23)

By (14.17) and (14.20), for almost every t 2 Œ0; T0 , hz  x.t/; .t/i  f .x.t// C f .z/  0 :

(14.24)

It follows from (14.2), (14.22), (14.23), and (14.24) that for almost every t 2 Œ0; T0 , 0 .t/  20 .t/ C 2ıkz  x.t/k  20 1 C 2ıkz  x.t/k:

(14.25)

Relations (14.3), (14.16), and (14.18) imply that .0/  4M 2 : We show that for all t 2 Œ0; T0 , kz  x.t/k D .t/1=2  2M C 1: Assume the contrary. Then there exists  2 .0; T

(14.26)

14.5 Continuous Subgradient Projection Method

231

such that kz  x.t/k < 2M C 1; t 2 Œ0; /;

(14.27)

kz  x. /k D 2M C 1:

(14.28)

By (14.1) and (14.25)–(14.28), .2M C 1/2  4M 2  kz  x. /k2  kz  x.0/k2 Z  D ./  .0/ D 0 .t/dt Z  0

0



.20 1 C 2ıkz  x.t/k/dt

 .20 1 C 2ı.2M C 1//   0 1 ; a contradiction. The contradiction we have reached proves that kz  x.t/k  2M C 1; t 2 Œ0; T0 :

(14.29)

By (14.1), (14.25), and (14.29), for almost every t 2 Œ0; T0 , 0 .t/  20 1 C 2ı.2M C 1/  0 1 :

(14.30)

It follows from (14.26) and (14.30) that Z

2

4M  .0/  .T0 / D 

T0 0

0 .t/dt  T0 0 1 ;

T0  4M 2 . 1 0 /1 : This contradicts the choice of T0 (see (4.1)). The contradiction we have reached proves Theorem 14.1.

14.5 Continuous Subgradient Projection Method We use the notation and definitions introduced in Sect. 14.2. Let C be a nonempty, convex, and closed set in the Hilbert space X, U be an open and convex subset of X such that CU and f W U ! R1 be a convex locally Lipschitzian function.

232

14 Continuous Subgradient Method

Let x 2 U and  2 X. Set f 0 .x; / D lim t1 Œf .x C t/  f .x/;

(14.31)

@f .xI / D fl 2 @f .x/ W hl; i D f 0 .x; /g:

(14.32)

t!0C

It is a well-known fact of the convex analysis that @f .xI / 6D ;: Let M > 1; L > 0 and assume that C  B.0; M  1/; fy 2 X W d.y; C/  1g  U;

(14.33)

jf .v1 /  f .v2 /j  Lkv1  v2 k for all v1 ; v2 2 B.0; M C 1/ \ U:

(14.34)

We will prove the following result. Theorem 14.4. Let ı 2 .0; 1, 0 < 1 < 2 ; 1  1;

(14.35)

0 D 2ı.10M C 2 .L C 1// 1 1 ;

(14.36)

T0 > ı 1 .10M C 2 .L C 1//1 Œ.2 1 /1 4M 2 C L.2M C 2/ 1

(14.37)

and let

1   2 :

(14.38)

d.x.0/; C/  ı

(14.39)

Assume that x 2 W 1;1 .0; T0 I X/,

and that for almost every t 2 Œ0; T0 , there exists .t/ 2 X such that B..t/; ı/ \ @f .x.t/I x0 .t// 6D ;; PC .x.t/  .t// 2 B.x.t/ C x0 .t/; ı/:

(14.40) (14.41)

Then minff .x.t// W t 2 Œ0; T0 g  inf.f I C/ C 0 :

(14.42)

14.5 Continuous Subgradient Projection Method

233

Proof. Assume that (14.42) does not hold. Then there exists z2C

(14.43)

f .x.t// > f .z/ C 0 ; t 2 Œ0; T0 :

(14.44)

such that

For almost every t 2 Œ0; T0  set .t/ D x.t/ C x0 .t/:

(14.45)

It is clear that W Œ0; T is a Bochner integrable function. In view of (14.41) and (14.45), for almost every t 2 Œ0; T0 , B. .t/; ı/ \ C 6D ;:

(14.46)

Cı D fx 2 X W d.x; C/  ıg:

(14.47)

Define

Clearly, Cı is a convex closed set, for each x 2 Cı , B.x; ı/ \ C 6D ;

(14.48)

.t/ 2 Cı for almost every t 2 Œ0; T0 :

(14.49)

and in view of (14.46),

Evidently, the function es .s/, s 2 Œ0; T0  is Bochner integrable. We claim that for all t 2 Œ0; T0 , x.t/ D et x.0/ C et

Z

t 0

es .s/ds:

Clearly, (14.50) holds for t D 0. For every t 2 .0; T0  we have Z

t

Z e .s/ds D

t

s

0

0

Z D

0

t

es .x.s/ C x0 .s//ds .es x.s//0 ds D et x.t/  x.0/:

This implies (14.50) for all t 2 Œ0; T0 .

(14.50)

234

14 Continuous Subgradient Method

By (14.50), for all t 2 Œ0; T0 , t

t

t 1 t

x.t/ D e x.0/ C .1  e /.1  e / e D et x.0/ C .1  et /

Z

t 0

Z

t 0

es .s/ds

es .et  1/1 .s/ds:

In view of (14.49), for all t 2 Œ0; T0 , Z t es .et  1/1 .s/ds 2 Cı :

(14.51)

(14.52)

0

Relations (14.39), (14.51), and (14.52) imply that x.t/ 2 Cı for all t 2 Œ0; T0 :

(14.53)

It follows from (14.48) and (14.53) that for every t 2 Œ0; T0 , there exists xO .t/ 2 C

(14.54)

kx.t/  xO .t/k  ı:

(14.55)

such that

By (14.55) and Lemma 2.2, for almost every t 2 Œ0; T0 , hOx.t/  PC .x.t/  .t//; x.t/  .t/  PC .x.t/  .t//i  0:

(14.56)

Inequality (14.56) implies that for almost every t 2 Œ0; T0 ; hx.t/  PC .x.t/  .t//; x.t/  .t/  PC .x.t/  .t//i  hx.t/  xO .t/; x.t/  .t/  PC .x.t/  .t//i:

(14.57)

It follows from (14.43) and Lemma 2.2 that hz  PC .x.t/  .t//; x.t/  .t/  PC .x.t/  .t//i  0:

(14.58)

In view of (14.32) and (14.40), for almost every t 2 Œ0; T0  there exists O 2 @f .x.t/I x0 .t// .t/

(14.59)

O x0 .t/i; f 0 .x.t/; x0 .t// D h.t/;

(14.60)

O k.t/  .t/k  ı:

(14.61)

such that

14.5 Continuous Subgradient Projection Method

235

In view of (14.32) and (14.59), for almost every t 2 Œ0; T0 , O f .z/  f .x.t// C h.t/; z  x.t/i  0 and O O x0 .t/i: f .x.t//  f .z/  h.t/; x.t/  z C x0 .t/i  h.t/;

(14.62)

By (14.41), for almost every t 2 Œ0; T0 , k.x.t/ C x0 .t//  PC .x.t/  .t//k  ı:

(14.63)

Relations (14.45) and (14.63) imply that for almost every t 2 Œ0; T0 , hz  x.t/  x0 .t/; x.t/  .t/  x.t/  x0 .t/i D hz  .t/; x.t/  .t/  .t/i D hz  .t/; x.t/  .t/  PC .x.t/  .t//i Chz  .t/; PC .x.t/  .t//  .t/i  hz  .t/; x.t/  .t/  PC .x.t/  .t//i Cıkz  .t/k:

(14.64)

In view of (14.33) and (14.53), for all t 2 Œ0; T0 , kx.t/k  M:

(14.65)

It follows from (14.33), (14.41), (14.45), and (14.65) that for almost every t 2 Œ0; T0 , k .t/k D kx.t/ C x0 .t/k  M;

(14.66)

kx0 .t/k  2M:

(14.67)

By (14.33), (14.43), (14.64), and (14.66), hz  x.t/  x0 .t/;  .t/  x0 .t/i  hz  .t/; x.t/  .t/  PC .x.t/  .t//i C 2Mı:

(14.68)

By (14.33), (14.34), (14.38), (14.45), (14.53), (14.58), (14.59), (14.61), (14.63), and (14.65), hz  .t/; x.t/  .t/  PC .x.t/  .t//i  hz  PC .x.t/  .t//; x.t/  .t/  PC .x.t/  .t//i Cıkx.t/  .t/  PC .x.t/  .t//k  ı.2M C 2 .L C 1//: (14.69)

236

14 Continuous Subgradient Method

In view of (14.68) and (14.69), for almost every t 2 Œ0; T0 , hx.t/ C x0 .t/  z; .t/ C x0 .t/i  ı.4M C 2 .L C 1//:

(14.70)

It follows from (14.45), (14.61), (14.62), and (14.66) that for almost every t 2 Œ0; T0 , f .x.t//  f .z/ O  .t/; x.t/  z C x0 .t/i  h.t/; x.t/  z C x0 .t/i C h.t/ O h.t/; x0 .t/i C h.t/  .t/; x0 .t/i  h.t/; x.t/  z C x0 .t/i  h.t/; x0 .t/i C 4Mı:

(14.71)

Relations (14.70) and (14.71) imply that for almost every t 2 Œ0; T0 , f .x.t//  f .z/  1 hx0 .t/; x.t/ C x0 .t/  zi C 1 .4M C 2 .L C 1//ı h.t/; x0 .t/i C 4Mı:

(14.72)

By (14.44) and (14.72), for almost every t 2 Œ0; T0 , 0 < f .x.t//  f .z/   1 kx0 .t/k2  1 hx0 .t/; x.t/  zi h.t/; x0 .t/i C ı.8M C 2 .L C 1// 1 1

(14.73)

and

1 kx0 .t/k2 C 1 hx0 .t/; x.t/  zi Ch.t/; x0 .t/i C f .x.t//  f .z/  ı.8M C 2 .L C 1// 1 1 :

(14.74)

In view of (14.61), (14.67), and (14.74),

1 kx0 .t/k2 C 1 hx0 .t/; x.t/  zi O Ch.t/; x0 .t/i C f .x.t//  f .z/  ı.10M C 2 .L C 1// 1 1 :

(14.75)

It follows from (14.60), (14.75), and Corollary 14.3 that for almost every t 2 Œ0; T0 , .2 /1 .d=dt/.kx.t/  zk2 / C f 0 .x.t/; x0 .t// Cf .x.t//  f .z/ C 1 kx0 .t/k2  ı.10M C 2 .L C 1// 1 1 :

14.5 Continuous Subgradient Projection Method

237

Using Proposition 14.2, the equality f 0 .x.t/; x0 .t// D .f ı x/0 .t/ and integrating the inequality above over the interval Œ0; t, we obtain that for all t 2 Œ0; T0 , .2 /1 kx.t/  zk2  .2 /1 kx.0/  zk2 Z t Cf .x.t//  f .x.0// C .f .x.s//  f .z//ds  ıt.10M C 2 .L C 1// 1 1 : 0

(14.76) Relations (14.33), (14.34), (14.48), and (14.53) imply that for all t 2 Œ0; T0 , f .x.t//  inf.f I C/  ıL:

(14.77)

By (14.36), (14.38), (14.73), (14.76), and (14.77), for all t 2 Œ0; T0 , .2 2 /1 kx.t/  zk2  .2 1 /1 kx.0/  zk2 C inf.f I C/  f .x.0//  ıL  ıt.10M C 2 .L C

1// 1 1

Z

t

 0

.f .x.s//  f .z//ds

1  ıt.10M C 2 .L C 1// 1 1  0 t D ıt.10M C 2 .L C 1// 1 :

The relation above with t D T0 implies that .2 1 /1 kx.0/  zk2 C inf.f I C/  f .x.0//  ıL  ıT0 .10M C 2 .L C 1// 1 1 :

(14.78)

In view of (14.33), (14.34), (14.48), and (14.53), f .x.0//  sup.f I C/ C ıL:

(14.79)

Relations (14.33), (14.34), and (14.79) imply that inf.f I C/  f .x.0//  inf.f I C/  sup.f I C/  ıL  L.2M C 1/: It follows from (14.33), (14.43), (14.65), and (14.78) that .2 1 /1 4M 2  L.2M C 2/  ıT0 .10M C 2 .L C 1// 1 1 ; ıT0  ..2 1 /1 4M 2 C L.2M C 2//.10M C 2 .L C 1//1 1 :

(14.80)

238

14 Continuous Subgradient Method

This contradicts (14.37). The contradiction we have reached completes the proof of Theorem 14.4. In Theorem 14.4 ı is the computational error. According to this result we obtain a point  2 Cı (see (14.47), (14.53)) such that f ./  inf.f I C/ C c1 ı [see (14.36), (14.42)], during a period of time c2 ı 1 [see (14.37)], where c1 ; c2 > 0 are constants depending only on 1 ; 2 ; L; M.

