Necessary Conditions for an Extremum 082471556X

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
FOREWORD TO THE RUSSIAN EDITION
FOREWORD TO THE AMERICAN EDITION
TABLE OF CONTENTS
INTRODUCTION. ELEMENTS OF FUNCTIONAL ANALYSIS AND CONVEX SETS
1. Some Basic Concepts of Functional Analysis
2. Convex Sets
3. Convex Functionals
CHAPTER 1. PROPERTIES OF CONVEX FUNCTIONALS
CHAPTER II. CONVEX PROGRAMMING IN BANACH SPACES
CHAPTER III. QUASI-DIFFERENTIABLE FUNCTIONALS
CHAPTER IV. NECESSARY CONDITIONS FOR AN EXTREMUM IN GENERAL MATHEMATICAL PROGRAMMING PROBLEMS
CHAPTER V. NECESSARY CONDITIONS FOR AN EXTREMUM IN CONCRETE PROBLEMS
1. The Classical Mathematical Programming Problem
2. Mathematical Programming with a Continuum of Constraints
3. Theorems for Minimax Problems
4. Chebyshev Approximation Problems
5. A Linear Optimal Control Problem with Phase Constraints
6. A Duality Principle in Convex Programming
7. Systems of Convex Inequalitíes. Helly's Theorem
8. The Moment Problem
9. A Discrete Maximum Principle
SHORT BIBLIOGRAPHY
LITERATURE
NOTES AND SUPPLEMENTARY BIBLIOGRAPHY TO AMERICAN EDITION
SUPPLEMENTARY LITERATURE
SUBJECT INDEX

Citation preview

Necessary Conditions for an Extremum

PURE AND APPLIED MATHEMATICS A Series of Monographs COORDINATOR OF THE EDITORIAL BOARD

S. Kobayashi UNIVERSITY OF CALIFORNIA AT BERKELEY

1.

2. 3. 4. 5.

KENTARO YANO. Integral Formulas in Riemannian Geometry (1970) S. KOBAYASHI. Hyperbolic Manifolds and Holomorphic Mappings (1970) V. S. VLADIMIROV. Equations of Mathematical Physics (A. Jeffrey, editor; A. Littlewood, translator) ( 1970) B. N. PsHENICHNYI. Necessary Conditions for an Extremum. (L. Neustadt, translation editor; K. Makowski, translator) (1971) L. NARICI, E. BECKENSTEIN, and G. BACHMAN. Functional Analysis and Valuation Theory (1971)

In Preparation: W. BOOTHBY and G. L. WEISS (eds.). Symmetric Spaces Y. MATSUSHIMA. translator) D. PASSMAN. L. DoRNHOFF.

Geometry and Harmonic Analysis of

Differentiable Manifolds (E. J. Taft, editor; E. T. Kobayashi,

Infinite Group Rings Group Representation Theory

Necessary Conditions for an Extremum B. N. PSHENICHNYI

Institute of Cybernetics Kiev, U.S.S.R.

Translation edited by LUCIEN W. NEUSTADT Translated by KAROL MAKOWSKI Department of Electrical Engineering University of Southern California Los Angeles, California

MARCEL DEKKER, INC.,

New York

1971

This book was originally published in Russian under the title "Neobkhodimye Usloviya Ekstremuma" by Nauka, Moscow, 1969. COPYRIGHT© 1971 by MARCEL DEKKER, INC. ALL RIGHTS RESERVED No part of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including Xeroxing, photocopying, microfilm, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 95 Madison Avenue, New York, New York 10016

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 76-152570 ISBN NO.: 0-8247-1556-X

PRINTED IN THE UNITED STATES OF AMERICA

FOREWORD TO THE RUSSIAN EDITION

The last ten to twelve years have been a period of extremely rapid development for the theory of extremal problems and for methods of solving such problems.

A large

number of problems, interesting from a theoretical standpoint and important from a practical standpoint, has attracted the attention of many mathematicians and engineers. And this is not surprising.

It is now difficult to name any field of

knowledge in which, in one form or another, extremal problems do not arise and in which it is not essential for the development of these fields that such problems be solved. Among such fields are automatic control theory, economics, and even biology.

Each of these sciences brings forth its own

extremal problems and awaits the answers to two questions: What is the qualitative character of solution, and how does one find a solution? The first of these questions - what is the qualitative character of a solution - urges mathematicians to look for the

v

Foreword most complete necessary conditions for an extremum, since it is precisely such conditions which permit us to foretell the general structure of a solution.

The chain-reaction charac-

ter of the stream of newly arising problems has begun to indicate clearly that we are indeed in need of general conditions for an extremum, i.e. , of conditions of a type which may be applied to a broad class of problems.

In this way, we can

eliminate the necessity of developing a new theory for every concrete case.

On the other hand, the particular problems

that had already been solved provided a basis of confidence that such conditions could be formulated.

Moreover,

they

showed that such conditions would not be too distant from concrete problems - so that the application of such conditions to a given problem would resemble an occupation of previously prepared positions rather than an assault on a fortress. Extremum problems are not new in mathematics. They have been encountered and solved during the entire history of mathematics.

But an intensive and systematic investigation

of such problems has been begun comparatively recently when, on the one hand, the demands of economics and automatic control made the solving of such problems an urgent matter, and, on the other hand, the appearance of electronic

vi

Foreword computers provided researchers with a powerful tool with whose aid problems could be solved to the point of obtaining a final numerical result.

If we do not now speak of the

Calculus of Variations and of problems of minimizing functions under equality type constraints, i.e., of problems for which necessary conditions for an extremum were obtained a long time ago, then the beginning of the new stage of development of extremal problem theory can be dated to 1939.

In

this year, the Soviet mathematician L. V. Kantorovich created methods for solving a new class of problems - linear programming problems.

After this, a theory for linear pro-

gram.ming was widely developed in the works of G. Dantzig and of many other authors, both abroad and in the U.S.S.R. The next stage in the development of necessary conditions for an extremum was the elaboration of convex programming theory.

A central place in this theory is held by

the Kuhn-Tucker theorem, which gives necessary conditions for an extremum, and which was the source for a number of algorithms.

The differential form of the Kuhn-Tucker theo-

rem can also be applied to non-convex programming problems in a finite-dimensional space, and makes it possible to formulate necessary conditions for an extremum in such

vii

Foreword problems. In elaborating on the necessary conditions for linear and convex programming problems, the principle which lies at the heart of all of the constructions was clarified.

This

principle was summed up by G. Zoutendijk in his monograph, "Methods of Feasible Directions".

The essence of this prin-

ciple consists in the entirely obvious fact that if, at a given point, there exists a direction which does not lead out of the admissible domain and along which the objective function decreases, then such a point cannot be a mininum point.

On

the basis of this principle, necessary conditions for an extremum were constructed for a broad class of problems with smooth constraints in a finite-dimensional space. At the same time that a theory of finite-dimensional extremal problems, i.e., of mathematical programming, was being developed, another class of problems, namely, optimal control problems, was being investigated.

The crucial

step in this investigation was the formulation of necessary conditions for an extremum in the form of the Pontryagin maximum principle.

There is no need to dwell in detail on

the importance of this formulation to the entire theory of automatic control, and on the large number of works which it

viii

Foreword

brought forth.

Taking into account the direction which we

shall take in this book, we should, in the first place, note that the proof of the maximum principle, which is due to V. G. Boltyanskii, was, to some extent, a sensation, since it made use of methods for which it is difficult to find an analog in the previously developed theory of mathematical programming.

In this connection, the following question arose: Is it

possible to prove the maximum principle with the aid of ideas and methods from the classical calculus of variations and mathematical programming?

An affirmative answer to the

preceding question would have not only a purely esthetic value, but would also have another, practical side.

Namely, it would

allow us to apply to optimal control problems the computational methods which have been developed for mathematical programming. The embedding of optimal control theory into a general theory of necessary conditions was first carried out by A. A. Milyutin and A. Y. Dubovitskii.

The great importance of

their work lies in the fact that they succeeded in formulating, in a refined form, necessary conditions for an extremum which can be applied to a broad class of problems.

More-

over, their work made clear which part of the proof of the

ix

Foreword maximum principle could be put in a general framework, and which part could be attributed to the specific character of the optimal control problems, i.e., to the presence of the ordinary differential equation constraints.

The specific charac-

ter of the constraints in the form of ordinary differential equations was most fully reflected in the works of R. V. Gamkrelidze, who formulated the ideas of sliding regimes and of quasi-convex sets,

Based on the notion of a quasi-

convex set, R. V. Gamkrelidze gave a new proof of the Pontryagin maximum principle which clearly distinguished between the general variational problem and the specific character of the differential constraints, The work in optimal control theory had an extremely rich influence on a general theory of necessary conditions for an extremum.

Indeed, this work made possible a revela-

tion of the underlying principles, and also made it possible to develop techniques for constructing necessary conditions and to look upon a broad class of problems from a unified viewpoint.

Moreover, a profound and systematic study per-

mitted one to construct necessary conditions for problems containing functions which are not differentiable in the usual sense.

In terms of directionally differentiable functions,

x

Foreword H. Halkin and L. W. Neustadt formulated a very general theorem concerning necessary conditions.

This theorem can

be applied for solving a broad class of problems, including optimal control problems. It is interesting to note that the proof both of the basic

theorems of A. A. Milyutin and A. Y. Dubovitskii and of the theorems of H. Halkin and L. W. Neustadt require tools which have been known in mathematics for a considerable time.

And the fact that these results have only been obtained

in the last five to six years points up the large amount of work which was carried out with regard to understanding some of the underlying principles and elaborating on some of the basic concepts. This book is devoted to a presentation of a theory of necessary conditions for an extremum.

Our method of pre-

sentation is deductive, i.e., first, general results are stated, and then we show how these results can be particularized to specific problems.

This manner of presentation seems to be

justified at the present time, since there is a great number of works devoted to the derivation of necessary conditions for specific problems, and these works fully paved the way for a development of an abstract presentation.

xi

Further, in the

Foreword

theory of necessary conditions, two parts can be earmarked. These parts may, somewhat conditionally, be named as follows:

Formal Conditions for an Extremum and Computa-

tional Methods. Formal Conditions for an Extremum are presented in this book in Chapter IV.

They consist of a collection of

theorems which assert that, if the function being minimized as well as the domain over which the minimization takes place satisfy certain hypotheses, then a certain relation holds at a minimum point.

But these theorems do not tell us

how to write down concretely, for a given problem, this relation.

To construct a useful relation, it is necessary to

develop an apparatus of techniques for evaluating certain quantities.

In order to make these general statements under-

standable, we shall explain them for the special case of the problem of finding the minimum of a function of a single variable. In order that the minimum be achieved at some point, it is necessary that the derivative of the function at this point be zero.

In the terminology which we have presented, this

is equivalent to a formal condition for an extremum.

But if

methods for evaluating derivatives of quite complicated

xii

Foreword functions had not been developed, then it would have been impossible to write down the stated condition in a useful form for any realistic problem.

