Quasidifferential Calculus (''Mathematical Programming Studies '')

M A T H E M A T I C A L P R O G R A M M I N G STUDIES

Founder and first Editor-in-Chief M.L. BALINSKI Editor-in-Chief R.W. COTTLE, Department of Operations Research, Stanford University, Stanford, CA 94305, U.S.A. Co-Editors L.C.W. DIXON, Numerical Optimisation Centre, The Hatfield Polytechnic, College Lane, Hatfield, Hertfordshire ALt0 9AB, England B. KORTE, Institut fiar Okonometrie und Operations Research, Universit~tt Bonn, Nassestrasse 2, D-5300 Bonn 1, W. Germany M.J. TODD, School of Operations Research and Industrial Engineering, Upson Hall, Cornell University, Ithaca, NY 14853, U.S.A. Associate Editors E.L. ALLGOWER, Colorado State University, Fort Collins, CO, U.S.A. W.H. CUNNINGHAM, Carleton University, Ottawa, Ontario, Canada J.E. DENNIS, Jr., Rice University, Houston, TX, U.S.A. B.C. EAVES, Stanford University, CA, U.S.A. R. FLETCHER, University of Dundee, Dundee, Scotland D. GOLDFARB, Columbia University, New York, USA J.-B. HIRIART-URRUTY, Universit6 Paul Sabatier, Toulouse, France M. IRI, University of Tokyo, Tokyo, Japan R.G. JEROSLOW, Georgia Institute of Technology, Atlanta, GA, U.S.A. D.S. JOHNSON, Bell Telephone Laboratories, Murray Hill, N J, U.S.A. C. LEMARECHAL, INRIA-Laboria, Le Chesnay, France L. LOVASZ, University of Szeged, Szeged, Hungary L. MCLINDEN, University of Illinois, Urbana, IL, U.S.A. M.J.D. POWELL, University of Cambridge, Cambridge, England W.R. PULLEYBLANK, University of Calgary, Calgary, Alberta, Canada A.H.G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands K. R1TTER, University of Stuttgart, Stuttgart, W. Germany R.W.H. SARGENT, Imperial College, London, England D.F. SHANNO, University of California, Davis, CA, U.S.A. L.E. TROTTER, Jr., Cornell University, Ithaca, NY, U.S.A. H. TUY, Institute of Mathematics, Hanoi, Socialist Republic of Vietnam R.J.B. WETS, University of Kentucky, Lexington, KY, U.S.A. Senior Editors E.M.L. BEALE, Scicon Computer Services Ltd., Milton Keynes, England G.B. DANTZIG, Stanford University, Stanford, CA, U.S.A. L.V. KANTOROVICH, Academy of Sciences, Moscow, U.S.S.R. T.C. KOOPMANS, Yale University, New Haven, CT, U.S.A. A.W. TUCKER, Princeton University, Princeton, N J, U.S.A. P. WOLFE, IBM Research Center, Yorktown Heights, NY, U.S.A.

MATHEMATICAL

PROGRAMMING

STUDY29 A PUBLICATION OF THE MATHEMATICAL PROGRAMMING SOCIETY

Quasidifferential Calculus

Edited by V.F. D E M Y A N O V and L . C . W . D I X O N

May 1986

N O R T H - H O L L A N D - AMSTERDAM

© T h e M a t h e m a t i c a l P r o g r a m m i n g Society, Inc. -

1986

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Submission to this journal of a paper entails the author's irrevocable and exclusive authorization of the publisher to collect any sums or considerations for copying or reproduction payable by third parties (as mentioned in article 17 paragraph 2 of the Dutch Copyright Act of 1912 and in the Royal Decree of June 20, 1974 (S. 351) pursuant to article 16b of the Dutch Copyright Act of 1912) a n d / o r to act in or out of Court in connection therewith.

This STUDY is also available to nonsubscribers in a book edition.

Printed in The Netherlands

To P.L. Chebyshev, the Godfather of Nonsmooth Analysis

PREFACE

F2Ia~Ko 6bI~O na 6yMare, ~a 3a6bi~rl npo oBparn, A no HHM XO~ltTb It was smooth on paper But ravines had been forgotten Where we should walk

The papers in the present Study deal with quasidifferentiable functions, i.e. functions which are directionally differentiable and such that at each fixed point the directional derivative as a function of direction can be expressed as the difference of two convex positively homogeneous functions. It turns out that quasidifferentiable functions form a linear space closed with respect to all 'differentiable' operations and (very importantly) with respect to the operations of taking the point-wise maximum and minimum. Many properties of these functions have been discovered, and we are now in a position to speak about Quasidifferential Calculus. But the importance of quasidifferentiable functions is not simply based on the results obtained so far. We can foresee a much greater role for these functions since (as far as the first-order properties are concerned) all directionally differentiable Lipschitzian functions can be approximated by quasidifferentiable functions. This is due to the fact that the directional derivative of any directionally differentiable Lipschitzian function can be approximated to within any given accuracy by the difference of two convex positively homogeneous functions. This Study reflects the state-of-the-art of Quasidifferential Calculus. The original idea of simply publishing English translations of a number of Russian papers on the subject was immediately rejected by the Editor-in-Chief, Professor R. W. Cottle; we are now grateful for this decision, since the authors obtained new results, thus leading to a much greater understanding of the subject. The Editors of this Study are greatly indebted to the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, which provided editorial and secretarial support for preparing the Study. We offer especial thanks to our language editor, Helen Gasking, whose role cannot be overestimated, and to Nora Avedisians, Edith Gruber and Elfriede Herbst for typing and retyping the papers. Thanks are also due to the referees, whose assistance, advice and criticism helped to improve many of the contributions. It is also necessary to note that the idea of such a Study was proposed by Professor Roger Wets and supported by Professor Andrzej Wierzbicki, then the Chairman of the System and Decision Sciences Program at IIASA. Some of the authors became vii

viii

Preface

involved in Quasidifferential Calculus through or at IIASA, and therefore this Study is in some sense a child of IIASA (although whether it is an offspring to be proud of is a question that can only be answered by the reader). Most of the Soviet authors of this Study are graduates a n d / o r staff members of Leningrad State University, where the first serious attempt to attack the problem of nondifferentiability was made more than a hundred years ago by P.L. Chebyshev, to whom this Study is dedicated. V.F. Demyanov L.C.W. Dixon

(Editors)

CONTENTS

Preface V.F. Demyanov, L.N. Polyakova and A.M. Rubinov, Nonsmoothness and quasidifferentiability V.F. Demyanov, Quasidifferentiable functions: Necessary conditions and descent directions L.N. Polyakova, On the minimization ofa quasidifferentiable function subject to equality-type quasidifferentiable constraints A. Shapiro, Quasidifferential calculus and first-order optimality conditions in nonsmooth optimization L.N. Polyakova, On minimizing the sum of a convex function and a concave function V.F. Demyanov, S. Gamidov and T.I. Sivelina, An algorithm for minimizing a certain class of quasidifferentiable functions K.C. Kiwiel, A linearization method for minimizing certain quasidifferentiable functions V.A. Demidova and V.F. Demyanov, A directional implicit function theorem for quasidifferentiable functions V.F. Demyanov and I.S. Zabrodin, Directional differentiability of a continual maximum function of quasidifferentiable functions D. Melzer, On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions S.L. Pechersky, Positively homogeneous quasidifferentiable functions and their application in cooperative game theory N.A. Pecherskaya, Quasidifferentiable mappings and the differentiability of maximum functions V.F. Demyanov, V.N. Nikulina and I.R. Shablinskaya, Quasidifferentiable functions in Optimal Control A.M. Rubinov and A.A. Yagubov, The space of star-shaped sets and its applications in nonsmooth optimization V.V. Gorokhovik, e-Quasidifferentiability of real-valued functions and optimality conditions in extremal problems

vii 1 20 44 56 69 74 85 95 108 118 135 145 160 176 203

Appendix A guide to the bibliography on quasidifferential calculus Bibliography on quasidifferential calculus (January 1985)

219 219

Mathematical Programming Study 29 (1986) 1-19 North-Holland

NONSMOOTHNESS

AND

QUASIDIFFERENTIABILITY

V.F. D E M Y A N O V Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9, Leningrad 199164, USSR, and International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria L.N. P O L Y A K O V A Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9, Leningrad 199164, USSR A.M. R U B I N O V Institute for Social and Economic Problems, USSR Academy of Sciences, ul. Voinova 50-a, Leningrad 198015, USSR Received 9 April 1984 Revised manuscript received 15 November 1984

This paper is an introduction to the present volume. It is first shown that quasidifferentiable functions form a very distinct class of nondifferentiable functions. This and other papers in this volume demonstrate that we do not need to consider any other class of nonsmooth functions at least from the point of view of first-order approximation. The heart of quasidifferential calculus is the concept of a quasidiiterential--this replaces the concept of a gradient in the smooth case and that of a subdifferential in the convex case.

Key words: Nondifferentiable Functions, Quasidifferentiable Functions, Quasiditterentials, Subdifferentials, Superdiilerentials, Optimization Problems, Directional Differentiability, Upper Convex and Lower Concave Approximations, Clarke Subdifferential.