Chapter 15

Penalty Methods

In this chapter we use the penalty approach in order to study constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. Since we consider optimization problems in general Banach spaces, not necessarily finite-dimensional, the existence of solutions of original constrained problems and corresponding penalized unconstrained problems is not guaranteed. By this reason we deal with approximate solutions and with an approximate exact penalty property which contains the classical exact penalty property as a particular case. In our recent research we established the approximate exact penalty property for a large class of inequality-constrained minimization problems. In this chapter we improve this result and obtain an estimation of the exact penalty.

15.1 An Estimation of Exact Penalty in Constrained Optimization Penalty methods are an important and useful tool in constrained optimization. See, for example, [25, 33, 43, 45, 49, 57, 80, 85, 117, 121] and the references mentioned there. In this chapter we use the penalty approach in order to study constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. The notion of exact penalization was introduced by Eremin [48] and Zangwill [114] for use in the development of algorithms for nonlinear constrained optimization. Since that time, exact penalty functions have continued to play an important role in the theory of mathematical programming. © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_15

239

240

15 Penalty Methods

In our recent research, which was summarized in [121] we established the approximate exact penalty property for a large class of inequality-constrained minimization problems. This approximate exact penalty property can be used for approximate solutions and contains the classical exact penalty property as a particular case. In this chapter we obtain an estimation of the exact penalty. We use the convention that   1 D 1 for all  2 .0; 1/,  C 1 D 1 and maxf; 1g D 1 for every real number  and that supremum over empty set is 1. For every real number  put C D maxf; 0g. We use the following notation and definitions. Let .X; k  k/ be a Banach space. For every point x 2 X and every positive number r > 0 put B.x; r/ D fy 2 X W kx  yk  rg: For every function f W X ! R1 [ f1g and every nonempty set A  X define dom.f / D fx 2 X W f .x/ < 1g; inf.f / D infff .z/ W z 2 Xg and inf.f I A/ D infff .z/ W z 2 Ag: For every point x 2 X and every nonempty set B  X define d.x; B/ D inffkx  yk W y 2 Bg:

(15.1)

Let n  1 be an integer. For every  2 .0; 1/ denote by ˝ the set of all vectors D . 1 ; : : : ; n / 2 Rn such that   minf i W i D 1; : : : ; ng and maxf i W i D 1; : : : ; ng D 1:

(15.2)

Let gi W X ! R1 [ f1g, i D 1; : : : ; n be convex lower semicontinuous functions and c D .c1 ; : : : ; cn / 2 Rn . Define A D fx 2 X W gi .x/  ci for all i D 1; : : : ; ng:

(15.3)

Let f W X ! R1 [ f1g be a bounded from below lower semicontinuous function which satisfies the following growth condition lim f .x/ D 1:

kxk!1

(15.4)

We suppose that there exists a point xQ 2 X such that gj .Qx/ < cj for all j D 1; : : : ; n and f .Qx/ < 1:

(15.5)

15.1 An Estimation of Exact Penalty in Constrained Optimization

241

We consider the following constrained minimization problem f .x/ ! min subject to x 2 A:

(P)

By (15.5), A 6D ; and inf.f I A/ < 1. For every D . 1 ; : : : ; n / 2 .0; 1/n define .z/

D f .z/ C

n X

i maxfgi .z/  ci ; 0g; z 2 X:

(15.6)

iD1

It is clear that for every vector 2 .0; 1/n the function W X ! R1 [ f1g is bounded from below and lower semicontinuous and satisfies inf. / < 1. We associate with problem (P) the corresponding family of unconstrained minimization problems .z/

! min; z 2 X

(P )

where 2 .0; 1/n . Assume that there exists a function h W X  dom.f / ! R1 [ f1g such that the following assumptions hold: (A1) (A2) (A3)

h.z; y/ is finite for every pair of points y; z 2 dom.f / and h.y; y/ D 0 for every point y 2 dom.f /. For every point y 2 dom.f / the function h.; y/ ! R1 [ f1g is convex. For every point z 2 dom.f / and every positive number r supfh.z; y/ W y 2 dom.f / \ B.0; r/g < 1:

(A4)

For every positive number M there is M1 > 0 such that for every point y 2 X satisfying f .y/  M there exists a neighborhood V of y in X such that if z 2 V, then f .z/  f .y/  M1 h.z; y/:

Remark 15.1. Note that if the function f is convex, then assumptions (A1)–(A4) hold with h.z; y/ D f .z/  f .y/, z 2 X, y 2 dom.f /. In this case M1 D 1 for all M > 0. If the function f is finite-valued and Lipschitzian on all bounded subsets of X, then assumptions (A1)–(A4) hold with h.z; y/ D kz  yk for all z; y 2 X. Let  2 .0; 1/. The main result of [118] (Theorem 15.2 stated below) imply that if  is sufficiently large, then any solution of problem .P / with 2 ˝ is a solution of problem .P/. Note that if the space X is infinite-dimensional, then the existence

242

15 Penalty Methods

of solutions of problems .P / and .P/ is not guaranteed. In this case Theorem 15.2 implies that for each  > 0 there exists ı./ > 0 which depends only on  such that the following property holds: If   0 , 2 ˝ and if x is a ı-approximate solution of .P /, then there exists an -approximate solution y of .P/ such that ky  xk  . Here 0 is a positive constant which does not depend on . It should be mentioned that we deal with penalty functions whose penalty parameters for constraints g1 ; : : : ; gn are  1 ; : : : ;  n respectively, where  > 0 and . 1 ; : : : ; n / 2 ˝ for a given  2 .0; 1/. Note that the vector .1; 1; : : : ; 1/ 2 ˝ for any  2 .0; 1/. Therefore our results also includes the case 1 D    D n D 1 where one single parameter  is used for all constraints. Note that sometimes it is an advantage from numerical consideration to use penalty coefficients  1 ; : : : ;  n with different parameters i , i D 1; : : : ; n. For example, in the case when some of the constrained functions are very “small” and some of the constraint functions are very “large.” The next theorem is the main result of [118]. Theorem 15.2. Let  2 .0; 1/. Then there exists a positive number 0 such that for each  > 0 there exists ı 2 .0; / such that the following assertion holds: If 2 ˝ ,   0 and if x 2 X satisfies  .x/

 inf.

 /

C ı;

then there exists y 2 A such that ky  xk   and f .y/  inf.f I A/ C : Note that Theorem 15.2 is just an existence result and it does not provide any estimation of the constant 0 . In this chapter we prove the main result of [119] which improves Theorem 15.2 and provides an estimation of the exact penalty 0 . In view of (15.4) and (15.5), there exists a positive number M such that if y 2 X satisfies f .y/  jf .Qx/j C 1; then kyk < M:

(15.7)

By (15.7), we have kQxk < M:

(15.8)

In view of (A4), there exists a positive number M1 such that the following property holds: (P1)

for every point y 2 X satisfying f .y/  jf .Qx/jC1 there exists a neighborhood V of y in X such that f .z/  f .y/  M1 h.z; y/ for all z 2 V.

15.2 Proof of Theorem 15.4

243

By (15.4), (15.5), and assumption (A3), there exists a positive number M2 such that supfh.Qx; z/ W z 2 X and f .z/  f .Qx/ C 1g  M2 : Remark 15.3. If the function f is convex, then by Remark 15.1, we choose h.z; y/ D f .z/  f .y/ for all z 2 X and all y 2 dom.f / with M1 D 1 for all M > 0 and then supfh.Qx; z/ W z 2 X and f .z/  f .Qx/ C 1g  supff .Qx/  f .z/ W z 2 X and f .z/  f .Qx/ C 1g D f .Qx/  inf.f /: Thus in this case M2 can be any positive number such that M2  f .Qx/  inf.f /. If the function f is finite-valued and Lipschitzian on bounded subsets of X, then by Remark 15.1, we choose h.z; y/ D kz  yk for all z; y 2 X and M1 is a Lipschitz constant of the restriction of f to B.0; M/. In this case supfh.Qx; z/ W z 2 X and f .z/  f .Qx/ C 1g  supfkQx  zk W z 2 B.0; M/g  2M and M2 D M. Let  2 .0; 1/. Fix a number 0 > 1 such that 

n X

2 2 .ci  gi .Qx// > maxf21 0 M1 M2 ; 80 M g:

(15.9)

iD1

We will prove the following result obtained in [119]. Theorem 15.4. For each  2 .0; 1/, each 2 ˝ , each   0 and each x 2 X which satisfies  .x/

 inf.

 /

C .20 /1 

there exists y 2 A such that ky  xk   and f .y/ 

 .x/

 inf.f I A/ C :

15.2 Proof of Theorem 15.4 Assume that the theorem does not hold. Then there exist  2 .0; 1/; D . 1 ; : : : ; n / 2 ˝ ;

(15.10)

  0 and xN 2 X

(15.11)

244

15 Penalty Methods

such that x/  .N

 inf.

C 21 1 0

 /

(15.12)

and  .y/

fy 2 B.Nx; / \ A W



x/g  .N

D ;:

(15.13)

By (15.12) and Ekeland’s variational principle [50] (see Theorem 15.19), there exists a point yN 2 X such that y/  .N



x/;  .N

(15.14)

1

kNy  xN k  2 

(15.15)

C 1 N k for all z 2 X: 0 kz  y

(15.16)

and y/  .N



 .z/

In view of (15.13)–(15.15), yN 62 A:

(15.17)

I1 D fi 2 f1; : : : ; ng W gi .Ny/ > ci g;

(15.18)

Define

I2 D fi 2 f1; : : : ; ng W gi .Ny/ D ci g; I3 D fi 2 f1; : : : ; ng W gi .Ny/ < ci g: By (15.17) and (15.18), we have I1 6D ;:

(15.19)

Relations from (15.3), (15.6), (15.10), (15.11), (15.12), (15.14), (15.18), and (15.19) imply that infff .z/ W z 2 Ag D inff 

 .z/

x/  .N X

W z 2 Ag  inf.

1

y/  .N

 /

 1 D f .Ny/

 i .gi .Ny/  ci /  1:

(15.20)

f .Ny/  infff .z/ W z 2 Ag C 1  f .Qx/ C 1:

(15.21)

C

i2I1

By (15.20), (15.18), and (15.5),

15.2 Proof of Theorem 15.4

245

In view of (15.21) and (15.7), kNyk < M:

(15.22)

Property (P1), (15.21), and (15.22) imply that there exists an open neighborhood V of the point yN in X such that V  B.0; M/; f .z/  f .Ny/  M1 h.z; yN / for each z 2 V:

(15.23) (15.24)

Since the functions gi ; i D 1; : : : ; n are lower semicontinuous it follows from (15.18) that there exists a positive number r < 1 such that for every point y 2 B.Ny; r/, gi .y/ > ci for each i 2 I1 :

(15.25)

By (15.21), (15.5), (15.14), (15.12), (15.25), and (15.16), for every point z 2 B.Ny; r/ \ dom.f /, we have X

 i .gi .z/  ci / C

i2I1



X

X i2I2 [I3

 i .gi .Ny/  ci / 

X

 i maxfgi .Ny/  ci ; 0g

i2I2 [I3

i2I1

D

 i maxfgi .z/  ci ; 0g

 .z/



y/  .N

 f .z/ C f .Ny/  1 y  zk  f .z/ C f .Ny/: 0 kN

Combined with (15.11) this relation implies that for every point z 2 B.Ny; r/, X

i gi .z/ C

i2I1



X

X i2I2 [I3

i gi .Ny/ 

i maxfgi .z/  ci ; 0g X

i maxfgi .Ny/  ci ; 0g

i2I2 [I3

i2I1

C1 .f .z/  f .Ny//  2 y  zk: 0 kN By this inequality, (15.23) and (15.24), for every point z 2 B.Ny; r/ \ V, X

X

i gi .z/ C

i2I1

i maxfgi .z/  ci ; 0g

i2I2 [I3

C1 M1 h.z; yN / C 2 Nk 0 kz  y 

X i2I1

i gi .Ny/ C

X i2I2 [I3

i maxfgi .Ny/  ci ; 0g:

(15.26)

246

15 Penalty Methods

In view of (A2), the function X

i gi .z/ C

X

i maxfgi .z/  ci ; 0g

i2I2 [I3

i2I1

C1 M1 h.z; yN / C 2 N k; z 2 X 0 kz  y is convex. Combined with the equality h.Ny; yN / D 0 [see (A1)] this implies that (15.26) holds true for every point z 2 X. Since relation (15.26) is valid for z D xQ relations (15.5), (15.10), (15.2), (15.11), (15.18), and (15.19) imply that X

i gi .Qx/ C 1 M1 h.Qx; yN / C 2 x  yN k 0 kQ

i2I1



X

i gi .Ny/ >

i2I1

X

i ci :

i2I1

Combined with (15.21), (15.5), (15.8), (15.22), (15.10), (15.2), assumption (A1) and the choice of M2 (see Sect. 15.1) this implies that 2 1 x; z/ W z 2 X and f .z/  f .Qx/ C 1g 42 0 M C 0 M1 supfh.Q X 2 1  2 x; yN /C /  i .ci  gi .Qx// 0 4M C 0 M1 .h.Q i2I1



n X

.ci  gi .Qx//

iD1

and 

n X

2 1 .ci  gi .Qx//  42 0 M C 0 M1 M2 :

iD1

This contradicts (15.9). The contradiction we have reached proves Theorem 15.4.