The first three chapters of

this book are devoted to an investigation of techniques for evaluating certain quantities.

Only after these techniques

have been sufficiently developed, are formal conditions for an extremum presented. Any general theory is of value only to the extent that it permits one to look upon a sufficiently broad class of problems from a unified viewpoint.

Therefore, the relatively

large fifth chapter is devoted to an illustration of how the constructed theory can be applied to specific problems. Each of the problems which we shall consider is far from trivial, and a quite large number of works has been devoted to investigating them.

Some of the problems which we shall investi-

gate, for example, the Chebyshev approximation problem and the moment problem, have by themselves a very general character and numerous applications in Economics and Optimal Control Theory. In Chapter V, we consider only those optimal control problems for which a result can be obtained without an investigation of the special features introduced by the differential

xiii

Foreword equations constraints,

This is because a consideration of

these constraints would slightly l~ad us away from the general direction of the book, To make the presentation complete, we state, in the introduction, some basic facts from Functional Analysis and from the theory of convex sets which we shall use in our presentation.

We have not confined ourselves to finite-di-

mensional spaces because to refuse to present material which makes use of more general spaces would have resulted in a considerable impoverishment in the material which we could present.

Also, it would then have been impossible to treat a

number of problems for which, as a matter of fact, the complicated theory had been constructed, Therefore, the reader who is not too familiar with Functional Analysis may acquaint himself, in the introduction, with all of the facts which are necessary to understand the remainder of the book, all the more since there are very few such facts.

The reader who is familiar with the basic results

of Functional Analysis can begin his reading with the first chapter, It is also necessary to inake a remark on the method

of presenting references to the literature which we adopted

xiv

Foreword

in this book.

In the main presentation, we make only the

most necessary references regarding results which are used, but not proved, in the book. On the other hand, at the end of the book, in the short bibliography, we present references which indicate in which works certain theorems were proved, what relation some given result has with others, etc. This book has been written on the basis of a series of lectures which the author gave at the Second All- Union School on Optimization Methods, in the city of Shemakha, July 6 -26,

196 7.

The author is sincerely grateful to the chairman of the

Organizing Committee of the School, corresponding member of the Academy of Sciences of the USSR, N. N. Moiseev, for his invitation to read these lectures, and also for numerous fruitful discussions, and for his attention to this work. I also consider it my pleasant duty to express my acnowledgement to my colleagues in the Institute of Cybernetics of the Academy of Sciences of the Ukrainian SSR, whose help with the work on this book can hardly be overestimated.

B. N. Pshenichnyi

xv

FOREWORD TO THE AMERICAN EDITION

The theory of necessary conditions for an extremum is a field of mathematics for which the greatest contribution was made by Soviet and American scientists.

Thus, I am

very pleased that a translation of my small book will be published in the USA.

I hope that it will be conducive to mutual

understanding and friendly contacts between Soviet and American scientists. I have added to the present edition a small bibliography of works with which I became acquainted after I had written the main text of the book.

B. N. Pshenichnyi

Note to the reader.

The symbol 111 will be used to denote the end of a proof.

xvi

TABLE OF CONTENTS

FOREWORD TO THE RUSSIAN EDITION FOREWORD TO THE AMERICAN EDITION

v

xvi

INTRODUCTION. ELEMENTS OF FUNCTIONAL ANALYSIS AND CONVEX SETS 1. Some Basic Concepts of Functional Analysis 2. Convex Sets 3. Convex Functionals

1 22 35

CHAPTER 1. PROPERTIES OF CONVEX FUNCTIONALS

39

CHAPTER II. CONVEX PROGRAMMING IN BANACH SPACES

54

CHAPTER III. QUASI-DIFFERENTIABLE FUNCTIONALS

68

CHAPTER IV. NECESSARY CONDITIONS FOR AN EXTREMUM IN GENERAL MATHEMATICAL PROGRAMMING PROBLEMS

82

CHAPTER V. NECESSARY CONDITIONS FOR AN EXTREMUM IN CONCRETE PROBLEMS

1. 2. 3. 4. 5. 6.

The Classical Mathematical Programming Problem Mathematical Programming with a Continuum of Constraints Theorems for Minimax Problems Chebyshev Approximation Problems A Linear Optimal Control Problem with Phase Constraints A Duality Principle in Convex Programming

xvii

120

121 122 126 130 142 150

Table of Contents 7. 8. 9.

Systems of Convex Inequalities. Helly' s Theorem The Moment Problem A Discrete Maximum Principle

157 166 185

SHORT BIBLIOGRAPHY

201

LITERATURE

210

NOTES AND SUPPLEMENTARY BIBLIOGRAPHY TO AMERICAN EDITION

217

SUPPLEMENTARY LITERATURE

223

SUBJECT INDEX

226

xviii

IN TROD UC TION ELEMENTS OF FUNCTIONAL ANALYSIS AND CONVEX SETS

Functional analysis is the mathematical apparatus on which the construction of the theory of necessary conditions for minimization problems is based.

As a matter of fact, only

a few basic concepts and a few theorems are used to construct the theory.

These concepts are, first of all, the ideas of weak

convergence, compactness, and a separation theorem for convex sets.

For the reader's convenience, we shall briefly, and

without proofs, state those basic facts of Functional Analysis which are necessary for an understanding of the subsequent material.

We shall take them from [ 1 ].

Incidentally, the

majority of the theorems stated in the sequel - with the exception of a few basic ones - are immediate consequences of the definitions, and may be proved by the reader himself, if he wishes to test whether he correctly understands the introduced definitions. 1.

Some basic concepts of Functional Analysis

Elements of Functional Analysis and Convex Sets

Definition 1.

A family

topology on this set if

T

T

of subsets of a set X forms a

contains the empty set(/), the set X

itself, any union (of any number) and any intersection of a finite number of its sets. logical space.

The pair (X, T) is said to be a topo-

The sets of

T

are said to be open sets.

Any

open set which contains a point p is said to be a neighborhood of p. A.

A neighborhood

of~

set A is any open set which contains

A point pEA is said to be an interior point of A if there

exists a neighborhood of p entirely contained in A. The following Lemma is an obvious consequence of Definition 1. Lemma 1. only if it

A set

contains~

Definition 2.

T,

topological space is open

i.£

and

neighborhood of each of its points. A family fl of subsets of X is said to be

a basis of the topology family

in~

T

if every set of fl is contained in the

and if every set in

T

is the union of sets from the

family fl. In order that a family fl be a basis of some topology, it is necessary and sufficient that, for every pair of sets U, VE13, and every xEunv, there exists a set WE13 such that xEWcunv and, furthermore, that the union of all sets from fl coincides with X.

2

Elements of Functional Analysis and Convex Sets

If a basis

~

is given, then the topology

sets which are arbitrary unions of sets

in~-

T

consists of all

For example,

the usual topology on an n-dimens ional space can be given by means of the family

~which

consists of the sets defined by the

inequalities n 2 2 6(x.-y.) x':' (x ) 1

-

1

0

E

then

+E.

Let us show that x':' (x) >0 1 -

for all x EK.

Indeed, if xf (x 1 ) < 0 for some x 1 EK, then,

since A.x 1 EK for all A.>0,

as A.-+oo.

Atthe same time, since A.x 1 EK, it follows from the

construction of

xy that

x*1 ( A.x ) > x* (x ) + E. 1 1 0 The contradiction which has just been obtained shows that

31

Elements of Functional Analysis and Convex Sets

x':'(x) > 0 1

for all x EK, and

xl~

-

is a non-zero functional belonging to the

2) xEK if and only if

x':'(x) :::_ 0 for all x':'E K':'. 3) If x is an interior point of K, then x':'(x) > 0 for all

non-zero

x~'E

Proof.

K. 1) Since B is a normed space, the closure K

of a cone K consists of the points in K and of the points x such that there exists a sequence x EK with x -+x .

n

n

o

0

Thus,

if x':'E K'~, x':'(x ) > 0 for all n, n

-

and, by the continuity of x':', x':'(x ) > 0 0

for any x EK. 0

K':'c(K)'~.

-

It follows from this that x':'E (K)':'.

Thus,

On the other hand, since K:::::iK, by definition of a

2) If x EK, then by the definition of

But we have just shown that K 0~

= (K)~'.

32

(K/',

Therefore, the

Elements of Functional Analysis and Convex Sets desired inequality holds if xE K. x~'(x

for all x*E K':'.

0

be such that

) > 0

0

Suppose that x

Now let x

-

0

f:. K. Then, as was shown in

proving that K'~ contains non-zero elements if

Kt

B, there

ex is ts a functional xr EK* such that

for some

E

>0 and for all xEKo

But, since AxEK for any xEK

and A> 0, then, by letting A tend to zero, we obtain that OE K. Substituting 0 into the preceding inequality, we obtain - E

> x* 1 (x 0 ) •

-

This contradicts the fact that x~ EK"" and that x*(x 0

)

> 0 for

all x*E K*. 3) Let x

0

be an interior point of K; i.e., suppose

that, for some r > 0, [ x:

II x-x 0 I\

< r} c K.

* *

For any non-zero x EK ,

for all xEK, and, in particular, for all x satisfying (3). By definition of the norm of a functional, \lx*I\

=

sup

\!xii~ i

33

x':'(x) ,

(3)

Elements of Functional Analysis and Convex Sets there exists an element e with

II ell::: 1

such that

Let us consider the point

This point satisfies (3) because

Therefore, x 1 EK and

or x*(x ) > .!:. x':'( e) > .!:. llx'~\\ > 0, 0 2 - 4 as was to be proved. 111 Lemma 5. weak'~

If K is

~convex cone, then the cone K':' is

closed. Proof.

By definition of a closed set, it is necessary

to show that the complement of K

* is

open.

For this purpose,

it is sufficient, by Lemma 1 of the introduction, to show that, if x* !/:. K~', then there is a neighborhood of x~' in the weak'~ 0

0

topology of B* which does not have any points in common with

Let x* !/:. K*. 0

Then, by definition of K*, there exists

34

Elements of Functional Analysis and Convex Sets an x EK such that 0

x*(x ) = a.< 0. 0

0

Let M= {x*:Za.

0

oe

is an arbitrary element of Z(x ),

lim A.-++ 0

0

µ(x (A.)) - µ(x ) A. o >

sup a.E Z(x )

ocp (x 'a.) 0

oe

( 3. 3)

0

Now suppose that cp (x, a.) is differentiable in the direction e at the point x , i.e. , that, for A.> 0, 0

cp(x + A.e, a.) o

= cp(x o , a.)

+A.

ocp(x ,a.) 0 ._, +A. y(A., a.) ue

and, in addition, that y(A., a.)-+ 0 as A.-++ 0 uniformly with

73

III. Quasi-Differentiable Functionals

Then the directional derivative

respect to a..

ocp(x ,a.) 0

(le

, as

the uniform limit of the continuous functions cp(x

0

+ )..e,a.)-

oqi(x ,a.)

qi(x , a.) 0

0

inf sup w:::>Z(x ) a.E w

and

oe

0

oqi (x • a.) 0

sup a.E Z(x )

oe

0

is continuous in a.. By virtue of Lemma 3. 2, we obtain from ( 3. 2) that µ(x (A.)) - µ(x ) A.