1. Introduction This is not the place to go into the m o t i v a t i o n s a n d origins of n o n d i f f e r e n t i a b i l i t y (although these are very i m l ~ r t a n t a n d interesting): for the p u r p o s e of this p a p e r it is only necessary to realize that although a n o n d i f f e r e n t i a b l e f u n c t i o n can often be a p p r o x i m a t e d by a differentiable one, this s u b s t i t u t i o n is u s u a l l y u n a c c e p t a b l e from a n o p t i m i z a t i o n v i e w p o i n t since some very i m p o r t a n t properties of the f u n c t i o n are lost (see E x a m p l e 2.1 below). We must therefore find some n e w analytical tool to apply to the p r o b l e m . Define a finite-valued f u n c t i o n f o n an o p e n set D c E,. I f f u n c t i o n f i s directionally differentiable, i.e., if the following limit exists:

af(x) = lim l [ f ( x + a g ) - f ( x ) ] Og ,,~+o ot

VgcE,, 1

(1.1)

2

V.F. Demyanov, L.N. Polyakova and A.M. Rubinov / Quasidifferentiability

then

f ( x + ag) = f ( x ) + ~ of(x) + o(~).

ag

Many important properties of the function can be described using the directional derivative. To solve optimization problems we must be able to (i) check necessary conditions for an extremum; (ii) find steepest-descent or -ascent directions; (iii) construct numerical methods. In general, we cannot solve these auxiliary problems for an arbitrary function f: we must have some additional information. In classical differential calculus it is assumed that af(x)/ag can be represented in the form

Of(x) ag -

(f'(x),

g),

where f'(x) ~ E, and (a, b) is the scalar product of vectors a and b. The function f is said to be differentiable at x and the vector f ' ( x ) is called the gradient o f f at x. Dif[erentiable functions form a well-known and important class of functions. The next cases that we shall consider are convex functions and m a x i m u m functions. It turns out that for these functions the directional derivative has the form

Of(x) Og

max (v, g),

(1.2)

vcOf(x)

where Of(x) is a convex compact set called the subdifferential of f at x. Each of these two classes of functions forms a convex cone and therefore their calculus is very limited (only two operations are allowed: addition, and multiplication by a positive number). The importance of (1.2) has led to m a n y attempts to extend the concept of a subdifferential to other classes of nondifferentiable functions (see, e.g., [1, 15, 16, 18, 22, 23, 28, 32]). One very natural and simple generalization was suggested by the authors of the present paper in 1979 [7, 13]. We shall say that a function f is quasidifferentiable at x if it is directionatty differentiable at x and if there exists a pair of compact convex sets Of(x)~ E. and -~f(x)c E. such that

Of(x) Og

max (v, g ) + re_in (w, g).

o~f(x)

wec~f(x)

(1.3)

The pair D f ( x ) = [_0f(x), 0f(x)] is called a quasidifferential o f f at x. It has been shown that quasidifferentiable functions form a linear space closed with respect to all algebraic operations and, even more importantly, to the operations of taking pointwise maxima and minima. This has led to the development of quasidifferential calculus, and many important and interesting properties of these

V.F. Demyanov, L.N. Polyakova and A.M. Rubinov / Quasidifferentiability

3

functions have been discovered (including a chain rule, an implicit function theorem, and so on). One very important property of these functions is that if f is directionally differentiable and its directional derivative a f ( x ) / a g at x is a continuous function of direction g (every directionally ditterentiable Lipschitzian function has this property), then a f ( x ) / a g can be approximated to within any prescribed accuracy by a function of form (1.3). Thus, the quasidifterential is an ideal tool for studying the first-order properties of functions. A more general approach, involving an extension of quasiditterential calculus, has been presented by Rubinov and Yagubov [29]. They proved that if a f ( x ) / a g is continuous in g then it can be represented in the form Of(x) =inf{h > 0 [ g ~ h U } + s u p { h < 0 [ g e AV}, Og

(1.4)

where U and V are what are known as star-shaped sets. If U and V are convex sets then eqn. (1.4) can be rewritten in the form (1.3). Thus, i f f is directionally ditterentiable it is natural to use this construction (the directional derivative) to study optimization problems. However, i f f is not directionally ditterentiable some other tool must be found. One approach is to generalize the notion of the directional derivative (1.1). We shall mention only the following two generalizations: 1. The Dini upper derivative o f f at x in the direction g, defined as Oof(x)~ _ ~ 1 [ f ( x + ag') - f ( x ) ] . Og s'~g ct a~+O

In the case of a Lipschitzian function this becomes: Oof(x)'~ _ ~ 1 [ f ( x + ag) - f ( x ) ]. Og ~+o a

(1.5)

2. The Clarke upper derivative o f f at x in the direction g, defined as Octf(x)~ - ~ l[f(x' + Og x'~x et.

ag) +f(x')].

(1.6)

Other generalizations and extensions are given in [18, 22, 28]. Equation (1.5) is a natural generalization of (1.1) and, in the case of a directionally ditterentiable function, the Dini upper derivative (1.5) coincides with the directional derivative (1.1). However, this is not the case for the Clarke upper derivative (1.6). The reason for this is that (1.6) describes not the local properties o f f at x but some 'cumulative' properties o f f in a neighborhood of x. It seems to the authors that for optimization purposes it is better to use the Dini derivative (and this idea has been exploited by B.N. Pschenichnyi [23]).

4

V..E Demyanov, L.N. Polyakova and A.M. Rubinov / Quasidifferentiability

The Dini and Clarke upper derivatives are used to study minimization problems: for maximization problems it is necessary to invoke the Dini and Clarke lower derivatives. These are defined analogously to (1.5) and (1.6) with the operation I - ~ replaced by lim. We shall discuss both these generalizations later in the paper: for now, note only that if the Dini upper derivative is continuous (which is always the case if f is Lipschitzian), then it can be approximated by a function of the form (1.3), so that quasidifferential calculus can be used here as well. In Section 2 we discuss directional ditterentiability. Section 3 is concerned with convex functions and m a x i m u m functions, as well as with the Clarke subdifferential and pschenichnyi upper convex and lower concave approximations. Quasidifferentiable functions are treated in Section 4. This should be seen as a survey paper: we hope that it will provide a general introduction to the subject of this Study and enable readers to make use of the results in their own research.

2. Directional differentiability Let S c E, be an open set and f be defined and finite-valued on $. Fix x E S and g e E,. The function f is said to be differentiable at x in the direction g if the following finite limit exists: af(x) = f ' ( g ) -- lim l [ f ( x + ag a~+o a

ag) - f ( x ) ] .

(2.1)

(It is naturally assumed that x + a g ~ S ; since S is open this is the case for all a ~ [0, O~o(g)], where a o ( g ) > 0). The limit (2.1) is called the (first-order) directional derivative of f at x in the direction g. I f f is differentiable in every direction g ~ E, it is said to be directionally differentiable at x. If J" is directionally differentiable at x and Lipschitzian in some neighborhood of x, then lim l[f(x~-otg(t~))-f(x)] ~+o a

g(~)~g

Of(x) ag

i.e., in this case it is sufficient to consider only 'line' directions. It is clear from (2.1) that i f f is directionally ditierentiable then

f ( x + ag) = f ( x ) + ct af(x) + o ( a ), ag

i.e., the directional derivative provides a first-order approximation o f f in a neighborhood of x.

V.F. Demyanov, L.N. Polyakova and A.M. Rubinov / Ouasidifferentiability

5

Let f be directionally differentiable at x, x e S. A direction g(x) is known as a

steepest-descent direction o f f at x if Of(x) Of(x) = inf~g(x) ~ s , Og ' where S, = {go E. I Ilgll = 1}. Here Ilgll is the euclidean norm g. A direction g'(x) is called a steepest-ascent direction o f f at x if

Of(x) Of(x) = sup Og'(x) ~ s . Og Directions of steepest descent or ascent need not necessarily exist and if they do, they are not necessarily unique. It is clear that for a point x* c E, to be a minimum point o f f it is necessary that

Of(x*) >-0 Og

Vg e En.

An analogous necessary condition for a m a x i m u m is Of(x**) ~O 0g

VgeE~, g#0,

then the point x* is a strict local minimum point o f f . One very important class is that of differentiable functions. In this case

Of(x) Og

- ( f ' ( x ) , g),

(2.2)

where f ' ( x ) is the g r a d i e n t . o f f at x. Applying the concept of a gradient, for example, to the optimization problem, it is possible to: 1. C o m p u t e the directional derivative. 2. Derive the following necessary condition for a minimum or a maximum: for a differentiable function f to attain its local minimum (or maximum) value at x* c S it is necessary that

f ' ( x * ) --- 0.

(2.3)

The point x* at which condition (2.3) is satisfied is called a stationary point o f f .

V.F. Derayanov, L.N. Polyakovaand A.M. Rubinot;/ Quasidifferentiability 3. Find directions of steepest descent and ascent as follows: I f f ' ( x o ) # 0 then the direction

g(xo) =

f'(xo)

(2.4)

Ilf'xo)][

is the direction of steepest descent of f at Xo, and the direction

g'(xo)

f'(xo)

IIf'(xo)ll

is the direction of steepest ascent o f f at Xo. In this case the directions of steepest descent and ascent both exist and are unique. 4. Construct numerical methods for finding an extremum. The concept of a gradient (a derivative in the one-dimensional case) has had a profound impact on the development of science. It is impossible to overestimate its importance and influence. From being an art, mathematics became a technical science. However, differential calculus is only applicable if the functions studied are smooth (i.e., ditterentiable). For most practical problems tackled in the past (and for many presently under study) it has been sufficient to consider only smooth functions. Nevertheless, an increasing number of problems arising in engineering and technology are of an essentially non-smooth nature. There are two very popular ways to avoid nonditterentiability. First, one tries to replace a non-smooth problem by a smooth one. For example, the problem of minimizing the function

f(x) = max 6,(x), i61 where ~b~'s are smooth nonnegative functions, I = 1 : N and x E E,, is often replaced by the minimization of

F(x) = E a,6,(x), where the a~ are positive coefficients. The function F is smooth but it now describes quite a different problem. The second possibility is to consider the function

Fp(x) = ( ~ [( qb,(x)]P) '/, instead

off.

It is well-known that F,(x) p---~-s

Vx. Note that in many cases

the computations process by which Fp(x) is minimized becomes unstable. Some very important properties of the original function can thus be lost in the pursuit of smoothness. We can illustrate this using a very simple example.

Example 2.1. Let x = (x ~'), x C2~)~ E2; f(x) = Ix ')l- Ix'='l, go = (0, 0). R e function f

E F. Demyanov, L.N. Polyakova and A.M. Rubino~/ Quasidifferentiability

7

is not ditterentiable at points where x (~) = 0 or x ~2~= 0. Take a direction g = (g{3~, g~2~). The function f is directionally differentiable with directional derivative

Of(xo) Og

-- iim l [ f ( x o + a~+0

ag) - f ( x o ) ]

Ig"'l-Ig'2'l 9

=

O/

It is clear that there are two steepest-descent directions of f at Xo: g, = (0, 1) and

g'l = ( 0 , - 1 ) . There are also two steepest-ascent directions: g2 = (1, 0), g ~ = ( - 1 , 0 ) . Let us try to s m o o t h the function f functions:

Take e > 0

and consider the following

f , , ( x ) = ~/ix"))2 + e - ~/(xl2))2+ e,

(1)

f2,(x) = x/(x~'; + e) 2 - ~/(x~Z) + e) z,

(2)

f3,. (x) = ~ / ( - ~ ) 2 + e - ~/(x'2) + e)2,

(3)

It is clear that

f~(x)

?,f(x)

~0

V i e 1:3.