15.3 Infinite-Dimensional Inequality-Constrained Minimization Problems In this section we use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. For these problems, a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce penalty functions, prove the exact penalty property, and

15.3 Infinite-Dimensional Inequality-Constrained Minimization Problems

247

obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems. Let X be a vector space, X 0 be the set of all linear functionals on X and let Y be a vector space ordered by a convex cone YC such that YC \ .YC / D f0g; ˛YC  YC for all ˛  0; YC C YC  YC : We say that y1 ; y2 2 Y satisfy y1  y2 if and only if y2  y1 2 YC . We add to the space Y the largest element 1 and suppose that y C 1 D 1 for all y 2 Y [ f1g and that ˛  1 D 1 for all ˛ > 0. For a mapping F W X ! Y [ f1g we set dom.F/ D fx 2 X W F.x/ < 1g: A mapping F W X ! Y [ f1g is called convex if for all x1 ; x2 2 X and all ˛ 2 .0; 1/, F.˛x1 C .1  ˛/x2 /  ˛F.x1 / C .1  ˛/F.x2 /: A function G W Y ! R1 is called increasing if G.y1 /  G.y2 / for all y1 ; y2 2 Y satisfying y1  y2 . Assume that p W X ! R1 [ f1g is a convex function. Recall that for xN 2 dom.p/, @p.Nx/ D fl 2 X 0 W l.x  xN /  p.x/  p.Nx/ for all x 2 Xg:

(15.27)

The set @p.Nx/ is a subdifferential of p at the point xN . Since we consider convex minimization problems we need to use the following two important facts of convex analysis (see Theorems 3.6.1 and 3.6.4 of [76] respectively). Proposition 15.5. Let p1 W X ! R1 [ f1g and p2 W X ! R1 be convex functions. Then for any xN 2 dom.p1 /, @.p1 C p2 /.Nx/ D @p1 .Nx/ C @p2 .Nx/: Proposition 15.6. Let F W X ! Y [ f1g be a convex mapping, G W Y ! R1 be an increasing convex function, G.1/ D 1 and let xN 2 dom.F/. Then @.G ı F/.Nx/ D [[email protected] ı F/.Nx/ W l 2 @G.F.Nx//g: In this paper we suppose that .X; k  k/ is a Banach space, .Y; k  k/ is a normed space and that .X  ; k  k / and .Y  ; k  k / are their dual spaces.

248

15 Penalty Methods

We also suppose that Y is ordered by a convex cone YC which is a closed subset of .Y; k  k/. For each function f W Z ! R1 [ f1g where the set Z is nonempty put inf.f / D infff .z/ W z 2 Zg and for each nonempty subset A  Z put inf.f I A/ D infff .z/ W z 2 Ag: For all y 2 Y put

.y/ D inffkzk W z 2 Y and z  yg:

(15.28)

It is not difficult to see that

.y/  0 for all y 2 Y;

.y/ D 0 if and only if y  0;

.y/  kyk for all y 2 Y;

(15.29)

.˛y/ D ˛ .y/ for all ˛ 2 Œ0; 1/ and all y 2 Y;

(15.30)

.y1 C y2 /  .y1 / C .y2 /

(15.31)

if y1  y2 ; then .y1 /  .y2 /:

(15.32)

for all y1 ; y2 2 Y

and

Set

.1/ D 1: The functional was used in [115, 116, 121] for the study of minimization problems with increasing objective functions. Here we use it in order to construct a penalty function. The following auxiliary result is proved in Sect. 15.4. Lemma 15.7. Let y 2 Y n .YC / and l 2 @ .y/. Then l.z/  0 for all z 2 YC ; l.y/ D .y/ and jjljj D 1: For each x 2 X, each y 2 Y, and each r > 0 set BX .x; r/ D fz 2 X W kx  zk  rg; BY .y; r/ D fz 2 Y W ky  zk  rg:

(15.33)

15.3 Infinite-Dimensional Inequality-Constrained Minimization Problems

249

A set E  Y is called ./-bounded from above if the following property holds: (P2)

there exists ME > 0 such that for each y 2 E there is z 2 BY .0; ME / for which z  y.

Let f W X ! R1 [ f1g be a bounded from below lower semicontinuous function which satisfies the following growth condition lim f .x/ D 1:

kxk!1

(15.34)

Assume that G W X ! Y [ f1g is a convex mapping and c 2 Y. Set A D fx 2 X W G.x/  cg:

(15.35)

We suppose that there exist MQ > 0, rQ > 0 and a nonempty set ˝  X such that G.x/  c and f .x/  MQ for all x 2 ˝

(15.36)

and that the following property holds: (P3)

for each h 2 Y satisfying khk D rQ there is xh 2 ˝ such that G.xh / C h  c. By (15.36) and (15.34), supfkxk W x 2 ˝g < 1:

Remark 15.8. Property (P3) is an infinite-dimensional version of Slater condition [58]. In particular, it holds if there exists xQ 2 A such that f .Qx/ < 1 and c  G.Qx/ is an interior point of YC in the normed space Y. In this case MQ is any positive constant satisfying MQ  f .Qx/ and ˝ D fQxg. We assume that G possesses the following two properties. (P4) (P5)

If fyi g1 iD1  Y, y 2 Y, limi!1 yi D y in .Y; k  k/ and G.y/ D 1, then the sequence fG.yi /g1 iD1 is not ./-bounded above. If fyi g1 iD1  Y, y 2 Y, limi!1 yi D y in .Y; k  k/ and G.y/ < 1, then for each given  > 0 and all sufficiently large natural numbers i there exists ui 2 Y such that G.yi /  ui and kui  G.y/j  :

Remark 15.9. Clearly, G possesses (P1) and (P2) if G.X/  Y and G is continuous. It is easy to see that G possesses (P4) and (P5) if Y D Rn , YC D fy 2 Rn W yi  0; i D 1; : : : ; ng, G D .g1 ; : : : ; gn / and the functions gi W X ! R1 [ f1g, i D 1; : : : ; n are lower semicontinuous. In general properties (P4) and (P5) are an infinite-dimensional version of the lower semicontinuity property.

250

15 Penalty Methods

We consider the following constrained minimization problem f .x/ ! min subject to x 2 A:

(P)

In view of (15.36) A 6D ; and inf.f I A/ < 1: For each > 0 define .z/

D f .z/ C .G.z/  c/; z 2 X:

(15.37)

The set , > 0 is our family of penalty functions. The following auxiliary result is proved in Sect. 15.4. Lemma 15.10. For each > 0 the function semicontinuous.



W X ! R1 [ f1g is lower

By (15.34) there is M1 > 0 such that kzk < M1 for each z 2 X satisfying f .z/  MQ C 1:

(15.38)

We use the following assumption. Assumption (A1) There is M0 > 0 such that for each x 2 X satisfying f .x/  MQ C 1 there is a neighborhood V of x in .X; k  k/ such that for each z 2 V, f .z/ is finite and jf .z/  f .x/j  M0 kx  zk:

(15.39)

Remark 15.11. Note that assumption (A1) is a form of a local Lipschitz property for f on the sublevel set f 1 ..1; MQ C 1/. The following theorem is the first main result of this section. Theorem 15.12. Assume that (A1) holds, M0 > 0 is as guaranteed by (A1) and let 0 > 1 satisfy 1 rQ > 4.2 0 C M0 0 /.supfkzk W z 2 ˝g C M1 /:

Then for each  2 .0; 1/, each   0 and each x 2 X which satisfies  .x/

 inf.

/

C .20 /1 

there exists y 2 A such that ky  xk   and f .y/  inf.

/

C   inf.f I A/ C :

(15.40)

15.3 Infinite-Dimensional Inequality-Constrained Minimization Problems

251

Corollary 15.13. Assume that (A1) holds, M0 > 0 is as guaranteed by (A1) and let 0 > 1 satisfy (15.40). Then for each   0 and each sequence fxi g1 iD1  X satisfying lim

i!1

 .xi /

D inf.

/

(15.41)

there exists a sequence fyi g1 iD1  A such that lim kyi  xi k D 0 and lim f .yi / D inf.f I A/:

i!1

i!1

(15.42)

Moreover, for each   0 inf.f I A/ D inf.

 /:

Corollary 15.14. Assume that (A1) holds, M0 > 0 is as guaranteed by (A1) and let 0 > 1 satisfy (15.40). Then if   0 and if x 2 X satisfies  .x/

D inf.

 /;

(15.43)

then x 2 A and f .x/ D

 .x/

D inf.

/

D inf.f I A/:

(15.44)

Theorem 15.12 is proved in Sect. 15.5. In our second main result of this section we do not assume (A1). Instead of it we assume that the function f is convex and that the mapping G is finite-valued. Theorem 15.15. Assume that G.X/  Y, the function f is convex and let 0 > 1 satisfy 1 Q rQ > 42 0 .supfkxk W x 2 ˝g C M1 / C 40 .M  inf.f //:

(15.45)

Then for each  > 0, each   0 and each x 2 X which satisfies  .x/

 inf.

/

C .20 /1 

there exists y 2 A such that ky  xk   and f .y/  inf.

/

C   inf.f I A/ C :

Corollary 15.16. Let the assumptions of Theorem 15.15 hold. Then for each   0 and each sequence fxi g1 iD1  X satisfying (15.41) there exists a sequence  A such that (15.42) holds. Moreover, for each   0 fyi g1 iD1 inf.f I A/ D inf.

 /:

252

15 Penalty Methods

Corollary 15.17. Let the assumptions of Theorem 15.15 hold. Then if   0 and if x 2 X satisfies (15.43), then x 2 A and (15.44) holds. Theorem 15.15 is proved in Sect. 15.6. Using our exact penalty results we obtain necessary and sufficient optimality conditions for constrained minimization problems (P) with the convex function f . Theorem 15.18. Assume that G.X/  Y, the function f is convex, 0 > 1 satisfies (15.45),   0 and xN 2 X. Then the following assertions are equivalent. 1. xN 2 A and f .Nx/ D inf.f I A/. 2.  .Nx/ D inf.  /. 3. There exist l0 2 @ .G.Nx/  c/; l1 2 @.l0 ı .G./  c//.Nx/ and l2 2 @f .Nx/ such that l1 C l2 D 0. Proof. The equivalence of assertions 1 and 2 follows from Corollaries 15.16 and 15.17. Therefore it is sufficient to show that assertions 2 and 3 are equivalent. It is clear that if at least one of assertions 2 and 3 holds, then xN 2 dom.f /. Therefore we may assume that xN 2 dom.f /. Clearly, the function  is convex. By Propositions 15.5 and 15.6, @

x/  .N

D @f .Nx/ C @. ı .G./  c//.Nx/ D @f .Nx/ C [[email protected] ı .G./  c//.Nx/ W l 2 @ .G.Nx/  c/g:

Now in order to complete the proof it is sufficient to note that assertion 2 holds if and only if 02@

x/:  .N

It should be mentioned that Theorem 15.18 is an infinite-dimensional version of the classical Karush–Kuhn–Tucker theorem [58]. Assume now that the assumptions of Theorem 15.18 hold and assertions 1, 2, and 3 hold. Let l0 , l1 , and l2 be as guaranteed by assertion 3. Then l0 .G.Nx/  c/ D .G.Nx/  x/ D 0 because G.Nx/  c. This is an infinite-dimensional version of the complementary slackness condition [58]. The results of this section were obtained in [123]. In the proof of Theorem 15.12 we use the following fundamental variational principle of Ekeland [50]. Theorem 15.19. Assume that .Z; / is a complete metric space and that W Z ! R1 [ f1g is a lower semicontinuous bounded from below function which is not identically 1. Let  > 0 and x0 2 Z be given such that .x0 /  .x/ C  for all x 2 Z:

15.4 Proofs of Auxiliary Results

253

Then for any  > 0 there is xN 2 Z such that .Nx/  .x0 /; .Nx; x0 /  ; .x/ C .=/.x; xN / > .Nx/ for all x 2 Z n fNxg:

15.4 Proofs of Auxiliary Results Proof of Lemma 15.7. It is not difficult to see that klk  1; l.y/ D .y/ > 0; l.z/  0 for all z 2 YC :

(15.46)

We will show that klk D 1. By (15.46) and (15.28), l.y/ D .y/ D inffkzk W z 2 Y and z  yg:

(15.47)

Let  2 .0; .y/=4/ [see (15.46)]. By (15.47) there exists z 2 Y such that z  y and kzk  .y/ C :

(15.48)

kzk > 0

(15.49)

.y/ D l.y/  l.z/:

(15.50)

By (15.46) and (15.28)

and

It follows from (15.50) and (15.48) that l.z/  kzk  : Together with (15.49), (15.48), and (15.47) this implies that klk  l.z/kzk1  .kzk  /kzk1  1  kzk1  1   .y/1 : Since  is any positive number satisfying  < .y/=4 we conclude that klk  1: Combined with (15.46) this implies that klk D 1. Lemma 15.7 is proved.