0

< sup [ -

oqi(x • a.) 0

oe

a.Ew

+ y(A.,

for any w:::>Z(x ) and for sufficiently small)...

+ 0,

J

Passing to the

0

limit as )..-+

a.)-1

we conclude that µ(x (A.)) - µ(x ) 0

lim A.-+ +o

oqi(x ,a.) :'.::_ sup a.E w

0

(le

Because w is an arbitrary neighborhood of Z(x ), we finally 0

obtain lim A.-+ +o

µ(x(A.)) - µ(x ) 0


Z(x ) a.Ew

oql (x • a.) 0

Cle

0

oqi(x ,a.) 0

sup a.E Z(x )

(le

(3. 4)

0

A comparison of (3. 3) and (3. 4) leads us to the following

74

m.

Quasi-Differentiable Functionals

theorem. Theorem 3. 2.

Let cp (x, a.)

be~

functional which is

continuous in x and a., where xEB and a.E Z (Z

is~

compact

topological space). Moreover, let

cp(x + A.e,a.) = cp(x ,a.)+ A. o

o

ocp (x • a.) 0 .,. + A.y(x,a.) for A. >0, ue

where y(A., a.)-+ 0 uniformly in a. as A.-++ 0.

Then, at the point

x , the functional 0

µ(x)

max cp(x, a.) a.E Z

is differentiable in the direction e, and oµ(x ) 0

oe

sup a.E Z(x )

ocp(x ,a.) 0

oe

(3. 5)

0

Remark 1.

It is easy to see that the hypothesis that cp

is continuous in x and a. can, without any harm, be replaced by the hypothesis that the function cp(x + A.e, a.) is continuous 0

in A. and a.. ··Remark 2.

If cp (x, a.) is a functional which is convex

ocp (x • a.) 0-in x for each fixed a., then the theorem holds if - - oe depends continuously on a.. Indeed, a convex functional is always directionally

75

ID. Quasi-Differentiable Functionals differentiable, and the quotient (for fixed a.) qi(x + \e,a.)- qi(x ,a.) 0

0

A. tends to

oqi(x 'a.) 0

,

oe

decreasing monotonically [6] (also see

Lemma 6 in the introduction).

But then, by Dini' s theorem

[ 45], y(\, a.) tends to zero uniformly with respect to a.. Theorem 3. 3.

If qi(x + \e, a.) is a functional which is

-

0

--

----

continuous in A. and a., and, moreover, if the differential oqJ (x + \e, a.) 0 --0-A.--- exists and is continuous in A.E [O, l] and a.E Z,

then oµ(x ) 0

oe

max a.E Z(x )

ocp(x, a.) oe

(3. 6)

0

This may be proved analogously to the way in which we proved the preceding theorem, with one exception:

in

order to establish Inequality (3. 4), it is necessary to employ the mean-value theorem of the differential calculus to obtain the formula qJ

where

(x + \e, a.) - qi(x , a.) 0

0:::. s(A.):::.

0

oqi(x + s(A.)e, a.) 0

a\

A., and to note that oqi (x , a.)

oqi(x 0 +A.e.a.)I

o\

0

A.= 0

76

oe

III. Theorem 3. 4.

Quasi-Differentiable Functionals

If, for every a., cp (x, a.) is

which is quasi-differentiable at x 0

,

~functional

and if the hypotheses of

Theorem 3. 2 are satisfied for every e, then also µ(x) functional which is quasi-differentiable at x 0

=

M(x ) 0

co

•

is~

Moreover,

M(x ,a.),

U

a.E Z(x )

(3. 7)

0

0

where M(x 0 , a.) is the set of support functionals to cp(x, a.) at x0

,

M(x 0

)

is the set of support functionals to µ(x) at x 0

,

and

co A denotes the weak* closure of the convex hull of A.

On the basis of Theorem 3. 2 and the defini-

Proof.

tion of a quasi-differentiable functional, we obtain oµ(x ) 0

oe

sup a.E Z(x )

ocp(x ,a.) 0

sup a.E Z(x )

oe

0

0

sup x~'EM(x

0

, a.)

x* (e).

(3. 8)

But, as is easily seen, the maximum in the right-hand side of this equation equals the maximum of x* ( e) as x* ranges over the set M(x ) defined by Formula (3. 7). 0

is the

weak~'

Indeed, M(x ) 0

closure of the set

[x*: x*=L::A..x~, 11

L:A..= 1

l, x~'EM(x ,a..) and A..> 0 for all i}. 1

0

1

1-

Therefore, x*(e) = L::A..x~(e) 0

i=l

0

1

1

(4. 8)

-

for all eEKM. Lemma 4. 2.

If the sets M.(x ) are bounded, then,

------

l

0

--

---

for Condition (4. 8) to be satisfied, it is necessary and sufficient that there exist functionals x'.i'EM.(x ) such that

--- --- - - - ---

l

89

l

0

-- --

IV. Necessary Conditions for Extremum in Mathematical Programming k

.

"" LJ

Proof.

*

*

A.X. EK.- ,

l=-m

l

(4. 9)

--M

l

Consider the set N*

={ x * :

x *·

k_ = L:.1--m A..x.* , l l

x~ EM. (x ), i < 0} • Obviously, this set is convex, since l

l

0

-

the sets M. (x ) are. l

0

Moreover, it is weak* closed and weak*

compact since, by assumption (see Definition 3, 1), the M. (x ) are weak* closed and bounded and, thus, weak* l

0

compact.

Let us suppose that (4. 9) does not hold, i.e., that

N* and ~ have an empty intersection. regularly convex, because

Then

KM:- N * ls. ' ~~

~ is weak* closed, N* is weak*

closed and compact, and, thus, ~- N* is also weak* closed. Since ~and N* have no points in common, ~- N* does not contain the zero functional.

Therefore, there exists an

eE B such that (4. 10) for all y~'E ~ and x'~E N*. The inequality which has just been obtained shows that y':'(e) is bounded for all y"~E ~·

But,

since ~ is a cone, inf

y*E~

*

y (e)

= 0 •

This implies that eE ~· Furthermore, a comparison of (4.10) and (4. 11)

90

(4. 11)

IV. Necessary Conditions for Extremum in Mathematical Programming

o for

shows that x* (e) ::::_ -

all x*EN*.

But eE KM and (4. 8) imply that there exists an x*EN* such that

o

x*(e) :::_

x*EN*.

2 '

Thus, we have obtained a contradiction.

This means

that N* and K~ have a non-empty intersection and, therefore, (4. 9) holds.

The sufficiency is obvious.

Let cp.(x), for i = -m, ... , 0, ... , k, be

Theorem 4, 2. functionals

on~

Ill

--1

--

Banach space.

--

If cpi(x)

satisfies~

Lipschitz

condition and is quasi-differentiable for each i ::::_ 0, and satisfies~ Lipschitz condition and has~ Gateaux differential

x7' for each i > 0, then, in order that x 1 -----

---

-------

be a solution to o--

Problem (4.1), it is necessary that there exist numbers A..

1

.

k ~

i=-m

A..x7=EKM* , 1

1

A. > 0 for i < 0 and A. cp.(x ) = 0 for i i-

-

1

1

0

# 0.

The theorem actually follows from the preceding arguments.

It is only necessary to prove some details.

First, if, for each i > 0, cp.(x) satisfies a Lipschitz 1

condition, then

91

IV. Necessary Conditions for Extremum in Mathematical Programming cp.(x(A.)) - cp.(x ) 1

lim A. .... + 0

1

0

x'!' (e),

A.

1

where x(A.) is given by formula (4. 2). The proof of this fact is trivial.

From this it follows

that Condition 3 of the basic assumptions is satisfied. Second, let us show that M.(x ) is bounded, as is 1

required in Lemma 4. 2.

0

If we assume the contrary, then,

for every integer n > 0, there exist an element e lie

n

II=

n

with

1 and a functional x>: n n

for some

E

cp.(x i

o

> 0.

E,

Thus,

+ A.e n ) A.

n

- cp.(x ) i o

sup x*(e ) x*EM.(x ) n 1

+ o(\) >

A.(n - E)

+ o(\).

0

On the other hand, by virtue of the Lipschitz condition, lcp.(x 1

o

+ A.e n )-

cp.(x ll L, then the last inequality cannot hold for suffi-

ciently small A..

The contradiction which we have just

obtained shows that M.(x ) is bounded. l

0

92

This completes the

IV. Necessary Conditions for Extremum in Mathematical Programming

\\I

proof of the theorem.

Theorem 4. 3 (Kuhn-Tucker).

Let cp.(x) be convex 1

-

bounded functionals on B, and, moreover, suppose that cpi(x) is linear for each i > 0. vex.

Also suppose that the set M is con-

Then, in order that x

---

------

be a solution to the minimization

o--

Problem ( 4. 1 ), it is necessary that there exist numbers A. , 1

with A.. > 0 for i < 0, such that i-

--

-

----

k

.

L:

l=-m

A..cp.(x) > 11

-.

k L;

i=-m

A..cp.(x) for allxEM. 110

( 4. 12)

Moreover, A.cp.(x) = 0 for i#O. 1

1

( 4. 13)

0

If A. 0 > 0, then the conditions of the theorem are sufficient. Proof. K

M

We define the cone KM as follows:

= {e: e= A.(x-x )forxEM, x#x, A.>0}. o o

It is easy to see that if eE KM, then

x(A.) for small A.

x + A.eEM 0

Further, cp.(x + A.e) - cp (x )

ocp(x )

A.

oe

1

lim A.-+ +o

0

0

0

< cp.(x +e)-cp(x) =h.(e). -

1

0

0

93

1

IV. Necessary Conditions for Extremum in Mathematical Programming For i > 0, the inequality in an obvious way becomes an equality.

For i :::_ 0, this inequality follows from the fact that

the quotient c.p.(x 1

0

+ A.e)

-

('j)(X

0

)

A. is monotonically decreasing for decreasing A. (see Lemma 6 of the introduction).

If we take into account the corollary to

Theorem 4. 1, we can see that all of the hypotheses of Thus, there exist numbers A.,

Theorem 4. 1 are satisfied.

1

with A.. > 0 for i :::_ 0, such that i-

k

.

6

A..h.(e)

i=-m

1

1

for all eE KM. Setting he re e k

.

6

=x

- x , where xEM, we obtain 0

A. c.p.(x) >

i=-m

1

1

k

.

6

i=-m

A..c.p.(x ) 1

1

0

for all xEM. Now suppose that (4. 12) and (4. 13) are satisfied and that A.

0

> 0.

Then x

0

is a solution to Problem ( 4. 1 ).

Indeed,

for any x which satisfies all of the constraints, we have, by virtue of (4. 12) and (4. 13),

94

IV. Necessary Conditions for Extremum in Mathematical Programming k

.