Find the gradients o f these functions at xo:

~f,~.(x) ox

(

of,,(Xo)

- - =

x I'~ x ''-~ ) 4[.~,~")~+ ~' ,/(x%2 + ~ '

(0,0)

Ox

'de>O,

,/(x (11 +~) 2 t 4(x (2) +~)12

ox of~,(Xo) Ox 3f~(x)

(-1,1)

(

re>O,

- x (~)

-

of3,(Xo) ax

x(2'+e

) '

(0, 1~) VE>O.

We can then make the following deductions: forfl~: Xo is a stationary point. forf2~ : the steepest-descent direction at Xo is g3 = (,f2/2, - x / 2 / 2 ) and the steepestascent direction is g~ = ( - x / 2 / 2 , x/2/2). forf3~ : the steepest-descent direction at Xo is g4 = (0, - 1 ) and the steepest-ascent direction is g~ = (0, 1). Thus, all three s m o o t h i n g functions provide incomplete or even misleading information about stationarity or directions o f steepest descent and ascent. The reason

8

V.F. Demyanov, L.N. Polyakova and A.M. Rubinov / Quasidifferentiability

is that these smoothing functions are zeroth-order approximations while steepestascent and -descent directions reflect first-order properties of the function. Since it appears that we cannot avoid nondifferentiability, we should rather study the properties of special classes of nonsmooth functions with the aim of developing analytical tools to handle these problems.

3. The subdifferentiai and its generalizations

3.1. Maximum functions Let

f ( x ) = max ~b(x, y), yeG

(3.1)

where ~b(x, y) is continuous in x and y on S x G and continuously differentiable in x on S; G is a compact set. The function f described above is not necessarily continuously differentiable. However, it is directionally differentiable on S and

of(x) = max (~b'(x,y), g), Og y~R~)

(3.2)

where g ( x ) = {y E G I ~p(x, y) =f(x)}. The set R ( x ) is closed and bounded. We can rewrite (3.2) in the form

of(x) = max (v, g), ag v~af(x)

(3.3)

Of(x) = co{~b'(x, y)Jy c R(x)}.

(3.4)

where

It is not difficult to see that the set af(x) described by (3.4) can be used for several purposes [2, 6]: 1. To compute the directional derivative (see (3.3)). 2. To derive the following necessary condition for an unconstrained minimum: for x*~ S to be a local minimum point o f f defined by (3.1) it is necessary that

0 ~ Of(x*).

(3.5)

A point x*~ S at which (3.5) is satisfied is called a stationary point o f f (note that S is an open set). 3. If xo is not a stationary point then the direction

g(xo) =

V(Xo)

IIv(xo)ll'

where V(Xo) c Of(xo), IIV(xo) ll = min o~oy~xo)IIv II, is a steepest-descent direction o f f at Xo. This direction is unique.

V.F. Demyanov, L.N. Polyakovaand A.M. Rubinov / Quasidifferentiability

9

If we find vl(Xo) e af(xo) such that IIv,(Xo)ll = maxv~or 0, then the direction g~(xo) = v~(xo)/II v,(xo)II is a steepest-ascent direction of f at Xo. Note that this direction is not necessarily unique. The set Of(x) can also be used to construct numerical methods for minimizing f on E, or on a bounded set (see, e.g., [6]).

3.2. Convex functions Let S c E, be a convex open set and f be a convex function defined on S, i.e.,

f(ax,+(1-a)x2)~o vver(x*)

(1.2)

where (a, b) denotes the scalar product of a and b, and f ' ( x ) represents the gradient of f a t x. Unfortunately it is difficult to use this trivial condition in practice. Let A c F ( x ) be a convex cone such that 0 9 A and let M(x) be a family o f convex cones such that

AcF(x)

VA 9

U

A=F(x).

(1.3)

Ac.~t(x)

In [1] cones o f this type are called 'tents'. It is always possible to find a family .d(x) defined as above (take, for example, ~t(x) = {l I l = {v = Xvol A/> 0}, Vo9 F(x)}). We denote by A + the cone conjugate to A: A + = {w 9 Enl (v, w) >I 0 Vv 9 A}. Theorem 2. Condition (1.2) is equivalent to

f'(x*) 9

§

VA 9

(1.4)

A point x * 9 c l / 2 which satisfies (1.4) (or, equivalently, (1.2)) is called a stationary

point o f f on /2. In what follows we shall suppose that 12 is a closed set. Assume that x 9 is not a stationary point o f f o n / 2 . Then there exists A 9 M(x) such that

f ' ( x ) Z A 4. Let us find min IIf'(x)-wll w~m

= IIf'(x)-w(m)ll =- Ilv(a)[I.

(1.5)

§

It is not difficult to see that

v(A) = w(A) - f ' ( x ) 9 A and that v ( A ) is a descent direction o f f o n / 2 at x, i.e.,

(f(x), v(A)) < 0 . It is also clear that the direction go = Vo/I[volh where IIVo[[ = maxa~.~r direction o f steepest descent o f the function f on the s e t / 2 at x, i.e.,

Of(x) Of(x) inf Ogo g~s,~r~) c3g

[[v(A)ll, is a

22

V.F. Demyanov / Necessa~ conditions and descent directions

Here St = { g e E,I

Ilgll =

3f(x) f(x + ag) - f ( x ) = lira Og , , ~ + o a

1},

A steepest descent direction m a y not be u n i q u e . Note that

3f(x) Of(x) - rain 3g(A) s~s,~a 3g where g ( A ) =

(1.6)

v(A)/IIv(A)II.

R e m a r k 1. C o n d i t i o n (1.4) is equivalent to

f'(x*) e ~(x*)

(1.7)

where ~ ( x ) = Oa~,~(x)A+. If ~ ( x * ) = {0} then we obtain the well-known condition f ' ( x * ) = 0. Example 1. Let

X=(X ('),x (2))e E2,

Xo = (0, 0),

/2 = Ii w 12tO 13,

where

I, = {x = (c~,O)l~ ~0},

12-~{x = (0, ~ ) l a ~0},

/3={x=(-a,-a)la~>O}. It is clear that F(xo) = 12 and M(Xo)= {It, 12, 13}, i.e., M(Xo) = {A,, A2, A3}, where At = It, A2 = 12, A3 = 13. NOW we have

A~={xcE21(x, ft)>~O}, F t = ( 1 , 0 ) , A~={x~E2I(x, F2)~>O}, / 2 = ( 0 , 1 ) , Af={xeE21(x, /3) ~> 0},

T3 = ( - - 1 , --1).

It can be seen from Fig. 1 that Z/'(Xo) = f"qi~t :3 A ~ - = {0} and therefore f ' ( x o ) = 0 is a necessary condition at Xo.

s

\

\,,A~ \x

~a Fig. 1.

\

V.F. Demyanov / Necessary conditions and descent directions

23

Remark 2. If x c $2 is not a stationary point then min

IIv-f(x)ll

= Ilv(x)-f(x)tl >0.

vc~(x,)

However, note that the direction

v(x) - f ( x ) g- IIv(x)-f(x)ll has nothing to do with descent directions (it m a y not even be feasible). Thus, the necessary condition (1.7) provides no information about descent directions if Xo is not a stationary point. In contrast, condition (1.4) is more workable because it allows us to construct descent and even steepest descent directions. For a continuously differentiable function f,

Of(x) = ( f ( x ) , g). Og Thus the p r o b l e m of finding steepest descent directions of f o n / 2 at x is reduced to that o f solving (1.6) for all A e s C ( x ) . For this reason we are interested in constructing a family s f ( x ) containing as few cones as possible. I f / 2 is a convex set the cone F ( x ) is convex and therefore ~r consists of only one set. L e t / 2 be described by inequalities /2 - - { x 9 E, lh,(x) 0, where V,?

g, (F(x)) =

IIv, II

and

IIv~ II = minll vii,

L, = co{if(x); ~/h~(x) I i ~ Q(x)}.

v~ L n

The foregoing analysis reveals the importance of having several (possibly equivalent) necessary conditions, in that this enables us to develop different numerical methods. Remark 3. It is not difficult to show that, in (1.11), gn (A) ~ ~ ~o g(A), where g(A) is the steepest descent direction o f f on A at x.

1.2. Convex programming problems Similar considerations can be applied to constrained non-differentiable convex programming problems of the form

m i n { f ( x ) l x ~ [2}

V.F. Demyanov / Necessary conditions and descent directions

25

where

/2 = {x ~ E~l h(x) !0 Vx ~ E,, and ~b~(x*) = 0, x* is a minimum point of ~b, on E,. However, ~b~ is a convex function and so cg~n (x*) = co{Of(x*) w [r/h(x*)]}. Applying a necessary and sufficient condition for an unconstrained minimum of a convex function, we immediately obtain (1.15'). Assume that x e / 2 is not a minimum point o f f on /2, and find the direction

g,n(x)-

V'n(X)

IIv,,(x)lt

(1.16')

where

iiv,,(x)ll

=

min

vcLt,~(x)

Ilvll.

It can be shown that the direction gl~ (x) defined by (1.16') is a descent direction and gin (x) e int F(,x). Note also that

g,n(x)

, g(x),

where g(x) is the steepest descent direction o f f on /2.

Remark 4. Condition (1.15') is applicable even i f / 2 is an arbitrary convex compact set (not necessarily described explicitly by a convex function).