254

15 Penalty Methods

Proof of Lemma 15.10. It is sufficient to show that the function .G./  c/ W X ! R1 [ f1g is lower semicontinuous. Assume that y 2 Y, fyi g1 iD1  Y and lim kyi  yk D 0:

i!1

(15.51)

It is sufficient to show that lim inf .G.yi /  c/  .G.y/  c/: i!1

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that there exists lim .G.yi /  c/ < 1:

i!1

(15.52)

We may assume without loss of generality that .G.yi /  c/ is finite for all integers i  1. Let  > 0. By (15.28) for any integer i  1 there exists zi 2 Y such that zi  G.yi /  c; kzi k  .G.yi /  c/ C =4:

(15.53)

In view of (15.52) and (15.53) the sequence fkzi kg1 iD1 is bounded. Together with (15.53) this implies that the sequence fG.yi /  cg1 iD1 is ./-bounded from above [see (P2)]. It follows from (P4) and (15.53) that G.y/ < 1:

(15.54)

By (15.51), (15.54), and (P5) there exists a natural number i0 such that for each integer i  i0 there is ui 2 Y which satisfies G.yi /  ui and kui  G.y/k  =4:

(15.55)

In view of (15.55) and (15.53) for all integers i  i0 G.y/  c D .G.y/  ui / C ui  c  .G.y/  ui / C G.yi /  c  .G.y/  ui / C zi :

(15.56)

It follows from (15.55) and (15.53) that for all integers i  i0 kG.y/  ui C zi k  kG.y/  ui k C kzi k  =4 C .G.yi /  c/ C =4: By (15.56) and (15.57), for all integers i  i0 ,

.G.y/  c/  kG.y/  ui C zi k  .G.yi /  c/ C =2

(15.57)

15.5 Proof of Theorem 15.12

255

and

.G.y/  c/  lim .G.yi /  c/ C =2: i!1

Since  is any positive number we conclude that

.G.y/  c/  lim .G.yi /  c/: i!1

Lemma 15.10 is proved.

15.5 Proof of Theorem 15.12 We show that the following property holds: (P6)

For each  2 .0; 1/, each   0 and each x 2 X which satisfies  .x/

 inf.

/

C .20 /1 

there exists y 2 A for which  .y/

ky  xk   and



 .x/:

(It is easy to see that (P6) implies the validity of Theorem 15.2.) Assume the contrary. Then there exist  2 .0; 1/;   0 ; xN 2 X

(15.58)

such that x/  .N

 inf.

/

fy 2 BX .Nx; / \ A W

C 21 1 0 ;

 .y/



x/g  .N

(15.59) D ;:

(15.60)

It follows from (15.59), Lemma 15.10, and Theorem 15.19 that there is yN 2 X such that y/  .N

y/  .N



x/;  .N

(15.61)

kNy  xN k  21 ;

(15.62)

 .z/



C 1 N k for all z 2 X: 0 kz  y

(15.63)

By (15.60), (15.61), and (15.62), yN 62 A:

(15.64)

256

15 Penalty Methods

It follows from (15.37), (15.35), (15.59), and (15.61) that infff .z/ W z 2 Ag D inff 

 .z/

y/  .N

W z 2 Ag  inf.

/



x/  .N

1

 1  f .Ny/  1

and in view of (15.36), f .Ny/  infff .z/ W z 2 Ag C 1  MQ C 1:

(15.65)

By (15.65) and (15.38) kNyk < M1 :

(15.66)

In view of (A1) and (15.65) there exists a neighborhood V of yN in .X; k  k/ such that for each z 2 V, f .z/ is finite and jf .z/  f .Ny/j  M0 kz  yN k:

(15.67)

It follows from (15.61), (15.59), (15.67), (15.37), (15.63), and (15.58) that for each z2V  .G.z/  c/   .G.Ny/  c/ D

 .z/



y/  .N

 f .z/ C f .Ny/

 1 N k  f .z/ C f .Ny/  1 N k  M0 kz  yN k 0 kz  y 0 kz  y and 1

.G.z/  c/  .G.Ny/  c/  kz  yN k.2 0 C M0 0 /:

This implies that for each z 2 V 1 N k  .G.Ny/  c/:

.G.z/  c/ C .2 0 C M0 0 /kz  y

(15.68)

Clearly the function 1 N k; z 2 X

.z/ Q D .G.z/  c/ C .2 0 C M0 0 /kz  y

(15.69)

is convex. By (15.68) and (15.69) 0 2 @ .N Q y/:

(15.70)

By (15.70), (15.69), and Proposition 15.5 there is l0 2 @. .G./  c//.Ny/

(15.71)

1 kl0 k  .2 0 C M0 0 /:

(15.72)

such that

15.5 Proof of Theorem 15.12

257

It follows from (15.71) and Proposition 15.6 that there exists l1 2 @ .G.Ny/  c/

(15.73)

l0 2 @.l1 ı .G./  c//.Ny/:

(15.74)

such that

In view of (15.74) for each z 2 X l0 .z  yN /  l1 .G.z/  c/  l1 .G.Ny/  c/:

(15.75)

By (15.73), (15.64), and Lemma 15.7 kl1 k D 1; l1 .z/  0 for all z 2 YC ;

(15.76)

l1 .G.Ny/  c/ D .G.Ny/  c/: Let h 2 Y and khk D rQ :

(15.77)

By (15.77) and property (P3) there is xh 2 ˝

(15.78)

G.xh / C h  c:

(15.79)

such that

It follows from (15.78), (15.66), (15.72), (15.75), (15.76), and (15.79) that 1 2 1 yk/ .2 0 C M0 0 /.M1 C supfkxk W x 2 ˝g/  .0 C M0 0 /.kxh k C kN

 kl0 k .kxh k C kNyk/  l0 .xh  yN /  l1 .G.xh /  c/  l1 .G.Ny/  c/  l1 .G.xh /  c/  .G.Ny/  c/  l1 .G.xh /  c/  l1 .h/:

(15.80)

Since (15.80) holds for all h satisfying (15.77) we conclude using (15.76) that 1 .2 0 C M0 0 /.M1 C supfkxk W x 2 ˝g/

 supfl1 .h/ W h 2 Y and khk D rQ g D rQ kl1 k D rQ : This contradicts (15.40). The contradiction we have reached proves (P6) and Theorem 15.12 itself.

258

15 Penalty Methods

15.6 Proof of Theorem 15.15 We show that property (P6) (see Sect. 15.5) holds. (Note that (P6) implies the validity of Theorem 15.15). Assume the contrary. Then there exist  2 .0; 1/;   0 ; xN 2 X

(15.81)

such that (15.59) and (15.60) hold. It follows from (15.59), Lemma 15.10 and Ekeland’s variational principle [50] that there is yN 2 X such that (15.61)–(15.63) hold. By (15.60), (15.61) and (15.62), yN 62 A:

(15.82)

Arguing as in the proof of Theorem 15.12 we show that (15.37), (15.35), (15.59), (15.61), (15.36), and (15.38) imply that f .Ny/  MQ C 1; kNyk < M1 :

(15.83)

It follows from (15.37) and (15.63) that for each z 2 X f .z/ C  .G.z/  c/  f .Ny/   .G.Ny/  c/ D

 .z/



y/  .N

 1 Nk 0 kz  y

and 1 .f .z/  f .Ny// C .G.z/  c/  .G.Ny/  c/  1 1 N k: 0 kz  y This implies that for all z 2 X 1 f .z/ C .G.z/  c/ C 1 1 N k  1 f .Ny/ C .G.Ny/  c/: 0 kz  y

(15.84)

Put N k; z 2 X

.z/ Q D .G.z/  c/ C 1 f .z/ C 1 1 0 kz  y

(15.85)

In view of (15.85) the function Q is convex. By (15.84) and (15.85), 0 2 @ .N Q y/:

(15.86)

By (15.85), (15.86), (15.81), and Proposition 15.5 there exist l1 2 @. ı .G./  c//.Ny/; l2 2 @f .Ny/

(15.87)

15.6 Proof of Theorem 15.15

259

such that kl1 C 1 l2 k  2 0 :

(15.88)

It follows from (15.87), (15.30), (15.31), and Proposition 15.6 that there exists l0 2 @ .G.Ny/  c/

(15.89)

l1 2 @.l0 ı .G./  c//.Ny/:

(15.90)

such that

In view of (15.90) for each z 2 X l1 .z  yN /  l0 .G.z/  c/  l0 .G.Ny/  c/:

(15.91)

By (15.89), (15.82), and Lemma 15.7 kl0 k D 1; l0 .z/  0 for all z 2 YC ;

(15.92)

l0 .G.Ny/  c/ D .G.Ny/  c/: Let h 2 Y and khk D rQ :

(15.93)

By (15.93) and property (P3) there is xh 2 ˝

(15.94)

G.xh / C h  c:

(15.95)

such that

It follows from (15.94), (15.83), (15.88), (15.91), (15.87), (15.92), (15.36), and (15.95) that 2 yk/ 2 0 .M1 C supfkxk W x 2 ˝g/  0 .kxh k C kN

 kl1 C 1 l2 k .kxh k C kNyk/  .l1 C 1 l2 /.xh  yN / D l1 .xh  yN / C 1 l2 .xh  yN /  l0 .G.xh /  c/  l0 .G.Ny/  c/ C 1 .f .xh /  f .Ny//  l0 .G.xh /  c/  .G.Ny/  c/ C 1 .MQ  inf.f //  l0 .h/ C 1 .MQ  inf.f //

260

15 Penalty Methods

and in view of (15.81) 2 Q l0 .h/  1 0 .M  inf.f // C 0 .M1 C supfkxk W x 2 ˝g/:

(15.96)

Since the inequality above holds for all h satisfying (15.93) it follows from (15.92) and (15.96) that rQ D rQ kl0 k D supfl0 .h/ W h 2 Y and khk D rQ g 2 Q  1 0 .M  inf.f // C 0 .M1 C supfkxk W x 2 ˝g/:

This contradicts (15.45). The contradiction we have reached proves (P6) and Theorem 15.15 itself.

15.7 An Application Let X be a Hilbert space equipped with an inner product h; i which induces the complete norm k  k. We use the notation and definitions introduced in Sect. 15.1. Let n be a natural number, gi W X ! R1 [ f1g, i D 1; : : : ; n be convex lower semicontinuous functions and c D .c1 ; : : : ; cn / 2 Rn . Set A D fx 2 X W gi .x/  ci for all i D 1; : : : ; ng: Let f W X ! R1 [ f1g be a bounded from below lower semicontinuous function which satisfies the following growth condition lim f .x/ D 1:

kxk!1

We suppose that there is xQ 2 X such that gj .Qx/ < cj for all j D 1; : : : ; n and f .Qx/ < 1:

(15.97)

We consider the following constrained minimization problem f .x/ ! min subject to x 2 A:

(P)

For each vector D . 1 ; : : : ; n / 2 .0; 1/n define .z/

D f .z/ C

n X

i maxfgi .z/  ci ; 0g; z 2 X:

(15.98)

iD1

Clearly for each 2 .0; 1/n the function W X ! R1 [ f1g is bounded from below and lower semicontinuous and satisfies inf. / < 1.

15.7 An Application

261

We suppose that the function f is convex. By Remark 15.1, (A1)–(A4) hold with h.z; y/ D f .z/  f .y/, z 2 X, y 2 dom.f /. In this case M1 D 1 for all M > 0. There is M > 0 such that [see (15.7)] if y 2 X satisfies f .y/  jf .Qx/j C 1; then kyk < M:

(15.99)

Clearly, (P1) holds with M1 D 1. In view of Remark 15.3, the constant M2 can be any positive number such that M2  f .Qx/  inf.f /: We suppose that M2 is given. Let  2 .0; 1/ Choose 0 > 1 [see (15.9)] such that 

n X 2 2 .ci  gi .Qx// > maxf21 0 M2 ; 80 M g: iD1

By Theorem 15.4, the following property holds: (P7)

for each  2 .0; 1/, each 2 ˝ , each   0 and each x 2 X satisfies  .x/

 inf.

 /

C 21 1 0

there exists y 2 A such that ky  xk   and f .y/ 

 .x/

 inf.f I A/ C :

Property (P7) implies that for each 2 ˝ and each   0 ,  /

inf.

D inf.f I A/:

(15.100)

In order to obtain an approximate solution of problem (P) we apply the subgradient projection method, studied in Chap. 2, for the minimization of the function 0 , where is a fixed element of the set ˝ . We suppose that problem (P) has a solution x 2 A

(15.101)

f .x /  f .x/ for all x 2 A:

(15.102)

such that

By (15.100), (15.101), and (15.102), f .x / D

0 .x /

D inf.