2::

k

2:: \.l cp.(x) < l -

A..cp.(x ) < l

i=-m

l

0

i=-m

because cp.(x) < 0 for each i < 0 and cp.(x) l

-

l

A. cp (x) 0

0

0 for each i > 0.

Thus, cp (x ) < cp (x), 0

which implies that x

0

0

-

0

is a solution to Problem (4. 1).

Ill

Let us now establish the relationship between the results which have been obtained and the theory of Dubovitskii and Milyutin [10]. Suppose that a functional cp (x) is given on B, and that we wish to minimize cp on L subject to the constraints xEO., l

i = 1, ... , n. x0

,

Let x

0

be a minimum point.

Assume that, at

there exists, for every Oi, a convex cone Ki such that,

whenever eEK., l

(4. 14)

for all sufficiently small A.> 0 and all e' which satisfy the inequality lie' - ell < e: , where e: > 0 depends one. e e

More-

over, suppose that there exists a subspace Z tangent to L at x , i.e., for every eEL there exists a r( A.) such that 0

x(A.)

= x 0 + A.e + r(A.)EL

whenever A. is sufficiently small.

95

( 4. 15) as

IV. Necessary Conditions for Extremum in Mathematical Programming A.-t

+o. Further, suppose that cp (x) is such that there exists

a cone of "forbidden variations", i.e., a convex cone K such that (4. 14) holds for 0

0

0

0

0

, where

= [x: cp(x) < cp(x )}. 0

Now the first theorem of Milyutin and Dubovitskii is almost obvious. Theorem 4. 4.

In order that x be a solution to the ------ o--

problem min cp(x), xEO. for i=l, ... ,n, l

xEL

empty intersection. Proof.

Suppose that K , K 1 , ... , K and Z have a o n

non-empty intersection, i.e., that there exists an e such By definition of Z, there

that eEZ and eEK. for each i. l

exists a function r (A.) such that x= x

0

+ A.e + r(A.)EL

and

lim II r( 1..Jll A. )._-t 0

96

o.

IV. Necessary Conditions for Extremum in Mathematical Programming -Further, since eE K., 1

x

x+A.e'EO. 0

1

whenever A. is sufficiently small, and

II e' - ell


0

for

n eE n i=O

y*(e) < 0

for

e E Z •

y~'(e) 0 0

-

98

Ki']

(4.17)

IV. Necessary Conditions for Extremum in Mathematical Programming We set

max x* (x) for i x*EM.

0, 1, ••• , n ,

l

where M.= (-K~)ns*, ands* is the unit ball in B*, i.e., l

l

s* = [ x*: \Ix*\\_:: 1}. [x: µ.(x) < O} = l

-

It is not difficult to see that the

Kl.•

Indeed, this follows from the fact that xE K. if and l

only if

*

*

..

x (x) _::: 0 for all x E K~ • Now it is clear that the set

0 = [x: µ.(x) < 0 for i = O, ••• ,n} l

coincides with n~=O Ki.

-

Inequalities (4. 17) imply that the

functional µ 0 (x) achieves its minimum on 0 at x 0 = 0, We now note the following.

First, since all of the K. l

are open, their intersection is also open, and there exists an x 1 which is an interior point of n~-o K .• i-

l

But then µi(x 1 ) < 0 for each i = O, ••• ,n. Second, by virtue of Theorem 1. 6 of Chapter I, for each i > O, the set Mi(O) of support functionals to µi(x) at x 0 =o coincides with M .• l

99

IV. Necessary Conditions for Extremum in Mathematical Programming On the basis of Theorem 2.4, we can now assert that

there exist numbers A.. < 0 such that i-

Yo*

n

2::

==

i=O

*

A..y. l

l

preceding equation is transformed into X

But x*E x.~; l

z>:'

+ x*l + • • • + x':'n + x*

>:'

0

by virtue of (4. 17).

. ,. '" . < "' . y:", 11

1-

o,

= 0

Further, x::'E K~', since l

l

s*c - K~:::: an d y.~:;E ( - K*)n . .• l

l

l

The necessity of the conditions of the theorem has been proved. The sufficiency of Condition (4. 16) is easy to prove by contradiction. Indeed, suppose that (4. 16) is satisfied, and suppose that there exists an e 0 such that e 0 EKi for i such that e 0 E z.

= O, ••• ,n,

and

Then

by definition of the dual cone, and at least one number xt(e 0 ) is positive, because, for each i

= O, ••• ,n,

e 0 EK. and K. is l

l

open (and, thus, e 0 is an interior point of Ki), and, moreover, not all of the x'.:' are zero. l

100

IV. Necessary Conditions for Extremum in Mathematical Programming This implies that

which contradicts (4. 16). 111 Thus, we see that the preceding results imply the theorem of Dubovitskii and Milyutin.

This theorem yields

very general conditions for an extremum.

In particular, it

is in some respects more general than Theorem 4. 1, since it does not require the equality constraints (whose role is

played by the set L) to be given in the form of a finite system of functionals which must equal zero.

On the other hand,

Theorems 4. 4 and 4. 5 by themselves do not permit us to construct necessary conditions in the general case, because they leave open the question of how to effectively construct the cones and the subspace

z.

Our method of presentation in this work has been as follows:

First, we developed ways of evaluating directional

derivatives, or equivalently, of constructing the cones K .• l

Then we formulated the necessary conditions.

Because of

our success in the first task, and within the limits thereof, we could immediately write down these conditions in concrete form.

101

IV. Necessary Conditions for Extremum in Mathematical Programming We shall now consider a generalization of the problem studied at the beginning of this chapter,

This generalization

permits us to obtain, for some problems (e.g., the discrete optimal control problem), conditions for a minimum which are finer than those which are provided by Theorem 4. 1. Let L be a linear space, let X be a subset of L, and let Ube some arbitrary set.

Let there be given functionals

cp.(x,a)fori=-m, ••• ,-1,0, l, ••• ,k, onXXU, 1

Weare

seeking necessary conditions for the points x 0 , u 0 to be a solution to the problem min cp 0 (x, u) , cp. (x, u) < 0 for i < 0 , 1

-

cp. (x, u) 1

(4. 18)

= 0 for i > 0 ,

xEX ,

uE U •

We shall state some basic assumptions under which the problem will be solved. l,

If cp(x,u) = (cp_m(x,u), ••• ,cp0 (x,u), ••• ,cpk(x,u)),

then for every fixed x, the set cp (x, U) = [cp(x, u): uE U} is convex. 2,

There exists at x 0 a convex cone

eE KX implies that

102

~

such that

IV. Necessary Conditions for Extremum in Mathematical Programming k

L

x(A.)=x 0 +:>..e+

(4. 19)

r.(A.)a.EX j= 1 J J

for sufficiently small A. > 0 and for any a.EL, and for any J

functions r. (A.) which satisfy the condition J r .(A.) J lim -:>..-A. .... +O

3.

0.

For i ::;_ 0 and for any uE U,

lim :>.. .... +o

cp.(x(A.),u) - cp.(x 0 ,u) l l A < hi (u, e) ,

where x(A.) is given by (4.19), and h.(u,e) is a functional l

which is convex in e and is such that h. (u, A.e) = A.h. (u, e) for l

l

every e and A. > O. 4.

For each i > 0, there exists a functional h. (u, e) l

which is linear in e and is such that cp. (x(A.), u) - cp. (x 0 , u)- A.h. (u, e) l

l

l

k

:2::

j=l

r.h. (u, a.) J l J

where x(A.) is given by x(A.) = x 0 + A.e

k

+I:

j=l

and r(z) is a function such that

103

r .a. , J J

(4. 20)

IV. Necessary Conditions for Extremum in Mathematical Programming

lim

r(z) z

z->O

Moreover, cp.(x 0 + A.e 1'..

0,

k

+ Z:

r.a.,u) is, for each i, a continuous j= 1 J J

function of A. and r .• J

Condition 4 can be interpreted in a different way as follows,

If we substitute Expression (4. 20) into cp(x, u), then

we obtain a function of a finite number of variables:

A. and r .• J

Condition 4 then means that this function is differentiable at x 0 , i, e., this function can be approximated in some neighborhood of x 0 by a linear function of A. and rj' where the error consists of terms of higher order than A. and r .• J

Let us now discuss what Condition 1 implies.

Indeed,

it follows fro= this condition that, if u.EU for each j=l, ... , n,

J

and if the numbers y. satisfy the condition J

n

L; y.=l,

j=l

Y. > 0 for each j, J-

J

then there exists a uEU such that

tp(x,

u)

n

=

L; y .tp(x, u.) ,

j= 1

J

J

(4.21)

Further, if u 0EU and iiEU, then there exists a u(x, A.) such that (4.22)

104

IV. Necessary Conditions for Extremum in Mathematical Programming for 0 O, l

minimum value for cp 0 (x, u), subject to the given constraints. The contradiction which we have just obtained shows that co K and P have an empty intersection.

But co K and P

are convex sets in an (m+k+ !)-dimensional space.

There-

fore, there exist constants A.., i= -m, ••• , k, not all zero, l

such that

k

I:

i=-m

s

Ai i > 0 >

k

_L:

i=-m

Ai (i for all sE co Kand

c E P.

The right-hand side of this inequality implies that

A.. > 0 for each i < 1 -

o.

The left-hand side yields, by defini-

tion, that

109

IV. Necessary Conditions for Extremum in Mathematical Programming k

.

L:

i=-m

for all eE

A..[h.(u 0 ,e) l

~

l

+[

cp.(x 0 ,u) - cp.(x 0 ,u 0 )]] > l l -

o

and uE U.

But since e and u vary independently, the preceding inequality is equivalent to the following two relations: k

L

i=-m

A.h. (u 0 , e) > 0 l l fo, all k oE I' J , =h.(u,e)+'-' -,-h.(u,a.) fl. l J fl. l . 1 J=

.... h.(u, e), l

if r.(A.)A. J

-1

.... 0 as A. .... 0.

(4. 3 0)

Thus, if Condition 4 of Theorem 4.6

holds, then Condition 3 of Theorem 4.1 also holds. Conversely, suppose that Condition 3 is satisfied.

We

shall show that Condition 4 of Theorem 4. 6 is then also satisfied.

In order to prove this, we assume the contrary, i.e.,

we assume that there exist sequences (A.

n

l and

n n [ r n }, ••• , ( rk} suchthatA.n ... Oandrj .... Oasn""'"'• 1

e:

n

j

1

~

A.2 + (r1'.1)2 n j=l J

k

- cp.(x 0 ,u) - A. h.(u,e ) n1 n 1 does not tend to zero,

J= 1

I e: n I -> o for 112

L: .

r1'.1 h.(u,a.)J Ji J

all n.

Let

But

IV. Necessary Conditions for Extremum in Mathematical Programming

A. n -= yn A.' n

k

+ L; (r~)2 J

j= 1

n r. _J_ A.' n

n w. J

Since, by definition,

Jy:

+

j~l (w~/

I '

we can choose a convergent subsequence of the sequence of n

n

vectors [ yn,w 1 , ... ,wk}.