V.F. Demyanov / Necessary conditions and descent directions

27

2. Quasidifferentiable functions

2.1. Definitions and some properties A f u n c t i o n f is called quasidifferentiable (q.d.) at a point x c E~ if it is directionally differentiable at x and if there exist convex compact sets Of(x) c E, and "~f(x) c E, such that

Of(x)_ lim f ( x + a g ) - f ( x ) = Og ~+o a

max (v,g)+ min (w,g). vr weaf(x)

The pair of sets Df(x)= [0f(x), 0f(x)] is called a quasi-differential o f f at x. Quasidifferentiable functions were introduced in [3] and have been studied in more detail in [7, 2]. A survey of results concerning this class of functions is presented in [4]. It turns out that q.d. functions form a linear space closed with respect to all differentiable operations and, more importantly, to the operations of taking pointwise maximum and minimum. A new form of calculus (quasidifferential calculus) has been developed to handle these functions, and both a chain rule for composite functions and an inverse function theorem have been established [5, 4]. In what follows we shall use only two results from quasidifferential calculus (see below). If Di = [Ai, B1], D2 -- [A2, B2] a r e pairs of convex sets (i.e., A~ c En, Bi c En are convex sets) we put

Dl + D2 = [Ai + A2, BI + B2] and if D = [A, B] then ~[AA, AB]

AD=[[AB, AA]

if ht>0, ifh O but it is not clear whether

v(x) + w(v(x))

IIv(x)+w(v(x))ll

go=

is a descent direction. Let h(x)= O. The problem of finding a steepest descent direction is equivalent to the following problem: minimize

0

(2.16)

subject to af(x) ~0, ag

Oh(x) ag

(2.17)

~ 0, and/'2~n c En+l is described by max (v,g)+ rain (w,g) 0

(9)

holds. Since

h(x + ctg) = h(x) + othx(g) + o(a, g) and

o(,~, g)

O,

then without loss of generality we can assume that hx(g)I>0= h x ( g ) V g e S~o(r From the assumptions of the theorem the function hx(g) has no strict local minimum at/~ and therefore inequality (9) is not satisfied. In the same way it can be shown that there exists an a~ > 0 such that for every O/ E (0, Otfl] and any g e S~,(~), g # ~, the inequality h(x + a g ) < 0 is also not satisfied. The contradiction means that yo(x) c F ( x ) and thus proves the theorem.

Theorem 2. I f the function h has a quasidifferential Dh(x)=[_0h(x), ah(x)] at x e 12 such that -O_h( x ) c~ -ah(x ) = ~, then the function h satisfies the regularity condition at point x. Proof. Since 0 e yo(X), it follows from the properties o f a quasidifferential of function hx(g) at 0 (see (8)) and the assumptions of the theorem that neither the necessary condition for a minimum nor that for a m a x i m u m is satisfied for quasidifferentiable function hx on En at any point g e En. Thus, it follows from Theorem 1 that function h satisfies the regularity condition at point x, and Theorem 2 is proved. This regularity condition is first-order and therefore it possesses all the deficiencies characteristic of first-order conditions. We shall now consider an example in which this condition is not satisfied.

Example 2. Let t2 -- {x e El I h(x) = 0},

L N. Polyak.ova/ Equality-typequasidifferentiableconstraints

48 where

(x-l)

,

x>l,

h(x)= 0,

-1 0 and 8 > 0 such that f ( x ) >~f(xo) + e IIx- xoll Vx ~ 12 n S~(xo). Proof. Take F > 0 and set (see [1])

A~(xo)={g~E,,IUgll=l,

Oh(xo) -~g 0 such that min

af(xo)

8~Ao(~o)

Og

= r(xo) > 0,

and therefore we can find g > 0 and F(Xo)> 0 such that rain

g c Ai(xo)

Of(xo) = F(xo) > 0. ag

Fix a > 0 and choose an arbitrary x ~/2 c~ S~(xo). IfA = then

IIx- xoll and g = ( l / A ) ( x

- Xo)

(af(xo) + o(,~,x g)~,/ ag

f(x)-f(xo) = A \

(17)

dg where

o(,~, g) A

A~+o

o,(,~, g)

; O,

h

A---+O

~0

uniformly with respect to g, Ilgll = 1. Set F--- min{f, 89 such that max{l~,

~

}~ 0

V A t ( O , 8].

(18)

Given such a 8, eqs. (17) are valid for any x s / 2 n S~(xo). This gives us

f ( x ) - f ( x o ) >i ]Ix - xoll ~, where e = F(Xo) Example 4. Consider the same f u n c t i o n f and set O as in Example 3. I f the inclusion

af(xo) c i n t cone(co{(-1, - 1 ) , (1, -1)}) is satisfied at Xo = (0, O) c/2, then Xo is a strict local minimum point of function f on set 12.

6. Reduction to the unconstrained case

Consider the function

F ( x ) = m a x { f (x) - f * , h(x), - h ( x ) } ,

L.N. Polyakova / Equality-type quasidifferentiable constraints

54

where f * = infxr162 Function F is quasidifferentiable on En. It is clear that if a point Xo is a solution to problem (12) then Xo is also a minimum point of F on En. We shall now write down a necessary condition for F to have a minimum on En at Xo. Since

~_F(Xo)= co{A, B, C], where

A = O_f(xo) - -~h(xo) + O_h(xo), B = 2_0h(Xo) --~f(xo),

C = -2~h(xo) --Of(xo)

and

-~F ( xo) :- ~f( Xo) + "~h( xo) - ~_h( xo), then the following result holds. Proposition. For a point Xo ~ 0 to be a minimum point o f f on 1"1 it is necessary that

- ~ F ( x o ) c OF(xo).

(19)

Remark. In some cases condition (19) is a worse requirement for an extremum than condition (13). This can be illustrated by means of an example. Example 5. Consider the came function h as in Example 3:

h(x)=llxd+ x=l,

x=(x"),xr

E=.

Let Xo = (0, 0). It is not difficult to check that

O_h(xo) --~h(xo) = co{(1, 1), ( - 1 , 1), (3, 3), ( - 3 , 3)}, 2Oh(xo) = co{(0, 0), (4, 4), ( - 4 , 4)} and

O_h(xo) - - ~ h ( ~ ) ~ 20_h(xo).

(20)

However, inclusion (20) implies that any quasidifferentiable function f satisfies (19) (the necessary condition for a minimum on the set 12) at the point Xo = (0, 0). Theorem 7. I f functions f and h are quasidifferentiable, the convergence in (15) is

uniform with respect to g e EM, IIg[[ = 1, and -'~F(xo)C int ~_F(xo), then Xo~ O is a strict local minimum point o f f on the set aq described by (2). The p r o o f is analogous to that of Theorem 11 in [4].

L. N. Polyakova / Equality-type quasidifferentiable constraints

55

References [1] V.A. Daugavet and V.N. Malozemov, "Nonlinear approximation problems" (in Russian), in: N.N. Moiseev, ed., The state-of-the-art of operations research theory (Nauka, Moscow, 1979) pp. 336-363. [2] V.F. Demyanov and A.M. Rubinov, Approximate methods in optimization problems (American Elsevier, New York, 1970). [3] V.F. Demyanov and L.V. Vasiliev, Nondifferemiableoptimization (in Russian) (Nauka, Moscow, 1981). [4] V.F. Demyanov, "Quasidifferentiable functions: Necessary conditions and descent directions", Working Paper WP-83-64, International Institute for Applied Systems Analysis (Laxenburg, Austria, 1983). [5] L.N. Polyakova, "Necessary conditions for an extremum of quasidifferentiable functions", Vestnik Leningradskogo Universiteta 13 (1980) 57-62 (translated in Vestnik Leningrad University Mathematics 13 (1981) 241-247). [6] L.N. Polyakova, "On one problem in nonsmooth optimization" (in Russian), Kibernetika 2 (1982) 119-122.

Mathematical ProgrammingStudy 29 (1986) 56-68 North-Holland

QUASIDIFFERENTIAL CALCULUS AND FIRST-ORDER OPTIMALITY CONDITIONS IN NONSMOOTH OPTIMIZATION Alexander S H A P I R O Department of Mathematics and Applied Mathematics, University of South Africa, P.O. Box 392, Pretoria 0001, South Africa

Received 22 September 1983 Revised manuscript received 28 October 1984 This paper is concerned with first-order optimality conditions for nonsmooth extremal problems. Local approximations are obtained in terms of positively homogeneous functions representable as the sum of sublinear and superlinear functions or, equivalently, as the difference of two sublinear functions (d.s.I. functions). The resulting optimality conditions are expressed in the form of set inclusions. The idea of such approximations is exploited through the detailed study of d.s.I, functions and the cones corresponding to nonpositive values of d.s.I, functions. Key words: Nonsmooth Optimization, Quasidifferentiable Functions, Optimality Conditions.

I. Introduction

In recent years much attention has been paid to nondifferentiable functions appearing in optimization theory. Efforts to find adequate tools for handling nonsmooth problems have resulted in the use of methods and techniques borrowed from convex analysis. Linear functions, which served quite well for the purposes of local approximation in differential calculus, have been replaced by sublinear or superlinear functions. Analogously, single gradient vectors have been substituted by convex compact sets (see, e.g., Rockafellar [16], Demyanov and Malozemov [4], Pshenichnyi [13] and Clarke [3]). These efforts have led to Clarke's theory of generalized gradients [ 1-3], which has been found to be a powerful instrument for investigating nonsmooth problems. However, although the method is good enough for global analysis it is often too rough for the purposes of local approximation, which is the main concern in optimization theory. The reasons for this are quite clear: if a function f ( x ) is essentially nonconvex then a sublinear function can provide only a rough bound on the local behavior o f f ( x ) . In this paper we consider an approach aimed at obtaining better local approximations of nondifferentiable functions. This approach was first put forward by Demyanov and Rubinov in [5] (see also [7] and the references therein). They introduced a class of functions (quasidifferentiable functions) which could be approximated to first order by a sum of sublinear and superlinear functions. In Section 2 we carry out a detailed investigation of positively homogeneous functions 56