0 /:

(15.103)

262

15 Penalty Methods

It follows from (15.97), (15.99), and (15.102) that kx k < M:

(15.104)

In view of (15.103) and (15.104), we consider the minimization of 0 on B.0; M/. We suppose that there exist an open convex set U  X and a number L > 0 such that B.0; M C 1/  U;

(15.105)

the functions f and gi ; i D 1; : : : ; n are finite-valued on U and that for all x; y 2 U and all i D 1; : : : ; n, jf .x/  f .y/j  Lkx  yk; jgi .x/  gi .y/j  Lkx  yk: In view of (15.98) and (15.106), the function x; y 2 U, j

0 .x/



0

(15.106)

is Lipschitzian on U and for all

0 .y/j

 jf .x/  f .y/j C n0 Lkx  yk  Lkx  yk.1 C 0 n/:

(15.107)

In this section we use the projection on the set B.0; M/ denoted by PB.0;M/ and defined for each z 2 X by PB.0;M/ .z/ D z if kzk  M; PB.0;M/ .z/ D Mkzk1 z if kzk > M: We apply the subgradient projection method, studied in Chap. 2, for the minimization problem of the function 0 on the set B.0; M/. For each ı > 0 set ˛.ı/ D 21 .2M C 1/2 .1 C L.0 n C 1//.8M C 2/1=2 ı 1=2 .1 C 0 n/1=2 Cı.1 C 0 n/.2M C 1/ C .8M C 2/1=2 .1 C L.0 n C 1//ı 1=2 .1 C 0 n/1=2 C.4M C 1/.1 C L.0 n C 1//.8M C 2/1=2 ı 1=2 .1 C 0 n/1=2 :

(15.108)

Let ı 2 .0; 1 be our computational error which satisfies ı.1 C n0 / < 1; 2˛.ı/0 < 1:

(15.109)

ı0 D ı.1 C n0 /

(15.110)

Set

and a D .2ı0 .4M C 1//1=2 .L.1 C 0 n/ C 1/1 : Let us describe our algorithm.

15.7 An Application

263

Subgradient Projection Algorithm Initialization: select an arbitrary x0 2 B.0; M/. Iterative step: given a current iteration vector xt 2 U calculate t 2 @

0 .xt /

C B.0; ı0 /

(15.111)

and the next iteration vector xtC1 2 X such that kxtC1  PB.0;M/ .xt  at /k  ı0 :

(15.112)

Let t  0 be an integer. Let us explain how one can calculate t satisfying (15.111). We find 0 2 X satisfying

0 2 @f .xt / C B.0; ı/: For every i D 1; : : : ; n, if gi .xt /  ci , then set i D 0 and if gi .xt / > ci , then we calculate

i 2 @gi .xt / C B.0; ı/: Set t D 0 C 0

n X

i :

iD1

It follows from the equality above, the choice of i , i D 0; : : : ; n, the subdifferential calculus in [84], (15.98) and (15.110) that B.t ; ı0 / \ @

0 .xt /

D B.t ; ı.1 C n0 // \ @

and (15.111) is true. By Theorem 2.6, applied to the function

0

.T C 1/

1

T X

0 ,

0 .xt /

6D ;

for each natural number T,

! xt 

0 .x /;

tD0

minf

0 .xt /

W t D 0; : : : ; Tg 

0 .x /

 21 .T C 1/1 .2M C 1/2 .L.1 C 0 n/ C 1/.2ı0 .4M C 1//1=2 Cı0 .2M C 1/ C21 .2ı0 .4M C 1//1=2 .L.1 C 0 n/ C 1/ Cı0 .4M C 1/.L.1 C 0 n/ C 1/.2ı0 .4M C 1//1=2 :

264

15 Penalty Methods

Now we can think about the best choice of T. It was explained in Chap. 2 that it should be at the same order as bı01 c D bı 1 .1 C n0 /1 c: Put T D bı01 c and obtain from (15.108) and (15.110) that 0

.T C 1/

1

T X

! 0 .x /;

xt 

tD0

minf

0 .xt /

W t D 0; : : : ; Tg 

0 .x /

 ˛.ı/

(15.113)

By (15.100), (15.101), (15.103), and (15.113),

0

.T C 1/1

T X

! xt

 inf.

0 /

C ˛.ı/:

0 .xt /

W t D 0; : : : ; Tg:

(15.114)

tD0

Let  2 f0; : : : ; Tg satisfy 0 .x /

D minf

In view of (15.101), (15.103), (15.106), and (15.113), 0 .x /

 inf.

0 /

C ˛.ı/:

(15.115)

It follows from (15.109), (15.114), (15.115), and property (P7) that there exist y0 ; y1 2 A such that   T   X   1 xt   2˛.ı/0 ; ky1  x k  2˛.ı/0 ; y0  .T C 1/   tD0

f .y0 / 

0

.T C 1/

1

T X

! xt

 inf.f I A/ C ˛.ı/;

tD0

f .y1 / 

0 .x /

 inf.f I A/ C ˛.ı/:

The analogous analysis can be also done for the mirror descent method.

Chapter 16

Newton’s Method

In this chapter we study the convergence of Newton’s method for nonlinear equations and nonlinear inclusions in a Banach space. Nonlinear mappings, which appear in the right-hand side of the equations, are not necessarily differentiable. Our goal is to obtain an approximate solution in the presence of computational errors. In order to meet this goal, in the case of inclusions, we study the behavior of iterates of nonexpansive set-valued mappings in the presence of computational errors.

16.1 Pre-differentiable Mappings Newton’s method is an important and useful tool in optimization and numerical analysis. See, for example, [8, 21, 22, 24, 32, 41, 46, 47, 63–65, 88, 94, 97, 99, 102] and the references mentioned therein. We study equations with nonlinear mappings which are not necessarily differentiable. In this section we consider this class of mappings. Let .X; k  k/ and .Y; k  k/ be normed spaces. For each x 2 X, each y 2 Y, and each r > 0 set BX .x; r/ D fu 2 X W ku  xk  rg; BY .y; r/ D fv 2 Y W kv  yk  rg: Let IX .x/ D x for all x 2 X and let IY .y/ D y for all y 2 Y. Denote by L.X; Y/ the set of all linear continuous operators A W X ! Y. For each A 2 L.X; Y/ set kAk D supfkA.x/k W x 2 BX .0; 1/g: Let U  X be a nonempty open set, F W U ! Y, x 2 U and > 0. We say that the mapping F is . /-pre-differentiable at x if there exists A 2 L.X; Y/ such that © Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7_16

265

266

16 Newton’s Method

 lim sup kF.x C h/  F.x/  A.h/kkhk1 h!0

D lim supfkF.x C h/  F.x/  A.h/kkhk1 W h 2 BX .0; / n f0gg: (16.1) !0C

If A 2 L.X; Y/ satisfies (16.1), then A is called a . /-pre-derivative of F at x. We denote by @ F.x/ the set of all . /-pre-derivatives of F at x. Note that the set @ F.x/ can be empty. We say that the mapping F is . /-pre-differentiable if it is . /-pre-differentiable at every x 2 U. If G W U ! Y, x 2 U and G is Frechet differentiable at x, then we denote by G0 .x/ the Frechet derivative of G at x. Proposition 16.1. Let G W U ! Y, g W U ! Y, x 2 U, > 0, G be Frechet differentiable at x and let kg.z2 /  g.z1 /k  kz2  z1 k for all z1 ; z2 2 U: Then G0 .x/ is the . /-pre-derivative of G C g at x 2 X. Proof. For every h 2 X n f0g such that khk is sufficiently small, khk1 k.G C g/.x C h/  .G C g/.x/  .G0 .x//.h/k  khk1 kG.x C h/  G.x/  .G0 .x//.h/k C khk1 kg.x C h/  g.x/k  khk1 kG.x C h/  G.x/  .G0 .x//.h/k C : This implies that lim sup khk1 k.G C g/.x C h/  .G C g/.x/  .G0 .x//.h/k  : h!0

Proposition 16.1 is proved. In our analysis of Newton’s method we need the following mean-valued theorem. Theorem 16.2. Assume that U  X is a nonempty open set, > 0, a mapping F W U ! Y is . /-pre-differentiable at every point of U and that x; y 2 U satisfy x 6D y and ftx C .1  t/y W t 2 Œ0; 1g  U: Then there exists t0 2 .0; 1/ such that for every A 2 @ F.x C t0 .y  x// the following inequality holds: kF.x/  F.y/k  kA.y  x/k C ky  xk:

16.1 Pre-differentiable Mappings

267

Proof. For each t 2 Œ0; 1 set .t/ D kF.x/  F.x C t.y  x//k; t 2 Œ0; 1:

(16.2)

Since the mapping F is . /-pre-differentiable, the function is continuous on Œ0; 1. For every t 2 Œ0; 1 set .t/ D .t/  t. .1/  .0//; t 2 Œ0; 1: Clearly, the function

(16.3)

is continuous on Œ0; 1 and .0/ D .0/ D 0;

.1/ D 0:

(16.4)

By (16.4), there exists t0 2 .0; 1/ such that either (a) t0 is a point of minimum of

on Œ0; 1

(b) t0 is a point of maximum of

on Œ0; 1:

or

It is easy to see that in the case (a) lim inf. .t/  .t0 //.t  t0 /1  0; lim sup. .t/  .t0 //.t  t0 /1  0 t!t0

t!t0C

(16.5)

and in the case (b) lim sup. .t/ .t0 //.t t0 /1  0; lim inf . .t/ .t0 //.t t0 /1  0:  t!t0

t!t0C

(16.6)

Assume that A 2 @ F.x C t0 .y  x//:

(16.7)

Let t 2 .0; 1/. By (16.2), j .t/  .t0 /j jkF.x/  F.x C t.y  x//k  kF.x/  F.x C t0 .y  x//kj  kF.x C t.y  x//  F.x C t0 .y  x//k  kF.x C t.y  x//  F.x C t0 .y  x//  .t  t0 /A.y  x/k Cjt  t0 jkA.y  x/k:

(16.8)

268

16 Newton’s Method

Let  > 0. Since the mapping F is . /-pre-differentiable on U, it follows from (16.1) that there exists ı./ 2 .0; / such that BX .x C t0 .y  x/; ı.//  U

(16.9)

and that for each h 2 BX .0; ı.//, kF.x C t0 .y  x/ C h/  F.x C t0 .y  x//  A.h/k  . C /khk:

(16.10)

Assume that t 2 .0; 1/ satisfies jt  t0 j  ı./.kxk C kyk C 1/1 :

(16.11)

h D .t  t0 /.y  x/:

(16.12)

Set

In view of (16.11) and (16.12), inequality (16.10) holds. By (6.8), (6.10), and (6.12), j .t/  .t0 /j  jt  t0 jkA.y  x/k C . C /jt  t0 jky  xk

(16.13)

for every t 2 .0; 1/ satisfying (16.11). Assume that the case (a) holds. Then (16.3), (16.5), and (16.13) imply that 0  lim inf. .t/  t!t0C

.t0 //.t  t0 /1

D lim inf. .t/  t. .1/  .0//  . .t0 /  t0 . .1/  .0////.t  t0 /1 t!t0C

D lim inf. .t/  .t0 //.t  t0 /1  .1/ t!t0C

 kA.y  x/k C . C /ky  xk  kF.x/  F.y/k: Since  is any positive number we conclude that kF.x/  F.y/k  kA.y  x/k C ky  xk: Assume that the case (b) holds. Then (16.2), (16.3), (16.6), and (16.13) imply that . .t/  0  lim inf  t!t0

.t0 //.t  t0 /1

D lim inf . .t/  t. .1/  .0//  . .t0 /  t0 . .1/  .0////.t  t0 /1  t!t0

16.2 Convergence of Newton’s Method

269

D lim inf . .t/  .t0 //.t  t0 /1  .1/  t!t0

j .t/  .t0 /jjt  t0 j1  kF.x/  F.y/k D lim inf  t!t0

 kA.y  x/k C . C /ky  xk  kF.x/  F.y/k: Since  is any positive number we conclude that kF.x/  F.y/k  kA.y  x/k C ky  xk: This completes the proof of Theorem 16.2. Let x 2 X and r > 0. Define Px;r .z/ 2 X for every z 2 X by Px;r .z/ D z if z 2 BX .x; r/ and Px;r .z/ D x C rkz  xk1 .z  x/ if z 2 X n BX .x; r/: Clearly, for each z 2 X, kz  Px;r .z/k D inffkz  yk W y 2 BX .x; r/g:

16.2 Convergence of Newton’s Method We use the notation and definitions of Sect. 16.1. Suppose that the normed space X is Banach. Let > 0, r > 0, xN 2 X, U be a nonempty open subset of X such that BX .Nx; r/  U

(16.14)

and let F W U ! Y be a . /-pre-differentiable mapping. Let L > 0; 1 2 .0; 1/:

(16.15)

Suppose that for each z 2 U there exists A.z/ 2 @ F.z/

(16.16)

such that for each z1 ; z2 2 BX .Nx; r/, kA.z1 /  A.z2 /k  Lkz1  z2 k:

(16.17)

kIX  A ı A.Nx/k  1 ;

(16.18)

Let A 2 L.Y; X/ satisfy

M D kAk

(16.19)

270

16 Newton’s Method

and let a positive number K satisfy kA.F.Nx//k  K  41 r:

(16.20)

In view of (16.15), (16.18), and (16.19), M > 0: Set h D MLK:

(16.21)

ht2  .1  1  M /t C 1 D 0

(16.22)

Let us consider the equation

with respect to the variable t 2 R1 . We suppose that 1 C M  21 ; MLK D h < 41 .1  1  M /2 :