Thus, we shall assume that

(4. 31)

and set

k

e'

Now E

n

E

n

yoe

0 w. a. J J

+ L:

j=l

can be written in the following form: {

q>. (x

iO

+

y )e + A'n e' + A'n(yn-O

k

) - qi.(x ,u) } (\') -1 0 n .._, LJ A.'(w.- w.)a.,u 0 n i J J j=l n J

+

- h. (ue 1 ) l

(y - y 0 )h. (u, e) n

i

k

+ L: (w~ j=l

J

0 w. )h. (u,a.). J i J

Condition 3 of Theorem 4.1 and (4.31) imply that, as n ""'"', i.e., as \

1

n

-o,

tends to h.(u,e'), and i

the first term of the formula for E

n

itself tends to zero.

contradicts the assumption that IE

113

n

I -> o.

E

But this This proves

n

IV. Necessary Conditions for Extremum in Mathematical Programming that Condition 3 of Theorem 4. 1 implies Condition 4 of Theorem 4. 6. In concluding this chapter, let us give a proof of the theorem on the solvability of a system of non-linear equations, which has been frequently referred to. Theorem 4. 7.

Let x be

let ¢i (A.,x), for each i,

be~

.. 2 + (T(A.)-:>.. 2 >2 ),::::Kr(HIT-:>.. 2

1), (4. 3 5)

where we have made use of the fact that r(z) is a nondecreasing function and that

for positive A. and w.

Further, since T(A.),::;: A. for small A.,

the right-hand side of (4,35) can be estimated as follows:

Thus, we obtain 2

T(A.) - A. _:::: Kr(2A.) , or

T(A.) < 2K r(2A.) +A. A. 2 A. '

117

IV. Necessary Conditions for Extremum in Mathematical Programming i. e.

I

T (A_) ..., 0 -A.-

as A. .... 0.

But then also

Thus, Inequality (4. 34) holds for sufficiently small A.. Now if

then, according to (4. 33),

This implies that the continuous mapping g(A.,x) maps the ball

into itself.

By virtue of the Brouwer fixed point theorem [ 1 ],

we can now assert that g(A.,x) has a fixed point, i.e., there exists a point x(A.) such that x(A.)

= g(A.,x(A.)),

\\x(A.)\\ ::;_ T':'(A_). But the definition of g (A., x) implies that 1(r(A. ,x(A.))

= 0,

i.e., the set of nonlinear equations under consideration has a solution.

Moreover,

118

IV. Necessary Conditions for Extremum in Mathematical Programming \\x(A.)\\ A.


O.

Ill

x(A.)\\ A.

_, 0

119

CHAPTER V NECESSARY CONDITIONS FOR AN EXTREMUM IN CONCRETE PROBLEMS

In this chapter, we shall illustrate how the general theory which has been developed in the preceding chapters may be applied to various extremal problems.

The power of

any general theory lies in the fact that it allows one to consider in a uniform way various particular problems and to obtain complete results by a single method. A number of the problems which we shall consider in the sequel has been considered previously.

A number of

works has been devoted to these problems.

As a rule, to

investigate each of these problems, some specific method has been used, with each method applicable just for solving the given, narrow problem.

We shall show that we can apply

the general theory which has been developed to every problem which we shall consider in the sequel, and that we can at once obtain, without any additional, lengthy reasoning, the complete results.

These results are often even more general

120

V.

Necessary Conditions for an Extremum in Concrete Problems

than those obtained by special methods.

1.

The classical mathematical programming

problem.

Let there be given continuously differentiable

functions

f. (x), i = -m, ••• , k, on an n-dimensional space. l

We are looking for conditions which a point x 0 that is a solution to the following problem: min f 0 (x), f.(x) < 0 for i

-m, ••• ,-1,

f. (x) = 0 for i

1, ••• 'k.

l

l

must satisfy. Since the functions f. (x) are differentiable, they belong 1

to the class of quasi-differentiable functions.

Moreover, for

these functions, the set Mi (x 0 ) consists of a single vector which coincides with the gradient of fi (x) at x 0 • this gradient by

ox f.l (x 0 ).

We denote

Indeed, by a known theorem of

analysis [ 44 J:

of.l

-.,.- = (o f. oe

x

l

(xo>·

e) •

It is easy to see that the conditions under which we can

apply Theorem 4. 2 are satisfied, where M is the entire space.

Thus, ~is also the entire space, and~ = [O}.

Now Theorem 4. 2 implies that there exist numbers

121

V.

Necessary Conditions for an Extremum in Concrete Problems

A.., with A.. > 0 for i _< O, such that l

l

-

k

L; A. i=-m

.o

i

f. (x 0 )

O,

xi

A.f. (x 0 ) l l

"f o.

0 for

We have obtained the well-known Lagrange multiplier rule, 2.

Mathematical programming with a_ continuum of

constraints.

We are to find the minimum of a continuously

differentiable function µ(x) of an n-dimensional argument x, subject to the constraints µ(a,x) < 0 for

aEO,

where µ(a,x) is a function which is continuous with respect to a and x and which has a continuous gradient

ox µ(a,x).

Here, O is a compact set. Let us introduce the function µ -1 (x)

max µ(a,x). aEO

Then the just formulated problem is equivalent to the problem min µ(x). µ_ 1 (x)_::::O According to Theorem 3. 3, µ _ 1 (x) is a quasi-

122

V.

Necessary Conditions for an Extremum in Concrete Problems

differentiable function, and, according to Theorem 3. 4, M _ 1 (x 0 ) for this function is given by the formula

continuously differentiable with respect to x, M(x 0 , Q') = { oxµ(Q',x 0 )}, and we can write

Here, the closure bar has been omitted because the convex hull of a compact set in a finite-dimensional space is compact, and therefore closed, and the set

is compact because 0 (xo) is compact and axµ(Q',xo) depends continuously on Q'. On the basis of Theorem 4. 2, we can assert that there exist non-negative constants A. 0 and /.. _ 1 , not both zero, such that

123

V.

Necessary Conditions for an Extremum in Concrete Problems

n-dimensional vectors oxµ(a,x 0 ), where aEO (x 0 ).

This

implies that the vector c can be represented in the form

c

=

n+l

L

i=l

A.a l

x

µ(a.,x 0 ), l

:l.: i=l n + 1 '''i =

where, for each i, ai E 0 (x 0 ), and Ai~ 0, and

1.

Thus, we finally obtain n+l A0 o µ(x 0 ) + L; y.o µ(a.,x 0 ) x i=l l x l where y. = AlA. > 0 and l

l -

n+l

L 1._ 1

y. =A l

-

(5. 1)

0.

Let us formulate

1•

the result which we have just obtained, Theorem 5. 1.

In order that x 0 be

mathematical programming problem constraints,

i:E

'!.

solution to the

with~

continuum of

is necessary that there exist constants AO

z0

such that Equation (5. 1) holds and such that y.= 0 for every --- -- - - --

l

-

---

Now suppose that µ(x) and µ(a,x), for each fixed a, are functions which are convex in x, and that there exists an x 1 such thatµ _ 1 (x 1 ) < O.

Then the problem under considera-

tion becomes a convex programming problem. Applying Theorem 2. 4 to the case of a single

124

V. Necessary Conditions for an Extremum in Concrete Problems constraint, we obtain that there exist a non-negative number

A. _ 1 and a vector cEM _ 1 (x 0 ) such that

and this condition is both necessary and sufficient. Now, arguing in the same way as before, we obtain the following corollary. Corollary 1.

g

µ(x) and µ(a ,x), for each fixed a,

convex functionals, and µ_ 1 (x 1 ) < 0

for~

~

x 1 , then, in

order that x 0 be ::_solution to the problem with::_ continuum of constraints,

g

necessary and sufficient that there exist

~

Corollary 2.

Under the assumptions of Corollary 1,

in order that x 0 be..'.:. solution to the problem with::_ continu~ ~constraints, it~

necessary and sufficient that there

problem min µ(x), µ (a.,x) < 0 for l

-

i

= 1, ••• ,n+l.

}

(5. 2)

Indeed, relation (5.1) (with A. 0 = 1) is at the same time a necessary and sufficient condition (by virtue of Theorem

12 5

V.

Necessary Conditions for an Extremum in Concrete Problems

2. 4) for x 0 to be a solution to Problem (5. 2). 111 Thus, Corollary 2 shows that, in the convex case, the problem with a continuum of constraints can be reduced to another, specially chosen problem which has at most (n+l) cons train ts. 3.

Theorems for minimax problems.

Let the re be

given a convex, compact set Min a space Band a convex, bounded, weak* closed set M* in the space B*.

Let us intro-

duce the function cp(x) = max x*EM* x*(x). Let x 0 EM be a minimum point of cp(x) on M.

Then,

according to Theorem 2. 1, there exists an x~EM(x 0 ), where M(x 0 ) is the set of support functionals to cp(x) at x 0 , such that x*E r * , where

xo

f

XO

r

xo

is the cone defined by the relation

= [ e: x 0 + A.e EM for sufficiently small A. > 0}.

In particular, e = x-x 0 with xEM, belongs to this cone. Further, by Theorem 1. 6,

Thus, there exists a functional x~EM(x 0 ) such that

and

126

V.

Necessary Conditions for an Extremum in Concrete Problems

Therefore, the two preceding inequalities imply that

Theorem 5. 2.

!;! Mis

'=.convex, compact set in B, and

min max x*{x) xEM x':'EM* Proof.

max min x*{x). x~'EM* xEM

(5. 4)

The theorem is an immediate consequence of

{5.3), since (5.3) {see [ 5]) is equivalent to {5.4).111 Corollary 1.

If X and Y ~compact sets in En, k{x,y)

is a continuous function of the variables xEX and yE Y, and x* and Y ""·denote, respectively, the sets o_! all regular, positive measures µ on X and v on Y such that µ{X)

= v{Y) = 1,

then

r

min max k{x, y)µ{dx)v{dy) µEx* v E Y~' ,,

= max min f' k{x, y)µ {dx) v {dy) • vEY':'µEx*,, Proof.

Let B be the space of all continuous functionals

on Y, let M be the set of all functionals c{y) which can be represented in the form c {y)

=

J

k{x, y)µ {dx) ,

127

V.

Necessary Conditions for an Extremum in Concrete Problems

= Y*.

and let M'~

Then the corollary follows immediately

from Theorem 5. 2. 111 Corollary 2.

!!_X and Y

~~compact,

convex sets in

En, k(x, y) is~ continuous function ~the arguments xEX and yEY, and k is, in addition,

~

i!:_x and concave in y,

then min max k(x, y) xEX yE Y Proof.

max min k(x, y) • yEY xEX

On the basis of the preceding corollary to

Theorem 5. 2, we can assert that there exist measures µ 0 and v 0 , with µ 0 (x)

J

= v 0 (Y) = 1,

k(x,y)µ(dx)v 0 (dy) ;::

J

such that

k(x,y)µ 0 (dx)v 0 (dy)

;:: Jk(x, y)µ 0 (dx)v (dy).