A. Shapiro / Quasidifferential calculus and optimality conditions

57

which can be represented as the sum of sublinear and superlinear functions or, equivalently, as the difference of two sublinear functions (d.s.I. functions). It will be shown that there is a close relation between d.s.l, functions and the class of functions which can be locally represented as the difference of two convex functions (d.c. functions). Section 3 is concerned with first-order approximations of locally Lipschitz functions. It is shown how Clarke's method of generalized derivatives can be used to obtain approximations in terms of d.s.i, functions. Finally, different optimality conditions involving d.s.1, approximations are discussed in Section 4. It is demonstrated that necessary conditions proposed in [6] hold for almost every perturbed inequality-constrained problem. We denote by SA(') the support function of a bounded set A d E,, i.e.,

SA(X) =--sup{(x, y)lY r A}. It is known that every support function is sublinear and conversely that every sublinear function ~ is the support function of a certain convex compact set. This set is unique and given by

A = {yI(x, y)0 then

a[A, B] ~- [aA, aB] while if or < 0 then

a[A, B] =- [aB, aA]. The space ~ with the norm imposed by the Hausdortt distance p(A, - B ) between the sets A and - B becomes a normed space considered by RadstrOm [14]. It can be shown that the correspondence

[ A, B]~--~SA(" )-- s_n(" )

(4)

A. Shapiro / Quasidifferential calculus and optimality conditions

60

between Radstr6m's space ~ and the DSL space, with the sup norm

II~ll~-- sup{l~(x)ll x ~ S~_,}, is isometric (cf. Demyanov and Rubinov [5]). Since the d.s.l, functions are continuous, their restrictions to S,_~ form a subspace of C (S~ _j), i.e., the Banach space of continuous functions on S,_ j with the sup-norm. A continuous function on Sn-~ can be approximated with given precision by a C2-smooth function on S~_~ with the sup-norm. We have seen that a positively homogeneous function with C2-smooth restriction to Sn_~ is d.s.l. This leads to the following result: Corollary 1. The normed spaces D and DSL (with the sup norm) are isometric to

each other and to a dense subspace of C(Sn_l). It has been shown by Demyanov and Rubinov [8, Theorem 2] that the superposition of d.s.l, functions is also a d.s.I, function (see also [18, Theorem 2.1]). In particular, the DSL space is closed under the operations of taking pointwise maximum and minimum of a finite family of d.s.l, functions. Moreover, if ~o~ are d.s.l, functions and [A~, B~] are the corresponding elements of D, i = 1. . . . . m, then the max function ~0ma~(X) --=max{~oi(x), i e 1 : m} is associated with

and the min function ~mm with

(cf. [7, Lemmas 2.2 and 2.3]). In the classical theory of convex analysis, every closed convex cone C is associated with another closed convex cone C o

C~

y)~O, V x ~ C},

which is said to be polar (or dual) to C. Note that (C~ ~ C, i.e., cone C can be considered as the polar of C o (e.g., [16, p. 121]). Therefore every d o s e d convex cone C can be represented in the form c = { x l s A ~ x ) t (v, z - x ) V z ~ E.},

~ f ( x ) = af2(x) = {w ~ t,:, If2(z) - A ( x ) ~ ( w, z - x ) V z ~ En}. In other words, a_f(x) is the subdif[erential of the convex f u n c t i o n ]'1 at x ~ E . (as defined in convex analysis) a n d a f ( x ) is the s u p e r d i t t e r e n t i a l o f the c o n c a v e f u n c t i o n

f2atx~E.. Translated from Russian at the International Institute for Applied Systems Analysis (IIASA), A-2361 Laxenburg, Austria. 69

70

L.N. Polyakova / The sum of a convex function and a concave function Consider the p r o b l e m of calculating inf f ( x ) .

(1)

x(: E n

Quasiditterential calculus shows that for x* e E. to be a m i n i m u m point o f f on E. it is necessary that -af(x*) c af(x*).

(2)

We shall now show that the p r o b l e m of m i n i m i z i n g f on the space E, can be reduced to that of minimizing a concave function on a convex set. Let /2 denote the epigraph of the convex function f~, i.e., $2 = epi f~ = {z = [x, IX] e E, x E~ ] h(z) =-f~(x) - IX 0

VweOf(xo),

Ilgll= 1 zew+Oef(x 0)

and thus for every g e E,, IIg[I = 1, we have min

max (z, g)/> 0.

wESf(xo) ULi2ff(Xo)

However, this means that min ilgll=l

a,f(xo)

I> 0

(9)

c~g

proving that the condition is necessary. That it is also sufficient can be d e m o n s t r a t e d in an a n a l o g o u s way, arguing backwards from the inequality (9). Note that since the m a p p i n g _0~f: E, •

+co]--, 2E-

is H a u s d o r t t - c o n t i n u o u s if e > 0 (see, e.g., [1]), then the following t h e o r e m holds.

Theorem 3. I f e > 0 then the function max~o,y(x)(v , g) is continuous in x on E. for any fixed g c E,.

L.N. Polyakova / The sum of a convex function and a concave function

72

Assume that Xo is not an e-inf-stationary point. Then we can describe the vector

O,.f(xo) g,(xo) = arg min - ~g

Ilgll= 1

as a direction o f e-steepest-descent o f function f at point Xo. It is not difficult to show that the direction

/' Vo.+Wo)

=

+

ll

'

where vo~ e O_~f(xo), woe-~f(xo) and -

max w (: ~f(xo)

min IIv+wll---IlVo~+woll--a~(xo), veO,:f(Xo)

is a direction o f e-steepest-descent of function f at point Xo. N o w let us consider the following m e t h o d o f successive approximations. Fix e > 0 and choose an arbitrary initial a p p r o x i m a t i o n Xor E~. Suppose that the Lebesque set

D(xo) = {Xo ~ E, If(x) O

where g~k----gE(Xk) is an e-steepest-descent direction o f f at Xk. Theorem 4. The following relation holds: lim a~(Xk) = O.

k~of~,

Proof. We shall prove the theorem by contradiction. Assume that a subsequence {Xk~} o f sequence {Xk} and a n u m b e r a > 0 exist such that

a~(xk,) 0 and ~z > 0 fixed. Choose an arbitrary xo c E~. Suppose that xk has been found. If

---Of(Xk) c O_,f(Xk)

(7)

then Xk is an e-inf-stationary point and the process terminates. If, on the other hand, (7) is not satisfied then for every we B~,(Xk) we find min

vc #,f(x~)

IIw+vll = IIw+v~(w)ll.

If w+ Vk( W) ~ 0 then let gk ( W) = --( W + v~ ( w ) ) / ll w + v~(w)ll and compute

min f(xk + agk( w) ) = f(Xk + ak( W)gk( W) ). c~~O If w + vk(w) = 0 then take ak(W)gk(W) = 0. Next find min

f(Xk +C~k(w)gk(w))=f(xk +ak(Wk)gk(Wk)).

wca~,(xk)"

We then set

Xk§ = Xk + ak(Wk)gk( Wk).

(8)

78

V.F. Demyanoo et al. / An algorithm for minimizing quasidifferentiable functions

It is clear that (9)

f ( x k + , ) K.

(11)

Now we have f ( x k , + agE.,) ; J ' ( x * + (xk, - x* + C~gk,)) = f(x*) 4

a/(x*) B[Xk, -- X* + Otgk,]

~-or IIx~, - x* + ~g~, II)

where

w* +%(w*,) gk, =- gk.,(w*,) = -- IIw~*+ v~,(w~*)I~'

c3f(x*) c3[xk - x* + a g j

(cgF(y(x*)) ) k -~x , Xk, -- X* + agk,

r 12)

V. t( Demyanov et al. / An algorithm for minimizing quasidifferentiable functions

,., -1- L . icl§

+

~,

L .

79

[OF(y(x*)) . , , , , "~ max. I 7-cpo[ x ), Xk -- X* + Otgk,]

)Je~i~ x ) \

.

oyi

[OF(y(x*)) . , , , ,

mm. |

iel..(x )J~-Ri(x ) \

m oyi

'~

q)ij[X ) , X k - - X * - l - O ~ g k , ] .

(13) Since max a~ + m i n b~ 0 and ks such that f ( X k ~ + Otogk,)

~O V d e E., ad

where

of(x) od

- f ' ( d ) = li~n If(x+ td) - f ( x ) ] / t 1~0

denotes the derivative o f f at x in the direction d. Points ~ satisfying (3) are called inf-stationary for f. For convenience, we shall use the following notation: f~(x) = m a x f j , ( x ) , jaJi

i e I,

F ( x ) = ( f l ( x ) , . . . , fm (x)), f ( x ) = g(x, F ( x ) ) .

K.C. Kiwiel/ A linearization method

87

Vg(x, y) the n - v e c t o r

F o r z = [x, y ] e E , x Em we d e n o t e by

while

3•(x,y) denotes

-

Og

ieL

az~+. (z),

Let a i ( x ) - - ag (x, r

F(x))

Vx e E,,, i e / ,

b(x) = Vg(x, F(x))

Vx.

Then

of(x) -(b(x), d ) + Y. a,(x) )at;(x .~ o(l Od i~t = (b(x), d ) + Y. ai(x) max (Vfi(x), d), ic!

Jc-Ji(x)

"

so that

af(x) =(b(x),d)+ ad +

•

~

max

(ai(x)Vfj,(x),d)

ic I~(x) ) ~ . J d x )

min

(a,(x)Vfj,(x), d),

(4)

ie t _ ( x ) j s

where

J , ( x ) = { j c J , ls

ie I,

I+(x)={i~1la,(x)>O},

l_(x)={i~1la,(x) 0, an activity tolerance 8 > 0 and a line search parameter m > 0. Set k = 1. Step 1 (Direction finding). For each we B(x k, 8), find d(w) from the solution (d(w); l,l i ( w ) , i E l+(xk)) to {he quadratic programming subproblem min d, ul

89

d)+

Y. ~ a,(xk)u,+(w, d),

icl§

)

subject to f~,(x k) - f ~ ( x k) + (Vfj,(xk), d) ~lim

mtuk=O,

t~.O

producing a contradiction. We may add that if B(X k, t$) is a singleton, e . g . , B(x k, c$) = {0} as considered in [2, 11], then only one search direction is used at the kth iteration. In this case Algorithm 1 may be regarded as a natural extension of well-known and relatively efficient linearization methods [9, 15], and is close to the method of [2].