(16.23)

Equation (16.22) has solutions .2h/1 .1  1  M C ..1  1  M /2  4h/1=2 / and .2h/1 .1  1  M  ..1  1  M /2  4h/1=2 /: Set t0 D .2h/1 .1  1  M  ..1  1  M /2  4h/1=2 /:

(16.24)

For every x 2 U define T.x/ D x  A.F.x//:

(16.25)

The following result is proved in Sect. 16.4. Theorem 16.3. For each x; y 2 BX .Nx; Kt0 /, kT.x/  T.y/k  .3=4/kx  yk; T.BX .Nx; Kt0 //  BX .Nx; Kt0 /; there exists a unique point x 2 BX .Nx; Kt0 / such that A.F.x // D 0 and for each x 2 BX .Nx; Kt0 /, kT i .x/  x k  2.3=4/i Kt0 ; i D 0; 1; : : : :

16.2 Convergence of Newton’s Method

271

If the operator A is injective, then F.x / D 0. Moreover, the following assertions hold. 1. Let ı > 0, a natural number n0 satisfy .3  41 /n0 2Kt0  ı

(16.26)

x; Kt0 / satisfy and let a sequence fxi g1 iD0  BX .N kT.xi /  xiC1 k  ı for all integers i  0: Then for all integers n  n0 , kxn  x k  5ı: 2. Let ı > 0, a natural number n0 > 4 satisfy (16.26) and let sequences x; Kt0 /; fyi g1 fxi g1 iD0  BX .N iD0  X satisfy for all integers i  0, kyiC1  T.xi /k  ı; xiC1 D PxN;Kt0 .yiC1 /:

(16.27)

Then for all integers n  n0 , kxn  x k  10ı: 3. Let  > 0, an integer n0 > 2 satisfy .3  41 /n0 16Kt0  

(16.28)

and let a positive number ı satisfy ı < 1281 ; 24.n0 C 1/ı < K; K  6kA.F.Nx//k:

(16.29)

0  X, x0 D xN and that if an integer i 2 Œ0; n0  1 satisfies Assume that fxi gniD0

xi 2 BX .Nx; Kt0 /; then kT.xi /  xiC1 k  ı: Then 0  BX .Nx; Kt0 /; fxi gniD0

kxn0 1  x k   and kxn0 1  xn0 k < =4:

(16.30)

272

16 Newton’s Method

16.3 Auxiliary Results We use the notation, assumptions, and definitions introduced in Sects. 16.1 and 16.2. Lemma 16.4. The mapping T W U ! X is . M/-pre-differentiable at every point of U and for every x 2 U, IX  A ı A.x/ 2 @M T.x/: Proof. Let x 2 U and  > 0. In view of (16.1) and (16.16), there exists ı > 0 such that BX .x; ı/  U

(16.31)

and for each h 2 BX .0; ı/, kF.x C h/  F.x/  .A.x//.h/k  . C .kAk C 1/1 /khk: By (16.19), (16.25), (16.31), and (16.32), for each h 2 BX .0; ı/, kT.x C h/  T.x/  .IX  A ı A.x//.h/k D kx C h  A.F.x C h//  x C A.F.x//  h C A..A.x//.h//k D k.A/.F.x C h/  F.x/  .A.x//.h//k  kAkkF.x C h/  F.x/  .A.x//.h/k  kAk. C .kAk C 1/1 /khk  .kAk C /khk D .M C /khk: Since  is any positive number, this completes the proof of Lemma 16.4. Lemma 16.5. Let r0 2 .0; r and x 2 BX .Nx; r0 /. Then kIX  A ı A.x/k  1 C MLr0 : Proof. By (16.17), (16.18), and (16.19), kIX  A ı A.x/k D kIX  A ı A.Nx/ C A ı A.Nx/  A ı A.x/k  kIX  A ı A.Nx/k C kAkkA.Nx/  A.x/k  1 C MLkx  xN k  1 C MLr0 : Lemma 16.5 is proved.

(16.32)

16.3 Auxiliary Results

273

Lemma 16.6. Let r0 2 .0; r and x; y 2 BX .Nx; r0 /: Then kT.x/  T.y/k  . 1 C MLr0 C M /ky  xk: Proof. By (16.14), Theorem 16.2, and Lemmas 16.4 and 16.5, there exists 0 2 .0; 1/ such that kT.x/  T.y/k  k.IX  A ı A.x C 0 .y  x///.y  x/k C M ky  xk  . 1 C MLr0 C M /ky  xk: Lemma 16.6 is proved. Lemma 16.7. Let r0 2 .0; r. Then for each x 2 BX .Nx; r0 /, kT.x/  xN k  K C 1 r0 C M. r0 C Lr02 /: Proof. Let x 2 BX .Nx; r0 /:

(16.33)

By (16.19), (16.20), and (16.25), kT.x/  xN k D kx  A.F.x//  xN k D kAŒ.A.Nx//.x  xN /  F.x/ C F.Nx/ A.F.Nx// C .IX  A ı A.Nx//.x  xN /k  kAkkF.x/  F.Nx/  .A.Nx//.x  xN /k CkA.F.Nx//k C kIX  A ı A.Nx/kkx  xN k  MkF.x/  F.Nx/  .A.Nx//.x  xN /k C K C 1 kx  xN k: (16.34) For every x 2 U define .x/ D F.x/  F.Nx/  .A.Nx//.x  xN /:

(16.35)

We show that the mapping is . /-pre-differentiable on U. By (16.35), for each x 2 U and each h 2 X satisfying x C h 2 U, .x C h/  .x/  .A.x/  A.Nx//.h/ D F.x C h/  F.Nx/  .A.Nx//.x C h  xN / F.x/ C F.Nx/ C .A.Nx//.x  xN /  .A.x/  A.Nx//.h/

274

16 Newton’s Method

D F.x C h/  F.x/  .A.Nx//.h/  .A.x/  A.Nx//.h/ D F.x C h/  F.x/  .A.x//.h/: Together with (16.1) and (16.16) this implies that the mapping is . /-predifferentiable at every point of U and that for all z 2 U, A.z/  A.Nx/ 2 @ .z/:

(16.36)

In view of (16.17), for all z 2 BX .Nx; r0 /, kA.z/  A.Nx/k  Lkz  xN k:

(16.37)

It follows from (16.35), (16.36), and Theorem 16.2 that for every z 2 BX .Nx; r0 /, k .z/k D k .z/  .Nx/k  kz  xN k C Lkz  xN k2 :

(16.38)

By (16.33)–(16.35) and (16.38), for every x 2 BX .Nx; r0 /, kT.x/  xN k  K C 1 r0 C M. r0 C Lr02 /: Lemma 16.7 is proved.

16.4 Proof of Theorem 16.3 Set r0 D Kt0 :

(16.39)

r0  r:

(16.40)

We show that

Indeed, in view of (16.20), (16.23), (16.24), and (16.39), r0 D Kt0 D 2KŒ.1  1  M / C ..1  1  M /2  4h/1=2 1  2.1  1  M /1 K  4K  r: By (16.21), (16.39), (16.40), and Lemma 16.6, for each x; y 2 BX .Nx; Kt0 /, kT.x/  T.y/k  . 1 C MLKt0 C M /kx  yk  . 1 C ht0 C M /kx  yk:

(16.41)

16.4 Proof of Theorem 16.3

275

It follows from (16.24) that t0  2.1  1  M /1 :

(16.42)

Relations (16.23) and (16.42) imply that 1 C ht0 C M  1 C M C 2.1  1  M /1 .1  1  M /2 =4 D 1 C M C 21 .1  1  M / D 21 C 21 . 1 C M /  3=4:

(16.43)

By (16.41) and (16.43), for each x; y 2 BX .Nx; Kt0 /, kT.x/  T.y/k  .3  41 /kx  yk:

(16.44)

x 2 BX .Nx; r0 /:

(16.45)

Let

It follows from (16.21), (16.22), (16.24), (16.39), (16.40), (16.45), and Lemma 16.7 that kT.x/  xN k  K C 1 Kt0 C M. Kt0 C LK 2 t02 /  K.1 C 1 t0 C M t0 C MLKt02 / D K.1 C 1 t0 C M t0 C ht02 / D Kt0 ; T.x/ 2 BX .Nx; r0 / and T.BX .Nx; r0 //  BX .Nx; r0 /:

(16.46)

Relations (16.45) and(16.46) imply that there exists a unique point x 2 BX .Nx; r0 / such that T.x / D x : In order to complete the proof of Theorem 16.3 it is sufficient to show that assertions 1, 2, and 3 hold. Let us prove assertion 1. For each integer i  0, kxiC1  x k  kxiC1  T.xi /k C kT.xi /  x k  ı C .3  41 /kxi  x k:

(16.47)

276

16 Newton’s Method

By induction we show that for all integers p  1, 1 p

kxp  x k  .3  4 / kx0  x k C ı

p1 X

.3  41 /i :

(16.48)

iD0

In view of (16.47), inequality (16.48) holds for p D 1. Assume that an integer p  1 and that (16.48) holds. It follows from (16.47) and (16.48) that kxpC1  x k  ı C .3=4/kxp  x k  .3  41 /pC1 kx0  x k C ı

p X

.3  41 /i :

iD0

Thus we showed by induction that (16.48) holds for all integers p  1. By (16.48) and the choice of n0 [see (16.26)], for all integer n  n0 , kxn  x k  .3  41 /n0 2Kt0 C 4ı  5ı: Assertion 1 is proved. Let us prove assertion 2. In view of (16.27), for each integer i  0, ı  kyiC1  T.xi /k  kyiC1  xiC1 k and kxiC1  T.xi /k  2ı: By the relation above, (16.26) and assertion 1, for all integers n  n0 , kxn  x k  10ı: Assertion 2 is proved. Let us prove assertion 3. In view of (16.24), t0 D 2..1  1  M / C ..1  1  M /2  4h/1=2 /1  .1  1  M /1 :

(16.49)

By (16.23),(16.25), (16.29), (16.30), and (16.49), kx1  x0 k D kx1  xN k  kx1  T.x0 /k C kT.x0 /  x0 k  ı C kA.F.Nx//k  ı C 61 K < K < K.1  1  M /1  Kt0

(16.50)

16.4 Proof of Theorem 16.3

277

and x1 2 BX .Nx; Kt0 /:

(16.51)

Relations (16.30), (16.44), (16.50), and (16.51) imply that kx2  x1 k  kx2  T.x1 /k C kT.x1 /  x1 k  ı C kT.x1 /  T.x0 /k C kT.x0 /  x1 k  2ı C .3  41 /kx1  x0 k  2ı C .3  41 /ı C 61 .3=4/K:

(16.52)

It follows from (16.23), (16.29), (16.49), (16.50), and (16.52), kx2  xN k  kx2  x1 k C kx1  x0 k  2ı C .3  41 /.ı C 61 K/ C .ı C 61 K/  2ı C 2.ı C 61 K/  4ı C K=3 < K  Kt0 :

(16.53)

Assume that an integer p satisfies 2  p < n0 ; xi 2 BX .Nx; Kt0 /

(16.54)

for all i D 1; : : : ; p and for all q D 2; : : : ; p, kxq  xq1 k  .3  41 /q1 kx1  x0 k C 2ı

q2 X

.3  41 /i :

(16.55)

iD0

(In view of (16.51), (16.52), and (16.53), our assumption holds for p D 2.) By (16.30), (16.44), (16.54), and (16.55), kxpC1  xp k  kxpC1  T.xp /k C kT.xp /  T.xp1 /k C kT.xp1 /  xp k  2ı C 3  41 kxp  xp1 k  .3  41 /p kx1  x0 k C .3  21 /ı

p2 X

.3  41 /i C 2ı

iD0

D .3  41 /p kx1  x0 k C 2ı

p1 X iD0

.3  41 /i

278

16 Newton’s Method

and (16.55) holds for q D p C 1. By (16.55) which holds for all q D 2; : : : ; p C 1, (16.29), and (16.50), kxpC1  x0 k 

pC1 X

kxq  xq1 k

qD1



pC1 X .3  41 /q1 kx1  x0 k C 8ıp qD1

 4kx1  x0 k C 8n0 ı  4ı C 2  31 K C 8n0 ı  8.n0 C 1/ı C 2  31 K  K  Kt0 and (16.54) holds for i D p C 1. Thus we showed by induction that our assumption holds for p D n0 . Together with (16.54) and (16.55) holding for p D n0 , (16.28) and (16.29) this implies that xn0 2 BX .Nx; Kt0 /; kxn0  xn0 1 k  8ı C .3  41 /n0 1 2Kt0  =8 C =16:

(16.56)

xQ 0 D xn0 1

(16.57)

xQ iC1 D T.Qxi /:

(16.58)

Set

and for all integer i  0 set

By (16.28), (16.30), (16.44), and (16.56)–(16.58), kQx0  xQ 1 k D kxn0 1  T.xn0 1 /k  kxn0 1  xn0 k C ı  =4

(16.59)

and for all integers i  0, kQxiC2  xQ iC1 k D kT.QxiC1 /  T.Qxi /k  .3  41 /kQxiC1  xQ i k: Together with (16.59) this implies that for all integers i  0, kQxiC1  xQ i k  .3  41 /i =4: Clearly, x D lim xQ i : i!1

(16.60)

16.5 Set-Valued Mappings

279

By (16.57) and (16.60), kx  xn0 1 k D kx  xQ 0 k  lim kQxq  xQ 0 k  q!1

1 X

kQxq  xQ qC1 k  :

qD0

Assertion 3 is proved. This completes the proof of Theorem 16.3.