Now set xµ

=

J

xµ (dx) and y \I

=

(5. 5)

Jyv(dy).

Then, since k(x, y) is convex in x and concave in y,

(5. 6)

J

k(x, y)µ(dx)v 0 (dy) ::;_

J

We shall show that

128

k(x, y

vo

)µ(dx).

( 5. 7)

V.

Necessary Conditions for an Extremum in Concrete Problems

(5. 8)

Indeed, on the basis of Inequality (5. 6), the right-hand side of the last equation cannot be greater than the left-hand side. Suppose that it is strictly less, so that

J

k(x, y)µ 0 (dx)v 0 (dy) >

J

k(x

µo

, y)v 0 (dy).

If we now choose the measure which is concentrated at the point x

µo

for µ in the left-hand side of (5. 5), we obtain a

contradiction. In exactly the same way it can be shown that

(5. 9) Indeed, since k(x, y) is concave in y,

Suppose that strict inequality holds.

It follows from (5. 5),

(5. 6), and (5. 8) that

J

k(x

µo

, y)v 0 (dy) >

-

J

k(x

µo

, y)v (dy).

Choosing for ·; the measure which is concentrated at y obtain a contradiction to our assumption.

129

\)0

, we

V.

Necessary Conditions for an Extremum in Concrete Problems Now, on the basis of (5. 5)-(5. 9), we obtain that

r k(x, y vo )µ(dx) -> k(x µo • y vo ) ->

"

If we choose forµ and

\I

Jk(x

µo

, y)v(dy).

measures which are concen-

trated at x and y, respectively, then we obtain k(x, y

) > k(x , y ) > k(x , y) vo µo v o µo

for all xEX and yE Y. 111 4.

Chebyshev approximation problems.

Let there be

given a function µ(x, a), where x ranges over some set O, and a ranges over a set En and that

a

a.

We suppose that 0 is a set in

is a compact set in Es.

The function µ(x,a)

is continuous and has a continuous gradient respect to x.

ox µ(x, a)

with

We shall consider the problem of minimizing

the function µ(x)

==

max µ(x, Q') aE 2

on a set O which has, at every point x, a non-empty, convex cone K : x K

x

==

[

e: x+A.eEO for sufficiently small A.> O}.

By virtue of what we proved in Chapter 3, µ(x) is quasidifferentiable, and M(x) is defined forµ by the formulas

130

V.

Necessary Conditions for an Extremum in Concrete Problems

M(x)

'2 (x) = (ct: otE a, µ(x, a) = µ(x)}. On the basis of the corollary to Theorem 4. 1 and Theorem 4. 2, we can assert that, if µ(x) achieves its minimum on O at x 0 , then the following condition holds: There exists a vector c 0 E~

0 such that c 0 EM(x 0 ).

But, since we

are dealing with an n-dimensional space, any element in the convex hull of a set can be represented as a convex combination of (n+l) vectors of this set.

Thus, there exist numbers

A. i' i=l, ••• , n+l, and points ai Ea (x 0 ), such that n+l

n+l

L:; A .0 µ(x 0 , a.), L; Ai= 1, A..i > O.

i=l

Theorem 5. 3.

l

x

i=l

l

(s. 10)

In order that the previously defined

function µ(x) achieve its minimum at x 0 Eo, !_!is necessary

i=l, ••• ,n+l, such that n+l

:B

A. 0 µ(xo, i=l 1 x

g

n+l Cl. )EK* ' 1 XO

µ(x, er) is convex in x for all

0t

stated conditions are sufficient.

131

:B A.= l

i=l

1.

(s. 11)

and 0 is convex, then the

V.

Necessary Conditions for an Extremum in Concrete Problems Proof.

Formula (5. 11) is an immediate consequence of

the relations c 0 E K*

and (5. 10).

XO

Moreover, if the additional

convexity assumptions are satisfied, then the sufficiency follows from the convexity of µ(x), so that Theorem 2. 1 can be applied to the problem.

Theorem 2. 1 in this case is

entirely equivalent to Theorem 4. 2 as far as the necessary conditions are concerned.

But this theorem in addition

implies that, in the convex case, the necessary conditions are at the same time also sufficient. Corollary.

Ill

If 0 coincides with the entire space, then

Condition (5. 11) of Theorem 3. 3 n+l

I; A. . o µ (x 0 , a- . )

i= 1

l

x

1

~be

written i!: the form

o•

Indeed, if 0 is the entire space, then K cides with En, and, thus, K* element O.

Ill

XO

xO

also coin-

contains only the single

The just obtained results stated in Theorem 5. 3 and its corollary can be considered as necessary conditions for a non-linear Chebyshev approximation problem.

Indeed, we

shall now show that the usual results for the classical Chebyshev approximation problem can be obtained from these results.

132

V.

Necessary Conditions for an Extremum in Concrete Problems Let there be given continuous functions cpk{a),

k =l, ••• , n, and a continuous function f{a) on a compact set E in an m-dimensional space.

It is required to approximate

this latter function by a generalized polynomial

F{a)

whose coefficients x. range over a closed, convex domain O. l

This approximation should be best in the sense that the quantity

µ(x) = max

a EE

I f{a)

should be minimized. Let us represent µ{x) in the following slightly different form which is more convenient for our investigation:

µ{x)

max

a EE I,::: s _:: : +l

S (f{a)

(5.12)

It is easy to see that µ(x) is convex in x.

Further, µ is the

maximum over a compact set of the family of functions

n

µ{x,a,s)=

s(f{a)-

~

i= 1

x.cp.{a)). l

l

These functions are obviously continuously differentiable with

133

V.

Necessary Conditions for an Extremum in Concrete Problems

respect to x, and

On the basis of Theorem 5. 3, in order that xOEO yield a minimum for µ(x), it is necessary and sufficient that there exist numbers A..> O and pairs (a., l -

n+l

- L; A.. i=l

l

l

C

• l(O'i)

s. • l

•

n

and all the pairs (a., l

l

L:

i=l

A.. l

= 1,

(5. 13)

i

are such that they yield a maximum

I f(a.) l

l

such that

n+l

XO

= x 0•

for the function (5. 12) for x

s.

EK* ,

( Q'.)

s.)

s.) l

This implies that

L; xk cpk(O'.)

n 1 k=

o

l

I•

) ( 5. 14)

n

sign (f(ai)

1: x~ cpk(ai>).

k=l

The following theorem summarizes this result. Theorem 5. 4. k = 1, .•• , n,

be~

0 In order that the coefficients xk,

solution to the previously stated Chebyshev

approximation problem,

~ ~

necessary and sufficient that

there exist numbers A..> 0 and points a.EE such that the

----

i-

-----

l

------

formulas (5. 14) and (5. 13) hold, where K:O is the dual~ to Kx •

-

0

134

V.

Necessary Conditions for an Extremum in Concrete Problems

g

Corollary. (5. 13)

O

the form

~be written~

n+l

:E >...s.

i=l

l

(

the entire space, then Condition

~

cp 1 ( 0 ) ) •

=0

•

l

•

•

( 5. 1 5)

cp (Cl.) n l We shall now apply the result which we have just obtained to the case where E is a subset of the real line, and cpk(et) =

Ct

k-1

0 µ(x )

•

=

Let Cti be points at which max a EE n 0 k-1 f (a.> - ~ xk a. l k=l l

s. l

= sign(f(a.)l

and suppose that

Ct

1

~

I,

( 5. 16)

n

~ x~a~- 1 ),

k=l

l

a 2 ~ • • • ,::;_ an+l •

The existence of

such numbers is guaranteed by Theorem 5. 4.

Then,

according to (5.15), there exist numbers ).. . > O, with

""n_ '-' +l i=

1

1-

," . = 1, such that l

0

a. l

n+l ~ n- 1

>...s. l l

1

=0

Ct. l

• n-1

a. l

135

•

(5. 1 7)

V.

Necessary Conditions for an Extremum in Concrete Problems

We now note that if some of the et. are equal, then the l

. 0 1 n-1 corresponding vectors S.(OI. ,OI. , ••• ,OI. ) are also equal. l

l

l

l

Collecting together the coefficients of such vectors in (5. 17), we conclude that there exist numbers Oli' with Oli < Oli+l' and numbers A.~ > 0 such that (5. 16) holds and such that l

f . -1

i-

-

A.~s.(:~)=O, l

l

r

1,

"n-1 OI. l

where r < n+l and r is the number of distinct points et.. -

But,

l

if the et. are distinct, then the vectors l

are linearly independent if r

:5. n,

1 n-1 s. (a.,0 OI., ••• , a. ) l

l

l

l

and this implies that

r = n+l, i.e., there exist exactly n+l points a. such that l

(5. 16) holds, and, moreover,

A.~ l

=A.. > 0 for each i. l

Thus, we can assume that all the a. in (5. 17) are l

distinct. Now, let us consider (5. 17) to be a system of equations for the unknowns

A..s .. l

l

Transposing the terms corresponding

to i = n+l to the right-hand sides of the equations, we obtain the system of equations

136

V.

Necessary Conditions for an Extremum in Concrete Problems

0

0/.

n

L:

i=l

A.. l

s.

l

l

n-1 an+l

n-1

0/. l

for the unknowns A.S., i=l, ••• ,n. l

l

According to Cramer's

rule, the solution of this system can be written in the form

A.s. l l

-A.n+l Sn+l

D (a 1' • • • ' a i-1' an+ 1' 0/ i + 1' • • • ' 0/ n) D( 0'1•••••0/n )

where

0

a1 1

D(011•••••0ln)

a1

0

a2

1

a2

..... n-1

a1

Q'

n-1 2

0 n 1 a n

0/

a

n-1 n

The determinant D(a 1 , ••• , Oln) is a Vandermonde determinant which, as is well known, is positive if a 1 < a 2 0 for all eE K

i=-m

1 l

-~

,

By the definition of a directional differential, the sets Mi(x 0 ) (i=O or -1) in the present problem consist of all functionals

x* i

of the form

where a(t)EM*(xo(. )). Since the equality-type constraints are linear, the functional x* has the form

~ (T)

T

I ., 0

~

-1

(t) BE(t) dt •

Applying Theorem 4. 2, we can assert the following: In order that the control u 0 (t) and the corresponding trajectory x 0 (t) be optimal in the stated problem, it is necessary that there exist non-negative numbers A. 0 and A. -l' a vector

147

V.

Necessary Conditions for an Extremum in Concrete Problems

nondecreasing functions o 0 (t) and o_ 1 (t) such that

+ for all E(t)E

c

A., HT)

~·

s

T

0

1

(5. 2 5)

iii- (t)BE(t)dt) .:: 0

= u(t)

or, equivalently, for all E(t)

where u(t) is an admissible control.

0 - u (t),

Moreover, the

following conditions must hold:

= 0,

o. (0) 1

J

T

0

=

g.(x 0 (t))do.(t) l

1

o. (T) = 1, l

µ,(x 0 (t)) for l

0 A._lµ-l(x (•))

=

i

= -1

and 0,

(5.26)

O.