3. Convergence In this section we shall establish the global convergence of the method. In the absence of convexity, we will content ourselves with finding an inf-stationary point for f. Naturally, we assume that the final accuracy tolerance es is set to zero. We start by analyzing the properties of search directions generated around nonstationary points.

K.C. Kiwiel/ A linearization method

9I

Lemma 2. Suppose that ~ ~ E,, if, ~ B( Y,) and d e E, are such that f ( d ; .~, ~, O)< O. Then there exist ~ > 0 and neighborhoods S(s and S( ff,) of ~ and if,, respectively, such that

f~(d(x, w))g V[x, w]eS(~)xS(Ce),

(13)

where d(x, w) denotes the solution of (7). Proof. By assumption, we have (see (6))

(b(.x'), d)+

~. ai(x.) max (Vf~,(.~), d)+(~, d ) < - e ie

I~(.~)

JcJt(x)

for some e > 0 and d ~ 0. Hence, using the continuity of a~, b , f , fj~ and Vfj~, we may choose bounded S(s and S(ff) such that I+(s c I+(x) and l+.(x)\l+(,2)c {i~ II a,(s =0},

(14a)

tl a,(~z) =0},

(14b)

I_(~)~ L ( x ) and l_(x)\l_(s (b(x), d)+

Y.

{ie

ai(x) max (Vfji(x), d ) + ( w , d ) < ~ - e / 2

ic I+(x)

J~'Ji (~)

for all [x, w ] c S(~) • Next, since f and fj; are continuous, we have J~(~)c Ji(x, 6) for 6 > 0 and f j ; ( x ) - f ( x ) ~ < - ~ for some fixed ~ > 0 i f x is close to ~ and j e J,(x, 6)\J~(.f). We may therefore shrink S(.~) and choose small / > 0 such that max (Vfj~(x), ?d)/> max -~llVf~,(x)ll Ildll i> - ~ / 2

j~Ji( ~ )

jE Ji(.x)

> f,,(x) -f,(x) + (vfj,(x), ~d) for a n y j e J i ( x , 8)\Ji(2), and (b(x), a~)+

~

ic I+(x)

ai(x) max [f~i(x)-f(x)+(Vfi,(x), tt)]+(w, a~)~ O such that

f ( x k + td k) --f(2 + td k) + f ( 2 ) - f ( x k ) + o( t, k) < e + it,

K.C. Kiwiel / A linearization method

93

and hence, for ~ = g - g > 0, (19)

f ( x k + td k) ~ f ( x k) + e - ~t

for all t ~ [0, t(g)] and large k ~ K. Let us choose e such that the interval [_t(e), ?(e)] of solutions to the inequality (20)

e - ~t 0. Since the functions f are quasidifferentiable, for any q 9 En we have, from (1),

f(xo+ otg, yo + aq) = f ( x o , Yo) + ct O[g, Of(zo) q] + oi(ct, q) =

af(zo) +o,(ot, q)

a O[g, q]

(3)

V.A. Demidova, V.F. Demyanov / An implicitfunction theorem

97

where

af,(zo) _

max [(vii, g)+(v2, q ) ] +

rain [(wli, g)+(w2/, q)].

(4)

Here Df~(z)=[O_fi(z),Of~(z)] is a quasiditterential of f~ at z; 9 f . . ( z ) c E m + . , ~f.(z) c E,.+. are respectively sub- and superditterentials off~ at z (convex c o m p a c t sets)" vi = [vii, w,], and wi = [wli, w2i]. Let q0 ~- E . be a solution to the quasi-linear system

of,( zo)

O[g, qo]=O

Vi 9

(5)

~+o 9 0

(6)

Suppose that in (3)

oi(a, q) a

uniformily with respect to q 9 S ~ ( q o ) = {q 9 E~

l llq- qoll ~< ~},

where 8 > 0 is fixed. Is it possible to find a vector function r ( a ) with ao>O such that

f~(xo+ ag, yo+ a [ q o + r

=0

Vi 9 1 : n, a 9 [0, ao]

(7)

where r ( a ) 9 E . V a c [0, ao]? Take e ~> 0 and introduce the sets R,. = {v, e 0f~(zo)[(v,,, g) +(v2,, qo) t> m a x [(~,i. g) + (v2,. qo)] - e}, /~i~ = {w, e0f~(zo)l (w,~ g)+(w2,, q o ) ~
O) such that y~.(O) --- lim 1 [ y ( a ) - y ( O ) ] = qoa ~ + 0 O/

We shall call Theorem 1 a directional implicit function theorem. Of course, there could be several solutions to (5), or none at all.

V.A. Demidova, V.F. Demyanov / An implicit function theorem

100

It is important to be able to solve systems o f equations o f the form max[(v1~,g)+(v2~,q)]+min[(w~,g)+(W2l, q)]=b~

oi~O'li

wlEo'2i

Viel:n

vi

where = [ v , i , v2~], wi =[wli, w2i], and t r , , c E,,+. and ~ 2 i c E,.+. are convex c o m p a c t sets. We shall call systems o f this type In some cases (for example, if cr~i and 0"2~ are convex hulls o f a finite n u m b e r of points) the problem o f solving quasilinear systems can be reduced to that o f solving several linear systems o f algebraic equations (we shall illustrate this later on).

quasilinear.

3. An inverse function theorem N o w let us consider a special case o f the problem, namely, where system (1) is o f the form x + d~(y) = 0

(13)

i.e.,

x(i)+~bi(y)=O

Vie l:n,

where x = (x (1). . . . , x (")) e E,,

y = (y(l) . . . . , yC~)) e E,

and the functions ~b~ are quasidifferentiable on E.. Suppose that Zo = [Xo, Yo] e E2,, is a solution to (13), i.e, Xo+ ~b(yo) = 0. C h o o s e and fix any direction g e E,. We n o w have to consider two questions: 1. What conditions are necessary for the existence o f a positive ao and a continuous at a = 0 vector function such that the expressions

y(a)

y(0)=yo, xo+ag+c~(y(a))=0

V a e [ 0 , ao]

(14)

are satisfied? 2. I f y ( a ) exists does ' y ' ( 0 ) =- lim 1 [ y ( a ) - y ( 0 ) ] a~-~O

necessarily exist? To answer these questions we turn to T h e o r e m 1 and its corollary. Let = [_0$~(y),~$i(y)] be a quasidifferential o f ~bi at y. We then have

Dcbi(y)

~bi(yo+aq)=c~i(yo)+a[L v,~_.o.~,(vo) max (vi, q)+ min (wi, q)]+oi(a,q). w, c ~,,(yo)

(15)

V.A. Demidova, V.E Demyanov / An implicit function theorem

101

In this case equation (4) takes the form max ( v , . q ) +

m in

vic-O. ~ , ( y o )

(wi, q ) = - g i

(16)

Vi~l:n.

wiCr)cbi(yo)

Suppose that qoe E. is a solution to (16) and that in (15) oi( a, q) Ol

a ~ +O

~0

uniformaly with respect to q c S~(qo). We now introduce the sets max (vi, q ) - e } ,

_Ri~ = {o,c~_4~,(yo)l(vi, q)>-

Vi~_d~i(Yo)

/~,. ={W~eO~b,(yo)l(wi, q)~
0. Theorem 2. I f f o r some e > 0 we have (17)

min det A > 0

A~ M~

then there exist an a o > 0 and a continuous vector function y( a ) such that y(O) = Yo, Xo+ ag + r

=0

and

y~(O):= qoRemark 1. In the case where each of the sets _0~bi(yo) and 0~bi(yo) (for all values of i) is a convex hull of a finite number of points, it can be shown that theorem 2 is valid if (17) holds for e = 0 . An analogous result can also be obtained for Theorem 1. Remark 2. Suppose that [Xo, Yo] is a solution to (14). Then to solve the directional inverse function problem it is necessary to find all the solutions to (16) and check whether condition (17) is satisfied. As an illustration of Theorem 2 and the use of the technique outlined above we shall now present a simple example.

V.A. Demidova, V.F. Demyanov / An implicit function theorem

102

Example. Let x = (x (1~, x t2)) ~ E2, y = (yU), y(2)) E E2, x o = (0, 0), Yo -- (0, 0). C o n s i d e r the following system of equations: x I1) + ly ~I)I - 2ly 12~l= O,

x~2~+ ly(t~_ y(2)] = 0.

(18)

This system is simple enough to be solved directly. It is not difficult to derive the following solutions: 1.

y(1} = x(I)-- 2x(2), (19)

y(2) = X(1)_X(2)

if y~/'21 = {y = (y('), yt2))ly~l) > O, y(2)> O, ytt)_y(2)>~ 0};

y(t) = X(I) + 2X(2),

2.

X(2)

y(2) = x ( 1 ) +

(20)

if y c O2 = {Y = (y~l), y(2))ly(i) > 0, y(2) >10, y(n _ y~2) 0 and therefore y~a(a) is not a directional inverse function o f (13) in the direction g~. From (22) y~4(a) = ( gaa , ~ a ) r V a > 0 , and therefore ylg(a) is also not a directional inverse function of (13) in the direction g~. Solutions (23) and (24) yield the functions yt5(a)=y~6(a)= ( - a , - a ) , where

yls(a)r

and

Yl6(Ot ) C ~"~6 ~Of ~ 0.

ThUS ylS(Ot)=y(ct) is a directional inverse function of (13) in the direction g~ and y'~s+(0) = ( - 1 , - 1 ) = q05. Thus there are two directional inverse functions o f (13) in the direction gl :yll(Ct) = (a, a ) and yls(a) = (-a, - a ) . N o w let us consider g2 = ( - 1 , 0). F r o m (19),

y21(Ot)=(--a,--ot)~Ol

Vot > 0,

and therefore y2~(a) is not a directional inverse function of (13) in the direction g~. In the same way we obtain (for a > 0):

yz2(a) = ( - a , - a ) ~ .02,

yz3(a) = (-~a, Jet) ~ J'~3,

y24(ct) = ( - l a , - ~ a ) ~ 124,

y25(a) = (a, a)~ 12s,

Y26(a) = (a, or)I~ J'~6.