16.5 Set-Valued Mappings Let .X; / be a complete metric space. For each z 2 X and each r > 0 set B.z; r/ D fy 2 X W .z; y/  rg: For each x 2 X and each nonempty set C  X define .x; C/ D inff.x; y/ W y 2 Cg: In Sect. 16.7 we prove the following result which is important in our study of Newton’s method for nonlinear inclusions. Theorem 16.8. Suppose that W X ! 2X , a > 0,  2 .0; 1/, xN 2 X, .x/ 6D ; for all x 2 B.Nx; a/;

(16.61)

.Nx; .Nx// < a.1   /;

(16.62)

for all u; v 2 B.Nx; a/, supf.z; .v// W z 2 .u/ \ B.Nx; a/g  .u; v/

(16.63)

and that the set graph. I B.Nx; a// WD f.x; y/ 2 B.Nx; a/  B.Nx; a/ W y 2 .x/g is closed. Then the following assertions hold. 1. Assume that a sequence fi g1 iD0  .0; 1/ satisfies 2.1  /1

1 X

P1

iD0 i

< 1,

i C .1   /1 .maxfi W i D 0; 1; : : : g C .Nx; .Nx///  a

iD0

and that a sequence fxi g1 iD0  X satisfies x0 D xN

(16.64)

(16.65)

280

16 Newton’s Method

and for each integer i  0 satisfying xi 2 B.Nx; a/, the inequalities .xiC1 ; .xi //  i ;

(16.66)

.xi ; xiC1 /  .xi ; .xi // C i

(16.67)

.xi ; xN / < a  i1 for all integers i  1;

(16.68)

hold. Then

for each integer k  1, .xk ; xkC1 /   ..Nx; .Nx// C 0 / C k

k1 X

j 

k1j

C

jD0

k X

j  kj ;

(16.69)

jD1

for each integer k  0, 1 X

.xi ; xiC1 /  .1   /1 .xk ; xkC1 / C 2.1   /1

iDk

1 X

i

(16.70)

iDk

and there exists x D lim xn 2 B.Nx; a/ n!1

satisfying x 2 .x /. 2. Let  2 .0; 1/, a natural number n0 > 2 satisfy 4 n0 .a.1   / C 1/ < .1   /

(16.71)

and let a number ı 2 .0; / satisfy ı < .8n0 /1 .1   /

(16.72)

and ı < Œa  .1   /1 .Nx; .Nx//.1   /.2n0 C 1/1 :

(16.73)

0  X satisfies Assume that a sequence fxi gniD0

x0 D xN and for each integer i 2 Œ0; n0  1 satisfying xi 2 B.Nx; a/, the inequalities .xiC1 ; .xi //  ı;

(16.74)

.xi ; xiC1 /  .xi ; .xi // C ı

(16.75)

16.6 An Auxiliary Result

281

hold. Then .xi ; xN / < a  ı for all i D 1; : : : ; n0 and there exists x 2 B.Nx; a/ such that x 2 .x / and .xn0 ; x / < : A prototype of Theorem 16.8, for which W X ! 2X n f;g is a strict contraction, was proved in Chap. 9 of [121].

16.6 An Auxiliary Result Lemma 16.9. Suppose that W X ! 2X , a > 0, P 2 .0; 1/, xN 2 X, (16.61)–(16.63) 1 hold and let a sequence fi g1  .0; 1/ satisfy iD0 i < 1 and (16.64). Then iD0 the following assertions hold. 1. Let x0 D xN and x1 2 X satisfy .x1 ; .x0 //  0 ; .x0 ; x1 /  .x0 ; .x0 // C 0 :

(16.76)

Then .x1 ; xN / < a  0 . 2. Assume that n  1 is an integer, fxi gnC1 iD0  X, x0 2 B.Nx; a/; .xi ; xN / < a  i1 ; i D 1; : : : ; n

(16.77) (16.78)

and that for each integer i 2 f0; : : : ; ng the inequalities .xiC1 ; .xi //  i ;

(16.79)

.xi ; xiC1 /  .xi ; .xi // C i

(16.80)

hold. Then for each integer k 2 Œ0; n  1 there exists ykC1 2 .xk / \ B.Nx; a/

(16.81)

.xkC1 ; ykC1 / < 2k ;

(16.82)

such that

for all integers k D 0; : : : ; n  1, .xkC2 ; xkC1 /  .xk ; xkC1 / C k C kC1

(16.83)

282

16 Newton’s Method

for each integer s satisfying 0  s < n and each integer k satisfying s < k  n, .xk ; xkC1 /   ks .xs ; xsC1 / C

k1 X

 is .ki1Cs C kiCs /

(16.84)

iDs

and n X

.xp ; xpC1 / 

pD0

n X

 p .x0 ; x1 / C

pD0

n1 X

0

n1i X

i @

iD0

1  jA C

n X

jD0

iD1

0 1 ni X i @  jA : jD0

(16.85) Moreover, if x0 D xN , then .xnC1 ; xN / < a  n . Proof. Let us prove assertion 1. By (16.64) and (16.76), .x1 ; xN /  .Nx; .Nx// C 0 < a  0 : Assertion 1 is proved. Let us prove assertion 2. Assume that an integer k 2 Œ0; n  1:

(16.86)

By (16.77)–(16.80) and (16.86), xk ; xkC1 2 B.Nx; a/; xkC2 2 X; .xkC2 ; .xkC1 //  kC1 ; .xkC1 ; .xk //  k ; .xkC2 ; xkC1 /  .xkC1 ; .xkC1 // C kC1 :

(16.87) (16.88)

We show that .xkC2 ; xkC1 /  .xk ; xkC1 / C k C kC1 : Let a positive number  satisfy  < minfk ; a  k  .xkC1 ; xN /g

(16.89)

[see (16.78)]. In view of (16.87), there exists ykC1 2 .xk /

(16.90)

.xkC1 ; ykC1 / < k C :

(16.91)

such that

16.6 An Auxiliary Result

283

By (16.89) and (16.91), .ykC1 ; xN / < k C  C .xkC1 ; xN / < a:

(16.92)

Relations (16.89)–(16.92) imply that ykC1 2 .xk / \ B.Nx; a/; .xkC1 ; ykC1 / < 2k :

(16.93)

Thus (16.81) and (16.82) hold. It follows from (16.63), (16.88), (16.91), and (16.93) that .xkC2 ; xkC1 /  kC1 C .xkC1 ; .xkC1 //  kC1 C .xkC1 ; ykC1 / C .ykC1 ; .xkC1 //  kC1 C k C  C supf.z; .xkC1 // W z 2 .xk / \ B.Nx; a/g  kC1 C k C  C .xk ; xkC1 /: Since  is an arbitrary positive number satisfying (16.89) we conclude that .xkC2 ; xkC1 /  .xk ; xkC1 / C k C kC1

(16.94)

for all integers k D 0; : : : ; n  1. Let an integer s satisfy 0  s < n: We show by induction that for each integer k satisfying s < k  n, (16.84) holds. In view of (16.94), .xsC2 ; xsC1 /  .xs ; xsC1 / C s C sC1 and (16.84) holds for k D s C 1. Assume that an integer k satisfies s 0 such that (16.64) holds, 2.1  /1

1 X

i < =4; i  ı for all integers i  0:

(16.104)

iDn0

Clearly, for every integer i  n0 C 1, there exists xi 2 X such that the following property holds: if an integer i  0 satisfies xi 2 B.Nx; a/, then (16.66) and (16.67) hold. It follows from (16.62), (16.64), (16.66)–(16.69), (16.71), (16.72), (16.103), (16.104), and assertion 1 that .xi ; xN / < a  i1 for all integers i  1; .xi ; xN / < a  ı for all integers i D 1; : : : ; n0 and .xn0 ; xn0 C1 /   n0 ..Nx; .Nx// C ı/ C 2n0 ı < 41 .1   / C 41 .1   /:

(16.105)

16.8 Pre-differentiable Set-Valued Mappings

287

By assertion 1, (16.64), (16.66), (16.67), (16.70), (16.104), and (16.105), 1 X

.xi ; xiC1 /  .1   /1 .xn0 ; xn0 C1 / C 2.1   /1

iDn0

1 X

i < =2 C =4

iDn0

(16.106)

and there exists x D lim xn 2 B.Nx; a/ n!1

(16.107)

satisfying x 2 .x /. It follows from (16.106) and (16.107) that .xn0 ; x / D lim .xn0 ; xn /  n!1

1 X

.xi ; xiC1 / < :

iDn0

Assertion 2 is proved. This completes the proof of Theorem 16.8.

16.8 Pre-differentiable Set-Valued Mappings Let .Z; k  k/ be a normed space. For each x 2 Z and each nonempty set C  Z define d.x; C/ D inffkx  zk W z 2 Cg: For each z 2 Z and each r > 0 set BZ .z; r/ D fy 2 Z W kz  yk  rg: Let IZ .z/ D z for all z 2 Z. For each pair of nonempty sets C1 ; C2  Z define H.C1 ; C2 / D maxfsupfd.x; C2 / W x 2 C1 g; supfd.y; C1 / W y 2 C2 gg: Let .X; k  k/ and .Y; k  k/ be normed spaces. Denote by L.X; Y/ the set of all linear continuous operators A W X ! Y. For each A 2 L.X; Y/ set kAk D supfkA.x/k W x 2 BX .0; 1/g: Let U  X be a nonempty open set, F W U ! 2Y n f;g, x 2 U and > 0. We say that the mapping F is . /-pre-differentiable at x if there exists A 2 L.X; Y/ such that the following property holds: (P1)

for each  > 0 there exists ı./ > 0 such that BX .x; ı.//  U

(16.108)

288

16 Newton’s Method

and that for each h 2 BX .0; ı.//, F.x/ C A.h/  F.x C h/ C . C /khkBY .0; 1/;

(16.109)

F.x C h/  F.x/ C A.h/ C . C /khkBY .0; 1/;

(16.110)

If A 2 L.X; Y/ and (P1) holds, then A is called a . /-pre-derivative of F at x. We denote by @ F.x/ the set of all . /-pre-derivatives of F at x. Note that the set @ F.x/ can be empty. Clearly, A 2 L.X; Y/ satisfies A 2 @ F.x/ if and only if for each  > 0 there exists ı./ > 0 such that BX .x; ı.//  U and that for each h 2 BX .0; ı.//, H.F.x/ C A.h/; F.x C h//  . C /khk:

(16.111)

We say that the mapping F is . /-pre-differentiable if it is . /-pre-differentiable at every x 2 U. Recall that if G W U ! Y, x 2 U and G is Frechet differentiable at x, then we denote by G0 .x/ the Frechet derivative of G at x. Proposition 16.10. Let G W U ! Y, g W U ! 2Y n f;g, x 2 U, > 0, G be Frechet differentiable at x and let H.g.z2 /; g.z1 //  kz2  z1 k for all z1 ; z2 2 U:

(16.112)

Then G0 .x/ is the . /-pre-derivative of G C g at x 2 X. Proof. Let  > 0. There exists ı./ > 0 such that BX .x; ı.//  U and that for each h 2 BX .0; ı.// n f0g, khk1 kG.x C h/  G.x/  .G0 .x//.h/k < =2:

(16.113)

By (16.112) and (16.113), for each h 2 BX .0; ı.// n f0g, G.x C h/ C g.x C h/  G.x/ C .G0 .x//.h/ C 21 khkBY .0; 1/ C g.x C h/  G.x/ C .G0 .x//.h/ C 21 khkBY .0; 1/ C g.x/ C . C 41 /khkBY .0; 1/  G.x/ C g.x/ C .G0 .x//.h/ C . C /khkBY .0; 1/

16.8 Pre-differentiable Set-Valued Mappings

289

and G.x/ C g.x/ C .G0 .x//.h/  G.x C h/ C 21 khkBY .0; 1/ C g.x/  G.x C h/ C 21 khkBY .0; 1/ C g.x C h/ C . C 41 /khkBY .0; 1/  G.x C h/ C g.x C h/ C . C /khkBY .0; 1/: Proposition 16.10 is proved. The next result is a mean-value theorem for pre-differentiable set-valued mappings. Theorem 16.11. Assume that U  X is a nonempty open set, > 0, a mapping F W U ! 2Y nf;g is . /-pre-differentiable at every point of U, x; y 2 U satisfy x 6D y and ftx C .1  t/y W t 2 Œ0; 1g  U and that xQ 2 F.x/:

(16.114)

Then there exists t0 2 .0; 1/ such that for every A 2 @ F.x C t0 .y  x// the following inequality holds: d.Qx; F.y//  kA.y  x/k C ky  xk: Proof. For each t 2 Œ0; 1 set .t/ D d.Qx; F.x C t.y  x///; t 2 Œ0; 1:

(16.115)

Since the mapping F is . /-pre-differentiable, the function is continuous on Œ0; 1. For every t 2 Œ0; 1 set .t/ D .t/  t. .1/  .0//; t 2 Œ0; 1: Clearly, the function

is continuous on Œ0; 1 and in view of (16.114) .0/ D .0/ D 0;

.1/ D 0:

Therefore there exists t0 2 .0; 1/ such that either (a) t0 is a point of minimum of

on Œ0; 1

(16.116)

290

16 Newton’s Method

or (b) t0 is a point of maximum of

on Œ0; 1:

It is easy to see that in the case (a) lim inf. .t/ .t0 //.tt0 /1  0; lim sup. .t/ .t0 //.tt0 /1  0 t!t0

t!t0C

(16.117)

and in the case (b) lim sup. .t/  t!t0C

.t0 //.t  t0 /1  0; lim inf . .t/   t!t0

.t0 //.t  t0 /1  0: (16.118)

Assume that A 2 @ F.x C t0 .y  x//:

(16.119)

By (16.115), for every t 2 Œ0; 1, .t/  .t0 / D d.Qx; F.x C t.y  x///  d.Qx; F.x C t0 .y  x///  H.F.x C t.y  x//; F.x C t0 .y  x///; .t0 /  .t/ D d.Qx; F.x C t0 .y  x///  d.Qx; F.x C t.y  x///  H.F.x C t.y  x//; F.x C t0 .y  x/// and j .t/  .t0 /j  H.F.x C t.y  x//; F.x C t0 .y  x///:

(16.120)

Let  > 0. Since the mapping F is . /-pre-differentiable on U, it follows from (16.119) that there exists ı./ 2 .0; / such that BX .x C t0 .y  x/; ı.//  U

(16.121)

h 2 BX .0; ı.//

(16.122)

and that for each

we have H.F.x C t0 .y  x// C A.h/; F.x C t0 .y  x/ C h//  . C =4/khk:

(16.123)

16.8 Pre-differentiable Set-Valued Mappings

291

Assume that t 2 Œ0; 1 satisfies jt  t0 j  ı./.kxk C kyk C 1/1 :

(16.124)

h D .t  t0 /.y  x/:

(16.125)

Set

In view of (16.124) and (16.125), relations (16.122) and (16.123) hold. By (16.123) and (16.125), H.F.x C t0 .y  x// C .t  t0 /A.y  x/; F.x C t.y  x///  . C 41 /jt  t0 jky  xk: It follows from the relation above and (16.120) that j .t/  .t0 /j  H.F.x C t0 .y  x//; F.x C t0 .y  x// C .t  t0 /A.y  x// CH.F.x C t0 .y  x// C .t  t0 /A.y  x/; F.x C t.y  x///  jt  t0 jkA.y  x/k C . C 41 /jt  t0 jky  xk:

(16.126)

Assume that the case (a) holds. Then (16.115)–(16.117), (16.124), and (16.126) imply that 0  lim inf. .t/  t!t0C

.t0 //.t  t0 /1

D lim inf. .t/  t. .1/  .0//  . .t0 /  t0 . .1/  .0////.t  t0 /1 t!t0C

D lim inf. .t/  .t0 //.t  t0 /1  . .1/  .0// t!t0C

 kA.y  x/k C . C 41 /ky  xk  d.Qx; F.y//: Since  is any positive number we conclude that d.Qx; F.y//  kA.y  x/k C ky  xk:

(16.127)

Assume that the case (b) holds. Then (16.114)–(16.116), (16.118), and (16.126) imply that . .t/  0  lim inf  t!t0

.t0 //.t  t0 /1

D lim inf . .t/  t. .1/  .0//  . .t0 /  t0 . .1/  .0////.t  t0 /1  t!t0

292

16 Newton’s Method

D lim inf . .t/  .t0 //.t  t0 /1  . .1/  .0//  t!t0

j .t/  .t0 /jjt  t0 j1  d.Qx; F.y// D lim inf  t!t0

 kA.y  x/k C . C 41 /ky  xk  d.Qx; F.y//: Since  is any positive number we conclude that d.Qx; F.y//  kA.y  x/k C ky  xk: Thus (16.127) holds in both cases. This completes the proof of Theorem 16.11.

16.9 Newton’s Method for Solving Inclusions We use the notation and definitions of Sect. 16.8. Suppose that the normed space X is Banach. Let > 0, r > 0, xN 2 X, U be a nonempty open subset of X such that BX .Nx; r/  U and let F W U ! 2Y n f;g be a . /-pre-differentiable mapping at all points of U such that F.x/ is a closed set for all x 2 U. Let L > 0; 1 2 .0; 1/: Suppose that for each z 2 U there exists A.z/ 2 @ F.z/

(16.128)

such that for each z1 ; z2 2 BX .Nx; r/, kA.z1 /  A.z2 /k  Lkz1  z2 k:

(16.129)

kIX  A ı A.Nx/k  1 ;

(16.130)

Let A 2 L.Y; X/ satisfy

there exists a continuous operator A1 W X ! Y, a positive number K satisfy inffkA.z/k W z 2 F.Nx/g  K  41 r and M WD kAk: In view of (16.130), M > 0:

(16.131)

16.9 Newton’s Method for Solving Inclusions

293

For every x 2 U define T.x/ D x  A.F.x//

(16.132)

and for every x 2 X n U set T.x/ D ;. The following result is proved in Sect. 16.11. Theorem 16.12. Let 1 C M  41 ; K  .16ML/1 ; r0 WD minf.4ML/1 ; rg:

(16.133)

Then for each x; y 2 BX .Nx; r0 /, H.T.x/; T.y//  21 kx  yk; d.Nx; T.Nx//  41 r0 and there exists x 2 BX .Nx; r0 / such that x 2 T.x / and 0 2 F.x /. Moreover, the following assertions hold. P1 1. Assume that a sequence fi g1 iD0 i < 1, iD0  .0; 1/ satisfies 4

1 X

i C 2 maxfi W i D 0; 1; : : : g  21 r0

iD0

and that a sequence fxi g1 iD0  X satisfies x0 D xN and for each integer i  0 satisfying xi 2 BX .Nx; r0 /, the inequalities d.xiC1 ; T.xi //  i ; kxi  xiC1 k  d.xi ; T.xi // C i hold. Then kxi  xN k < r0  i1 for all integers i  1; for each integer k  1, k

1

kxk  xkC1 k  2 .4 r0 C 0 / C

k1 X

kC1Cj

2

j C

jD0

k X jD1

for each integer k  0, 1 X iDk

kxi  xiC1 k  2kxk  xkC1 k C 4

1 X iDk

i

2kCj j ;

294

16 Newton’s Method

and there exists lim xn 2 BX .Nx; r0 /

n!1

satisfying limn!1 xn 2 T.limn!1 xn / and 0 2 F.limn!1 xn /. 2. Let  2 .0; 1/, a natural number n0 > 2 satisfy 2n0 .21 r0 C 1/ < 81  and let a number ı 2 .0; / satisfy ı < .16n0 /1 ; and ı < 41 .2n0 C 1/1 r0 : 0  X satisfies Assume that a sequence fxi gniD0

x0 D xN and for each integer i 2 Œ0; n0  1 satisfying xi 2 BX .Nx; r0 /, the inequalities d.xiC1 ; T.xi //  ı; kxi  xiC1 k  d.xi ; T.xi // C ı hold. Then kxi  xN k < r0  ı for all i D 1; : : : ; n0 and there exists x 2 BX .Nx; r0 / such that F.x / D 0 and kxn0  x k < :

16.10 Auxiliary Results for Theorem 16.12 Lemma 16.13. The mapping T W U ! 2X n f;g is . M/-pre-differentiable at every point of U and for every x 2 U, IX  A ı A.x/ 2 @M T.x/: Proof. Let x 2 U and  > 0. In view of (16.128), there exists ı > 0 such that BX .x; ı/  U and for each h 2 BX .0; ı/, H.F.x/ C .A.x//.h/; F.x C h//  . C .kAk C 1/1 /khk:

16.10 Auxiliary Results for Theorem 16.12

295

By the inclusion above, (16.131) (16.132), for each h 2 BX .0; ı/, H.T.x/ C .IX  A ı A.x//.h/; T.x C h// D H.x  A.F.x// C h  A..A.x//.h//; x C h  A.F.x C h/// D H.A.F.x/ C .A.x//.h//; A.F.x C h///  kAk. C .kAk C 1/1 /khk  .M C /khk: Since  is any positive number, this completes the proof of Lemma 16.13. Lemma 16.14. Let r0 2 .0; r and x 2 BX .Nx; r0 /. Then kIX  A ı A.x/k  1 C MLr0 : Proof. By (16.129) and (16.130), kIX  A ı A.x/k D kIX  A ı A.Nx/ C A ı A.Nx/  A ı A.x/k kIX  A ı A.Nx/k C kAkkA.Nx/  A.x/k  1 C MLkx  xN k  1 C MLr0 : Lemma 16.14 is proved. Lemma 16.15. Let r0 2 .0; r and x; y 2 BX .Nx; r0 /: Then H.T.x/; T.y//  . 1 C MLr0 C M /ky  xk: Proof. By Lemma 16.13, the mapping T is . M/-pre-differentiable and for every x 2 U, IX  A ı A.x/ 2 @ M T.x/: We may assume that x 6D y. Let xQ 2 T.x/: By Theorem 16.11 and Lemmas 16.13 and 16.14, there exists t0 2 .0; 1/ such that d.Qx; T.y//  k.IX  A ı A.x C t0 .y  x///.y  x/k C M ky  xk  . 1 C MLr0 C M /ky  xk:

296

16 Newton’s Method

Sine xQ is an arbitrary element of T.x/ this implies that supfd.Qx; T.y// W xQ 2 T.x/k  ky  xk. 1 C MLr0 C M /: Analogously, supfd.Qy; T.x// W yQ 2 T.y/k  ky  xk. 1 C MLr0 C M /: Therefore H.T.x/; T.y//  ky  xk. 1 C MLr0 C M /: Lemma 16.15 is proved.

16.11 Proof of Theorem 16.12 By (16.131)–(16.133), d.Nx; T.Nx// D inffkA.z/k W z 2 F.Nx/k  K  41 r0 : Let x; y 2 BX .Nx; r0 /: Lemma 16.15 imply that H.T.x/; T.y//  ky  xk. 1 C MLr0 C M /  21 kx  yk: Clearly, the set f.x; y/ 2 BX .Nx; r0 /  BX .Nx; r0 / W y 2 T.x/g is closed. It is not difficult to see that Theorem 16.12 follows from Theorem 16.8 applied to the mapping T.

References

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Index

A Absolutely continuous function, 154 Algorithm, 11, 13, 14 Approximate solution, 1, 44

B Banach space, 225 Bochner integrable function, 225

C Cardinality of a set, 137 Collinear vectors, 86 Compact set, 228 Concave function, 26, 36 Continuous subgradient algorithm, 225 Convex–concave function, 11 Convex cone, 247 Convex function, 1, 4, 6, 11, 20, 26, 35 Convex hull, 228 Convex minimization problem, 105 Convex set, 11, 12

E Ekelands variational principle, 244, 252 Euclidean norm, 167 Euclidean space, 86, 169 Exact penalty, 239 Extragradient method, 183, 205

F Fermat–Weber location problem, 85, 86

Fréchet derivative, 45, 59 Fréchet diferentiable function, 45, 59, 63

G Gâteaux derivative, 106 Gâteaux differential function, 106 Gradient-type method, 8, 73

H Hilbert space, 1, 4, 6, 11, 20

I Increasing function, 247 Inner product, 1, 4, 6, 20

K Karush–Kuhn–Tucker theorem, 252

L Lebesgue measurable function, 225, 227 Linear functional, 246 Linear inverse problem, 74 Lower semicontinuous function, 6, 137

M Maximal monotone operator, 169 Metric space, 149 Minimization problem, 2 Minimizer, 15, 16, 22, 42

© Springer International Publishing Switzerland 2016 A.J. Zaslavski, Numerical Optimization with Computational Errors, Springer Optimization and Its Applications 108, DOI 10.1007/978-3-319-30921-7

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304 Mirror descent method, 4, 5, 41 Monotone mapping, 8, 184 Monotone operator, 170

N Newtons method, 267 Nonexpansive mapping, 170 Norm, 1, 4, 6

P Penalty function, 239 Pre-derivative, 268 Pre-differentiable mapping, 268 Projected gradient algorithm, 59 Projected subgradient method, 119 Proximal mapping, 170 Proximal point method, 6, 137 Pseudo-monotone mapping, 8, 184

Q Quadratic function, 90

Index S Saddle point, 11 Strongly measurable function, 225 Subdifferential, 85, 120 Subgradient projection algorithm, 1, 11

T Taylor expansion, 90

V Variational inequality, 8, 183 Vector space, 247

W Weiszfelds method, 85 Well-posed problem, 140, 165

Z Zero-sum game, 25, 35