We shall not tire the reader by carrying out in detail the somewhat tedious manipulations which amount to interchanging the order of integration in (5. 2 5).

After such

manipulations, Inequality (5. 2 5) reduces to the form

J

T

>

(1\r (T), Bu(T))dT

0

T J' (1\r (T), Bu 0 (T))d·r ,

(5. 2 7)

0

where

1\r(T)

=

iii-

l':'

(T) [

0

-~ A.

i=-1

1

s

T ... 0 ili (t)o g.(x (t)do.(t)H*(T)A.], x l l T (5.28) 00

148

V.

Necessary Conditions for an Extremum in Concrete Problems

and the asterisk after the matrix denotes transposition.

But

it follows from Condition (5. 27) that the optimal control must satisfy the condition ($(T ), Bu 0 (T))

for almost all

T

with 0
O, then the conditions of Theorem 5. 6 are also sufficient. Remark 2.

Let us note some properties of $(T).

every point where the functions o 0 (T) and o _ 1 (T) are

149

At

V.

Necessary Conditions for an Extremum in Concrete Problems

differentiable, i.e., almost everywhere (since o 0 (T) and

o_ 1 (T) are non-decreasing functions), ljr(T) has a derivative given by the formula d ,1, dT

.::.;r_

0

= -A*,i."' - L;

i=-l

0 dol. A..o g.(x (T))-d • 1

X

T

1

This derivative can be obtained simply by differentiating

(5. 28).

At every point of discontinuity of o 0 (T) and o _ 1 (T),

ljr (T) has a jump, and, by definition of a Stieltjes integral,

0

0

L:

\jl(T -0) - \jl(T +0)

i=-1

A..110.(T)og.(x (T)). 1

1

1

Here, /10. (T) is the magnitude of the jump of o. at 1

1

T.

In conclusion, we note that, in the preceding problem, we can impose various more complicated constraints on the right-hand endpoint of the trajectory and on the phase coordinates.

Such constraints can be taken into account

much as we did before, without any particular difficulties.

6,

~duality

principle

0

~programming.

Let

L be a linear space, let cp.(x), i=-m, -(m-1), ••• ,-1,0, ... ,k 1

be functionals which are convex for each i < 0 and linear for each i > 0, and let X be a convex set. problem

150

We wish to solve the

V.

Necessary Conditions for an Extremum in Concrete Problems min cp0 (x),

o,

Cl'. (x) < 0 for i < l

cp. (x)

-

0 for

l

i > 0,

xEX.

)

(5.30)

Let us consider the following set in (m+k+ 1)dimensional space: M = [ z: zEE

m+k+l

, z.= cp.(x) for -m < i < k, xEX}. l

l

-

-

In what follows, we shall assume that Mis compact. We set µ(u)

k max L u. cp . (x) xEX i=-m 1 1

k

max L u.z .. zEM i=-m 1 1

(5. 31)

We shall consider the problem of minimizing this function on the set

0 =

r u:

uo= -1, u. < 0 for i < 0} • 1-

The function µ(u) is convex.

Since the function k

µ(u, z)

L:

i=-m

u.z. l

l

is differentiable with respect to u,

ou µ(u, z)

z,

z

(z -m, ••• , z 0 , ••• , zk),

then, according to Theorem 3.4 and Remarks 1, 2, and 3 to Theorem 3.2, the set of support functionals to µ(u) at u 0 is

151

V. Necessary Conditions for an Extremum in Concrete Problems the convex hull of the set of all vectors z which satisfy the condition

k ..,...

L.i

i=-m

0 0 ::: µ(u ), u. z. l

zEM,

l

or, equivalently, the convex hull of the set of all vectors i:p(x) such that k

I:

i=-m

0 u.

j=l

J

J -

o,

where every x. satisfies Condition (5. 32). J Now consider the set

0 for O at u ". vectors e

r u o of

all "admissible directions

It is easily seen that

= (e -m, •••

r

u

0 consists of all

, e 0 , ••• , ek) such that:

e. is arbitrary if u? < 0 and l

l

e. < 0 l-

e

0

0 if u. l

< O;

0 and i < O;

= O;

ei is arbitrary if i > 0, Therefore, the dual cone consists of all vectors

152

V.

Necessary Conditions for an Extremum in Concrete Problems

such that e ~' = 0 if u? < 0 and i < O; l l

e7'l -
0.

Ace or ding to Theorem 2. 1, the following condition holds at any point u 0 E 0 at which µ(u) achieves its minimum on 0:

M(u 0 )

n r~' 0 t- 0. u

This means that there exist points x.EX, j=l, ••• , £, and J £ numbers A.. > 0, with :2::: ._ 1 A.. = 1, such that the vector J JJ

c j= l

A. .cp(x.) J

J

belongs tor ':' 0 , and such that each x. satisfies Equation u J (5.32).

But, since

cEr~' 0 , u

this means, by virtue of what we

have just said about f '"' that O' u

l

L:; A. . cp • (x . ) = O if u.0 < 0 and

j=l

J

J

l

l

l

0

:2::: A. . cp • (x . ) < O if u.

·-1 Jl

L

·-1 J-

J

l

J -

A. . cp • (x . ) J

l

J

l

0 and

O if i > 0.

153


O.

Finally,

Thus, we have shown that x 0 satisfies all of the constraints of the original convex programming problem. We shall now show that x 0 is a solution to this problem, i.e. ,

that it yields a minimum for

qi 0

(x).

Since u~ < 0 for i < 0 and the functionals 0, k .;-' L...J

i=-m

u.0 Cl). (x 0 ) > 1

1

-

=

k

0 l u. ~ \.Cl). (x.) i=-m 1 j=l,, J 1 J

L;

l

0 u. Cl). (x.) J i =-m 1 1 J

L: \ .

j =I

k

L.;

154

l

L:

\.µ(uo)

j= I J

= µ(u

0

),

Necessary Conditions for an Extremum in Concrete Problems

V.

where we have made use of the fact that x. satisfies (5. 32). J But, since x 0 EX, it follows from the definition of µ(u) that k

0

0

L; u. cp. (x 0 ) < µ(u ).

i=-m

l

l

Thus, k

.

r;

i=-m

k

0 u. cp. (x. ) • i=-m l l J

0 0 u. cp. (x 0 ) = µ(u ) l l

L;

Multiplying each of the last relations by A.. and summing, and J

taking into account (5.33) and the relation

:L:~_ 1 A.. 1-

J

= 1, we

conclude that k

L

i=-m

0 u.cp.(x 0 ) = 1

1

k

:E

i=-m

0 u.( 1

Because cp 0 (x) is convex and

01,

J,

L; A..cp.(x.))= u 0 :L: A..cp 0 (x.).

j=l

u~

1

1

J

j=l J

J

= -1, we obtain, from the

last equation, that

or that

k

r; •

i=-m

0 u. cp . (x 0 ) < l

l

-

o.

(5. 34)

ii':O But we have already shown that cp. (x 0 ) < 0 for i < 0 and that l -

155

V.

Necessary Co~ditions for an Extremum in Concrete Problems

Because u? < 0 for i < 0, each term in the left-hand l

-

side of (5. 34) turns out to be non-negative.

Thus, Inequality

(5. 34) can hold only if

0 u.cp.(x 0 )=0 for l

1

iFO.

Now let x be an arbitrary point which satisfies the cons train ts (5. 3 0).

Then, by definition of µ(uo),

k

.

I;

i=-m

0 u. cp. (x 0 ) l 1

0 µ(u )

> -cp 0 (x) +

.

k

I:

i=-m

0 u. cp. (x) > -cp0 (x) , 1

1

-

iFO since u? < 0 for i < O, cp.(x) < 0 for i < O, and cp.(x) = 0 for 1 -

i > O.

1

-

1

Thus, the inequality cp 0 (x 0 ) ::::_ cp 0 (x) holds.

This

inequality shows that x 0 is a solution to the original convex programming problem. We have proved the following theorem. Theorem 5. 7.

~

the set M which

~

have defined be

compact, and let u 0 be~ minimum point ~ µ(u)(defined ~ (5. 31 ))

~

0.

Then

~solution

programming problem (5. 30)

t_£ the original convex ~be

which satisfy the condition

156

found among the points

V.

Necessary Conditions for an Extremum in Concrete Problems k "'"' ~

. i;;;-m

0

u.0 cp. (x) l

µ(u ),

l

xEX.

What is the basic essence of the result which we have just obtained?

This result shows that we can associate

another problem with the original convex programming problem.

This second problem has a more complicated

criterion function, but has simple constraints, and solving it is equivalent to solving the original problem.

It is known

from the classical theory of linear programming that the dual problem sometimes turns out to be more convenient to solve than the original one.

Incidentally, if we apply the

result which we have obtained to an ordinary linear programming problem, then we indeed obtain the problem which is dual in the sense of linear programming theory [ 32] • Finally, we point out the following important peculiarity of the dual problem:

it is finite-dimensional, whether or not

the original problem is.

It is precisely this fact which is

widely made use of in a number of algorithms for solving linear optimal control problems. 7.

Systems of convex inequalities.

Belly's theorem.

We shall show how the results in the theory of necessary conditions for an extremum which we have obtained may be

157

V.

Necessary Conditions for an Extremum in Concrete Problems

applied to investigate conditions regarding the compatibility of systems of convex inequalities, Let L be a linear space and let cp.(x), for i=l, ••• ,n, l

be convex functionals on L.

We shall consider the system of

inequalities cp i (x) _:::: 0,

i

=1 , ... , n,

where X is a convex set in L. cp(x)

x EX ,

( 5, 3 5)

Let

(cp 1 (x),.,., cpn (x)),

M= {z:zEEn,

z=cp(x),xEX}.

Suppose that Mis compact.

Then there exists the

following bounded function µ().. ) = max (A., z) zEM

max (A., cp(x)). xEX

Let us consider this function on the set

0 = { A.: A. < O,

n

:E

i=l

A.. l

This is a convex function on 0 which achieves its minimum . '0 at some point fl. • Theorem 5.8.

If the set

M={z:zEEn,z

158

cp(x);xEX}

V.

Necessary Conditions for an Extremum in Concrete Problems

is compact, then System (5. 3 5)

has~

solution

g and

only if

the function µ(A.)

max (A., cp (x)) xEX

is non-negative for all A.EO. Proof.

We have already dealt more than once with

functions of the form µ(A.), particularly in the preceding section of this chapter.

We showed there that the set M(A.)

of support functionals to this function is the convex hull of all vectors cp(x) which satisfy the condition (A.,cp(x))= µ(A.),

xEX.

(5. 36)

Now suppose that A. 0 Eo is a minimum point of µ(A.) and that µ(A. O) is non-negative.

On the basis of the corollary to

Theorem 2. 1, since the space is finite-dimensional, there must exist at A. O a vector c 0 E M(A.) such that 0 0 0 (A. , c ) _::: (A, c ) for all

A.E O ,

i.e., by the definition of 0,

0 = min (A. ,c ) = -

HO

0 max c .• l

161

i-

o.