This m e a n s that there is no directional inverse function of the system (13) in the direction g2. The same is also true for the direction g3 = (1, 1) since for a > 0 y31 (or) = (-- a, 0) ~ ,('21,

Y32 = (3a, 2 a ) ~ 122,

Y33(ot) : (--Or, 0) t~ ~'~3,

Y34(a) = (a, 0) ~ .O4,

y35(ct)=(-3a,-2a)~ 05,

Y36(0~) = (O~, O) ~ J~6.

In the same way we find that there are two directional inverse functions of (13) in the direction g4 = ( - 1 , - 1 ) :

y4~(ct)=Y4a(a)=(ct, O),

y44(ct)=Y46(a)=(-a,O),

and y~l+(0) = y~3+(0) = ( 1 , 0),

y h + ( 0 ) = y~s+(0) = ( - 1, 0).

N o w let us solve the p r o b l e m again using the results of T h e o r e m 2. System (18) can be rewritten in the following form (see (13)): x + 4,(y)=0

where 6 = (4h, ~b2), 6,(Y) = {Y~O

and

y',2+(O)=q,2=(-1,-1).

Moving to ql3 = (1, 1), and following the same line of argument we deduce the existence of a vector function y13(a) such that Xo+Otgl+gb(yl3(a))=OVot>~O

and

y'13+(O)=ql3=(1,1).

106

V.A. Demidova, V.F. Demyanov / An implicitfunction theorem

Thus there are two solutions to (15) for g ~ = ( 1 , 0 ) : duplicates the result o b t a i n e d earlier. For g2 = ( - 1, 0) we again arrive at the four vectors q2, = (~,-~),

qz2 = ( - 1 , - 1 ) ,

q23= (1, 1),

Y~2(~) and y13(ot). This

q2,= ( - ~ , - ~ )

calculated previously as solutions to (27)-(34); however, none of them satisfies (26). Thus the system (13) has no directional inverse function in the direction g2. For g3 = (1, 1), solving the systems (27)-(34) yields six different vectors: q3, = (1, 0)

(from (27) and (29)),

q32: (--1, 0)

(from (32) and (34)),

q33 = ( - ~ , 2)

(from (28)),

q34 = (89 - ] )

(from (33)),

q35 = ( - 3 , - 2 )

(from (30)),

q36 :

(3, 2)

(from (31)).

These values should then be tested by substituting them into (26) (for q O: y + a v r G ( x o + a g ) Vct c[O, ao]}. We shall denote the closure of y(y) by F ( y ) , i.e., F ( y ) =- F(xo, g, y) = cl y(y). The set F ( y ) is called the set of first-order feasible directions at the point y ~ G(xo) in the direction g. Remark 1. In the case where G does not depend on x, the set F ( y ) = F(xo, g, y) does not depend on Xo and g, and is a cone called the cone of feasible directions at y. A mapping G is said to allow first-order approximation at a point Xo in the direction g e E,, Igll-- 1, if, for an arbitrary convergent sequence {Yk} such that yk -~ y,

yk E G( Xo + akg ),

Otk "-~ + O

ask~,

the following representation holds: YR = Y + Otkl)k "~ O( Olk )

where Ok e F(Xo, g, y), akVk -~ O, y e G(xo). In what follows it is assumed that the mapping G is continuous (in the Hausdortt metric) at a point Xo and allows first-order approximation at Xo in any direction g e E,, Ilgll-- 1. It is also assumed that for every

xcS~(Xo)=(xcEnlllx-xoll~}, ~>0, the set G ( x ) is closed and sets G ( x ) are jointly bounded on S~(xo), i.e., there exists an open bounded set B c Em such that G(x) c B

Vx~S~(Xo).

V.E Demyanov, LS. Zabrodin / Differentiability of a maximumfunction

110

Let us consider the function f ( x ) = max

~(x,y)

ycG(x)

where function d~(z) = tb(x, y) is continuous in z = [x, y] on S~(xo) x B and differentiable on Zo in any direction r / = [ g , q]E E,+,,, i.e., there exists a finite limit c~&(Xo, y)

m

c~b(Xo,Yo)

On

O[g, q] = lim l [ ~ b ( X o + ag, yo + aq) - 6 ( X o , Yo)] ~+0

VZo = [Xo, Yo] E Zo.

Here

Zo :{XoIXlR(xo)},

R(x)={yEG(x)lck(x,y)=f(x)}.

Suppose that the following conditions hold: Condition 1. If qk --> q then

Ocb(Xo, y) 0, which is differentiable from the right at zero. Using Minkowski duality, we can reformulate Definition 1 in terms of support functions as follows: Let P ( E . ) be the set of real sublinear functions over E, with the t o p o l o g y induced by the C ( S . ) sup norm, and consider 4> :2 ~o ~ P(En) given by ~b(M,v)=max(z,v),

M ~ 2 ~~ v ~ E , ,

zcM

(the support-function m a p p i n g ) , and ~br :[0, 8 ) ~ P ( E , ) defined by rbr(a,v)=fb(l'(a),v),

veE,.

(8)

Then for all sets M, N e 2 e. we have

d,,(M, N ) = II~(M, ") - ~ ( N , . )11 = m a x l 6 ( M , v) - 4,(N, v)l,

(9)

t~s S n

which leads to Proposition 1 [20; 7, p. 131]. Proposition 1. 1":[0, 8 ) ~ 2 ~o is differentiable from the right at zero iff the limit lim ~ - l ( ~ b r ( a , v) - ~br(0, v)) =: ~b~-(0+, v)

(10)

~0+

exists uniformly with respect to v ~ S, and there are sets I'+, F - ~ 2 e" such that

4,~-(o +, . ) = ~ ( r +, 9) - , / , ( r , - ) . Let us n o w consider ~b~.(0+, .) : E, --> El. If F is differentiable from the right at zero then ~br(0 ' + , 9) is direetionally differentiable at zero in every direction g e E,, and we have - a- ~ , . r( o 0g

+

, o ) -- 6,-(0 ' , g) = ~b(l "+, g ) - ~b(F-, g).

(11)

But this m e a n s that ~b~-(0+, 9) is quasidifferentiable at zero with _a~b~-(0+, 0 ) = F +, a6~,(O +, O) = - Z ' - . 9

!

-i-

Proposition 1'. I ' : [0, 8) --} 2 E- is differentiable from the right at zero/ff~br(0 , v) exists uniformly with respect to v ~ S. and ~br(0 ' +, 9) is quasidifferentiable at zero.

D. Melzer / On the expressibility of piecewise-linear functions

123

Thus, if F is differentiable from the right at zero we have ~b~.(0§ ( C o ( E . ) - C o ( E . ) ) ; F is positively (negatively) conically differentiable from the right at zero if it is differentiable from the right and ~b~-(0§ 9) is a convex (concave) function. In the latter case, especially if we assume ~b~-(0+,v)>0, v e S.(cb'r(O ~, v) < O, v e S.), or, equivalently, 0 e int I "+, F - = {0} (0 e int F, I'* = {0}), it is clear that for sufficiently small positive a, F ( a ) is strictly increasing (decreasing) with respect to the set inclusion. In order to derive the desired close relation between differentiable mappings and ( C o ( E . ) - C o ( E . ) ) functions, l e t f e C ( E . ) be such that it can be represented as the difference of two Lipschitzian functions g, h e C o ( E . ) . The supposition that g e C o ( E . ) is Lipschitzian is equivalent to stating that

~(x,t)

~tg(x/t), = [ g0+(x),

t>0, t~ 0

mr(O,e):={y~E,](y,u,)f(Uio) then go to Step 2. Otherwise, for an arbitrary index ioc R(0, ujo), c o m p u t e ao := min{(u~o , y ~ - y , ) [ i~ R(0, Ujo)},

(44)

take a ~ (0, ao), and define /~o := Fort {yl(y, u~o) 0 sufficiently small,

r

+av) =max fi(x +av)= i r I :k

m a x fi(x +av),

i(R(x)

(49)

where R(x) = {i ~ 1 : k ]f~(x) = ~b(x)}. Therefore, from (48),

r

r

m a x Of/(X)+max o , . ~ ( a ) . icR(x)

OZ)

icl:k

On the other hand, we also have

~b(X+ctg)>~r162

max

of,(x)

i n oi,~(a). Ov + mi~l:k

These two inequalities imply (45), thus proving the theorem.

References [1] J.-P. Aubin, "'Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions", Mathematical Analysis and its Applications, Advances in Mathematics, Supplementary Studies, Part A 7 (1981) 159-228. [2] H.T. Banks and M.Q. Jacobs, "'A differential calculus for multifunctions', Journal of Mathematics and its Applications 24 (1970) 246-272. [3] V.V. Beresnev and B.N. Pschenichnyi, "On differential properties of the maximum function" (in Russian), Journal of Compulationol Mathematics and Mathematical Physics 14 (1974) 639-651. [4] M. Bradly and R. Datko, "Some analytic and measure theoretic properties of set-valued maps", SIAM Journal on Com~'ol and Optimization 15 (1977) 625-635. [5] V.F. Demyanov, Minimax: Directional differentiability (in Russian) (Leningrad University Press, Leningrad, 1974). [6] V.F. Demyanov and A.M. Rubinov, "'On quasidifferentiable functionals" (in Russian), Doklady Akademii Nauk SSSR 250 (1980) 21-25. [7] R.V. Gamkrelidze (ed.), "Progress in science and engineering," (in Russian), MathematicaIAnalysis 19 (1981) 127-230. [8] J. Gauvin and F. Dubeau, "Differential properties of the marginal functions in mathematical programming", Mathematical Programming Study 19 (1982) 101-119. [9] E.G. Golstein, Convex programming. Elements of the theory (in Russian) (Nauka, Moscow, 1970). [10] J.-B. Hiriart-Urruty, "Gradients g6n6ralises de fonction marginal", SIAM Journal on Control and Optimization (1978) 381-416.