V. Necessary Conditions for an Extremum in Concrete Problems We note that, since xEEP and (EP)*

= EP,

we can talk

about functionals x*E (EP)* as if they were p-dimensional vectors. Now, if Mi (x 0 ) denotes the set of support functionals to cpi (x) at x 0 , then the result which we have stated concerning the structure of the set of support functionals to µ(x) can also be formulated as follows:

Incidentally, this formula also follows immediately from Theorem 3. 4,

In this connection, the bar over co has been

deleted because the sets Mi (x 0 ) are closed, and, thus, their union (as the union of a finite number of closed sets) is also closed, and the convex hull of a compact set in a finitedimensional space is also closed,

Since the space under

consideration is p-dimensional, every vector in the convex hull of a set can be represented as a convex combination of at most p+l vectors of the original set.

Therefore, every

support functional (vector) to µ(x) can be represented in the form x::~

r

= L: A. .x~ • with r < p+l, l. j=l J J

162

(s. 3 9)

V.

Necessary Conditions for an Extremum in Concrete Problems

where i.E I(x 0 ) and x~ EM. (x 0 ). A convex function µ(x) J l. l. J J achieves its minim= on a compact set X at some point x 0 • According to the corollary to Theorem 2. 1, there exists a functional x~EM(x 0 ) such that

(s. 40) For our problem, this means that there exist an integer r :::;_ p+l, numbers A. > O, and functionals x'.~ EM. (x 0 ) such Jl. l. J J thatx~ (in (5.40)) can be represented in the form (5.39). Here, ijEI(x 0 ) and r

~ A.. j= 1 J

1.

But, according to Theorem 1.4, x~ is a support functional at x 0 to the convex function j:l(x)

=

max cp. (x) • 1 .::::_ j :::;_ r 1 j

According to the corollary to Theorem 2.1, the fulfillment of (5. 40) is necessary and sufficient for j:l(x) to achieve its minimum at x 0 •

Moreover,

since i/I(x 0 ), and, thus, at x 0 , cpi.(x 0 )= µ(x 0 )forj=l, ••. ,r, J

163

V.

Necessary Conditions for an Extremum in Concrete Problems

by definition of I(x 0 ). By the hypotheses of the theorem, the system of inequalities cp.(x) cr •

P

where

+

n

r: (:>...

cr =

l

sup :>.. + < 0 :>,.0

< II r:

or (if we introduce the notation account that when

:>..

positive numbers,

+ and :>.. -

:>..

a

:>..

-A~)OI. l

i=l n (A.: i=l l

- :>..

=

l

~) x.

l

l

:>.. +_ :>..

11

and take into

range over the set of all non-

ranges over the entire real axis)

=

sup

( 5. 4 5)

:>..

Thus, we can state the following corollary to Theorem 5. 10. Corollary.

The

minimum~ of~

satisfies the constraints (5.44) is

equal~

functional which

a as defined by

Eq. (5. 45), Remark.

Obviously, if not all of the

a =

inf :>..

171

; : :>..

i= 1

OI. l

.x.11 ·

l

l

are zero, then (5. 46)

V.

Necessary Conditions for an Extremum in Concrete Problems

where the infimum is taken over all numbers A. which satisfy the constraint n

6

i=l If the vectors

O'.

A.. l

1.

l

x., i=l, ••• , n, are linearly independent, l

then the infimum is achieved and is non-zero.

Indeed,

applying the Bunyakovskii-Schwarz inequality to (5. 47), we obtain that n

r;

1

i=l

A.O'. l

IAI. I I •


O. l

for any non-zero vector A. = (A. 1 , ••• , An)'

or

172

Then,

V.

Necessary Conditions for an Extremum in Concrete Problems

It follows from the last inequality that the domain of A for

which

II

L A .x. i= 1 l

l

JI

_::

\\

.L1 A~ x.

i=

l

l

JJ

and

n 0 ~ A..

i=l

l

O'. l

is contained in the domain

p

But it is clear that we must look for the minimum of n

A..x. l\ ll L._ 1- 1 l l

precisely in this domain.

Since this domain is

compact and the function to be minimized is continuous, the minimum is achieved.

Moreover, this minimum is different

from zero.

Indeed, as we have already seen, if A. satisfies

(5.47), then

IA. I > IQ' 1-1.

Therefore, for all such A,

The results which we have obtained permit us to establish a relation between the moment problem and the Chebyshev approximation problem.

173

V.

Necessary Conditions for an Extremum in Concrete Problems Let there be given elements x, x 1 , ••• , xn in a Banach

space.

Consider the moment problem min \\x~'\\, x*(~)

=1

,

x'~(x.) = 0 for i=l,. •• ,n. l

On the basis of the previous remark, the norm of a functional which is a solution to this problem equals a

L

where

L

min

A.l

~ +i=L1 A .x. \\ • l

l

i = 1, ••• , n.

Indeed, in the problem under consideration, a 0 = 1, a.=Ofori=l, ... ,n, and, thus, Condition(5.47), which can l

be written in the form n

6 A..l a.l

1,

i=O

is equivalent to the condition A. 0 = 1. Therefore, finding a functional of minimum norm in the moment problem under consideration is equivalent to solving the problem of finding a best approximation to an n element x by means of a linear combination '6._ 1 A..x .. 1-

174

l

l

V.

Necessary Conditions for an Extremum in Concrete Problems Conversely, if the numbers A.. are such that l

; +

i::

i=l

A.

.x.11

l

l

= L,

then, on the basis of Theorem 5. 5 and its corollary, there exists a functional x~ such that

x~

(-; +

n

L::

i=l

L,

A..x.) l

(5. 48)

l

with

II If L

x~

II

= 1 and x~(xi) = 0 for i=l, ••• ,n.

-/= 0 (and it is clear that only in this case is CJ
:' is a solution to the moment problem.

Therefore, we

have established that the moment problem and the Chebyshev approximation problem are the duals of one another, and that a solution to one of these problems can be obtained from a solution of the other. Let us consider the special case of the moment problem in which the space under consideration is the space C of all

17 5

V.

Necessary Conditions for an Extremum in Concrete Problems

continuous real-valued functions of an argument t which ranges over the interval [ O, T].

In this space, the norm is

given by the formula

\\x\\

=

max

0< t < T

I x I .

By a well-known theorem [ 2 J, the continuous functionals in c* are of the form

x*(x)

J

T

0

x(t)dg(t)

where g(t) is a function of bounded variation such that g(O) = 0, and llx*\\

Var

O O.

Now set

179

V.

Necessary Conditions for an Extremum in Concrete Problems

g. (t) J

g (t) =

{ 1 L

0 for

0 < t < t.,

1 for

t. < t < T, J-

-

.J

r

2::

v.sct.)g.Ctl J J J

j=l

Then, on the basis of the definition of the Stieltjes integral and (5.49), we have

J

T

0

cp(t) dg (t)

1

r

I L: y.s (t.)cp Ct.) j=l J

J

J

OI '

or, in a "by-component" form,

J

T

0

cp. (t) dg (t) l

= OI. l

for i = 1, ••• , n.

Further, g(t) is a piecewise-constant function which has jumps at the points t. of magnitude y . S (t. )L - l. J J J

Thus, the

total variation of this function is simply equal to the sum of the absolute values of the jumps.

1 L

But, r

L: j=l

y.

J

1 L =a.

Therefore, the norm of the functional which corresponds to g(t) is equal to a,

Thus, we can state, on the basis of the

remark to Theorem 5, 10, that the just-constructed functional is a solution to the moment problem.

180

V.

Necessary Conditions for an Extremum in Concrete Problems Theorem 5. 11.

..!!, n

functions cp 1 (t), ••• , cpn (t) in C

~

linearly independent, then the moment problem always has '.!_ solution in the form of a function of bounded variation which

discontinuity. The moment problem may be applied in a number of ways to solve linear optimal control problems.

In concluding

this section, we shall briefly consider one such problem. Let an object be described by the system of equations

X

Ax+ bu,

where A is an nx n matrix, b is an n-dimensional column vector, and u is a scalar-valued control function.

We wish

to transfer the system from an initial state x 0 to a final state x 1 in the time T, using an impulsive control with a minimum for the sum of the impulses.

The use of an impulsive control

means that u(t) has the form u(t) =

k ~ y .6(t-t.) j=l J J

where the t. are the times at which the impulses are applied,

I yj I

J

is the "strength" of the j-th impulse

the total "strength".

and~ ~=l I yj I is

The function 6(t) is the delta-function,

181

V.

Necessary Conditions for an Extremum in Concrete Problems

which formally satisfies the following condition

J13

cp(O) ,

cp (t) 0 (t) dt

Of

if

Of

:5:. 0 and 13.::: O, and a and 13 are not both zero. If

have the same sign, then the integral vanishes.

Of

and 13

Obviously,

if {

0 for t < 0, 1 fort_:::O,

then the value of the integral with the delta-function is quite the same as the value of the Stieltjes integral

J13

cp(t) dg 0 (t).

Of

Clearly, in connection with what we have just said, with every impulsive control we can associate a function of bounded variation of the form

g (t)

k ~ y.g 0 (t-t.), j= 1 J J

where Var g(t)

k ~ j=l

y.

J

1.

Let us return to the originally stated problem.

Every

solution of a system of linear differential equations can be

182

V.

Necessary Conditions for an Extremum in Concrete Problems

represented by virtue of the Cauchy formula in the form

=

x(t)

Ht)xo

+

J ~(t- T)bu(T)dT , t

0

where d~(t)/dt =A Ht) and HO) is the identity matrix.

Thus,

the given problem of transferring our system from x 0 to x 1 is equivalent to finding a control which satisfies the system of equations

x(T) or

=

HT)x

s

T

0

0

+

J

T

0

HT-T)bu(T)dT

tp(T)u(T)dT

=x

1

,

Ct,

where

cp (T) = HT - T)b, QI=

and

1 0 x - HT)x.

If we associate with u(t) the function g(t) according to the rules which we have just indicated, then the original problem is transformed into the following one:

Find a func-

tion g(t) of minimum total variation which satisfies the condition Ct •

183

V.

Necessary Conditions for an Extremum in Concrete Problems

But this is precisely the moment problem in the space of continuous functions which we considered before. (if

01

= 0 then the solution is trivial:

u(t)

=0),

If

OI

f. 0

and if the

components of the vector function qi(t) are linearly independent (which is also true if the "general position condition" holds, see [ 18, p. 116] ), then, on the basis of Theorem 5. 11, the moment problem which we have formulated has a piecewise-constant solution g(t) with at most n+l points of discontinuity.

But, to such a function there corresponds an

impulsive control with at most n+l impulses.

The "strengths"

of these impulses coincide with the magnitudes of the jumps of g(t), and the impulses are applied at the instants of time when g (t) is dis continuous.

Thus, if t 1 , ••• , tn are the points

of discontinuity of g(t), then r

0

u (t)

L:

i=l

(g(t.+ 0) - g(t.- 0)) o(t- t.), J J J

r < n+l.

The total strength of this control is equal to r

L:

i= 1

I g