N.A. Pecherskaya / Ouasidifferentiable mappings

159

[11] K.H. Hoffman and J. Kolumban, "'Verlagemeinerte Differentialbarkeitsbegriffe und Anwendung in der Optimierungs theorie", Computing 12 (1974) 17-41. [12] W.W. Hogan, "'Directional derivatives for extremal value functions with applications to the completely convex case", Operations Research 21 (1973) 188-209. [13] P. Huard, ed., "'Point-to-set maps and mathematical programming", Mathematical Programming Study 10 (1979) 1-190. [14] S.S. Kutateladze and A.M. Rubinov, Minkowski duality and its applications (in Russian) (Nauka, Novosibirsk, 1976). [15] V.L. Makarov and A.M. Rubinov, Mathematical theory of economic dynamics and equilibria (in Russian) (Nauka, Moscow, 1973). [16] L.I. Minchenko and O.F. Borisenko, "On the directional differentiability of a maximum function", Journal of Computational Mathematics and Mathematical Physics 23 (1983) 567-575. [17] E.A. Nurminski, "On the differentiability of set-valued mappings" (in Russian), Kibernetika 5 (1978) 46-48. [18] N.A. Pecherskaya, "On the directional differentiability of a maximum function subject to linked constraints" (in Russian), in: Yu.G. Evtushenko, ed., Operations research (models, systems, solutions) (Moscow Computing Center, Moscow, 1976) pp. 11-16. [19] N.A. Pecherskaya, "Differentiability of set-valued mappings" (in Russian), in: V.F. Demyanov, ed., Nonsmooth problems of control and optimization (Leningrad University Press, Leningrad, 1982) pp. 128-147. [20] N.A. Pecherskaya, "On the differentiability of set-valued mappings" (in Russian), Vestnik Leningradskogo Universiteta 7 ( 1981 ) I 15-117. [21] B.N. Pschenichnyi, Convex analysis and extremal problems (in Russian) (Nauka, Moscow, 1980). [22] B.N. Pschenichnyi, "Convex multi~'alued mappings and their conjugates" (in Russian), Kibernetika 3 (1972) 94-102. [23] A.M. Rubinov, Superlinear multivalued mappings and their application to economic and mathematical problems (in Russian) (Nauka, Leningrad, 1980). [24] S. Tagawa, "Optimierung mit mengenwerten Abbildungen", Operations Research Verfahren 31 (1979) 619-629. [25] Yu.N. Tyurin, "A mathematical formulation of a simplified model of industrial planning" (in Russian), Ekonomika i Mathematicheskie Metody 1 (1965) 391-409. [26] B.Z. Vulich, Special problems of the geometry of cones in normed spaces (in Russian) (Kalinin University Press, Kalinin, 1978).

Mathematical Programming Study 29 (1986) 160-175 North-Holland

QUASIDIFFERENTIABLE OPTIMAL CONTROL

FUNCTIONS

IN

V.F. D E M Y A N O V Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9, Leningrad 199164, USSR and International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria V.N. N I K U L I N A a n d I.R. S H A B L I N S K A Y A Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9, Leningrad 199164, USSR Received 14 October 1983 Revised manuscript received 16 November 1984

This paper is concerned with nonsmooth optimal control problems in which the functionals on the right-hand sides of the differential equations describing the controlled system are nondifferentiable (more specifically, quasidifferentiable). Several necessary conditions are derived. It turns out that different variations of a control produce different necessary conditions which are generally not equivalent. As a result we obtain several necessary conditions of different complexity which may be used to solve nonsmooth optimal control problems. Key words: Nonsmooth Optimal Control Problems, Quasidifferentiable Functions, Necessary Conditions, Variation of a Control, Variation of a Trajectory, Nonsingular Control.

1. Introduction N o n d i f f e r e n t i a b i l i t y in control t h e o r y a p p e a r s n a t u r a l l y on the r i g h t - h a n d side o f the system o f e q u a t i o n s as well as in the f u n c t i o n a l ( t h r o u g h s a t u r a t i o n functions, b y t a k i n g t h e m o d u l e s , etc.). (See, e.g., [2, 6, 7, 10].) In m a n y cases b o t h the system a n d the f u n c t i o n a l are des~afbed by quasidifferentiable functions, a class w h i c h is defined a n d i n v e s t i g a t e d in [3, 4, 8]. This p a p e r is c o n c e r n e d with the v a r i a t i o n s o f t r a j e c t o r y c a u s e d by using different v a r i a t i o n s o f the c o n t r o l for such q u a s i d i f f e r e n t i a b l e f i g h t - h a n d sides. W e c o n s i d e r five different t y p e s o f c o n t r o l variations. N e c e s s a r y c o n d i t i o n s for an e x t r e m u m o f a q u a s i d i f f e r e n t i a b l e f u n c t i o n a l are then stated. The m a i n i n t e n t i o n o f the a u t h o r s is to d r a w the a t t e n t i o n o f specialists in c o n t r o l t h e o r y a n d its a p p l i c a t i o n s to a new class o f p r o b l e m s which seems to be p r o m i s i n g a n d p r a c t i c a l l y oriented. A special case o f this class o f p r o b l e m s has a l r e a d y b e e n d i s c u s s e d in [5]. 160

V.F. Demyanov et al. / Optimal control problems

161

I. 1. Statement of the problem Let the object o f study be governed by the following system o f ordinary differential equations:

Yc(t) = f(x( t), u( t), t),

(1)

x(0) = Xoe E,

(2)

where x = (x (1). . . . , xr u = (u (n . . . . , utn),f = (f 0 . We wish to find

h(t) =-(h")(t),..., h(")(t))= lira l [ x ( t , u~)-x(t~ u*)], ~-,+0

(5)

E

where the vector function h is the variation of the trajectory x* caused by variation of the control u*. It is clear that h(t) = 0 Vt ~[0, 0). For t > O + e we have

x~(t) -- x(t, u~) = xo+ +

I'

L,

f(x*(r), u*(r), r) d r +

fo+

f ( x ~ r ) , y, r) dr

dO

f(xE(r), u*(r), ~') dT.

O+e

Invoking (5), and taking the limit as e-~ +0, we obtain (see [1]) h")(t) = f(~ +

O), y, O) -f(1)(x*( O), u*( O), O)

fr

max

(v,h(.r))+

min

(w,h(r))

]

dr

Viel:n

(6)

where _Of")(r) -_~fx 0, e > 0, and r is an arbitrary (but fixed) natural number. It is clear that x~(t) = x*(t) for t 01 then we have

x~(t)=-x(t,u~)=Xo +

Io'

f(x*(r),u*(r),r)dr+

I 2

+

+

01+ el I

f(x~(r),y,,r)dr

~ 02+el2

f(x~(r), u*(r), r) d r +

~ Oro~+el (r;

f ( x , ( r ) , Y2, 7") d r + " 9 9 dO 2

f ( x , ( r ) , y,~,), r) d r +

al Or(t)

f t

f(x~(r), u * ( r ) , r) d r

Or(t)+elr(O~

(10)

"

where r ( t ) c l : r is such that

O,m < t ~ 0,o)+ ~. If r( t ) = r then Or+~= T. Without loss of generality we can assume that t > Orm+ elr~,).

(11)

V.F. Demyanov et al. / Optimal control problems

164

From (10) it follows that

h(t) =- lim 1 [ x , ( t ) - x * ( t ) ] = l~[f(x*(Oi), Yl, 01)--f(x*(01), u*(O~), 01)] e~+O E

+

max (v, h(r))+ re_in (w, h(r))] d r weOf(r) I O!~ [ vcO.f(':)

+

12[f(x*(02), Y2, 02)-f(x*(02), u*(02), 02)]

+

I 02~ [ max

(v,h(r))+ re_in ( w , h ( r ) ) ] d r + . . . w~P,f(r)

vcOf(r

+ Ir(,)[f(x*(Or(,)), Yr~t), OrU))--f(x*(Or(t))), U*(Or(t),

+

I

0,(,))]

[ max (v, h ( r ) ) + m_in (w, h(r))] dr.

(12)

weOf(r)

0tit I Ve 0 f ( r )

Now let us introduce the functions ho(t)=0

Vt~[0, T],

hi(t)=0

Vt 0i the function hi(t) satisfies the differential equation /~,(t) = max (v, hi(t))+ m_in (w, hi(t)) re=Of(t)

w~.af(t)

(13)

with initialcondition

h,(0i) = hi-,(0i)+ li[f(x*(Oi), Yi, 0i)-f(x*(O,), u*(0i), 0i)].

(14)

From (12) it is clear that h(t) = hr(,)(t). Thus, h(t) (which depends on {yi}, {Oi}, and {li}) is a piecewise continuous function satisfying the system of differential equations (8) (or, equivalently (13)) with several 'jumps' as indicated by (14).

2.3. A bundle of variations Let ~yi--u*(t),

Au~(t)=(O,

tE[Oi, Oi+eli),

t~[O, O + e ) ,

where Yi ~ V, li ~> 0, ~. ~..~ li = 1, 01 = 0, 0i+~ = 0i + eli, 0r + el, = 0 + e and r is an arbitrary natural number. It is not difficult to check that h(t)=0

Vtc[0,0).

V.F. Demyanov et al. / Optimal control problems

165

For t/> 0, we have

h(t) = f l,[f(x*(O),yi, O)-f(x*(O), u*(O), 0)] i=l

+

;r

max (v, h(r))+

re_in (w, h(r))

weaf(r)

o L vc-~f(~')

]

dr.

(15)

The variation of trajectory h(t) satisfies the system o f ordinary differential equations (8) with initial condition

h(O) = Y. li[f(x*(O),yi, O)-f(x*(O), u*(O), 0)]. i=1

The vector function h(t) depends here on {Yi}, {li} and 0.

2.4. A multiple bundle of variations ( a bundle of variations at several points) Take

l Yo - u*(t), Aug(t) = ( 0 ,

t E

Oi-t- e

~

lik, Oi + e

k~O

lik

Vj E

I:Mi V/el:N,

k'~O

t~s U [0i, 0,+eli), i~l:N

,M.

where e > 0, 0~ e [0, T), Yo ~ V, lO>1O, and 1io= 0 for all i c 1 : N, j c 1 : Mi, ~j='~ 10 = 1 and Mi and N are natural numbers. Consider the functions ho(t)=0

h,(t)=O

V t E [ 0 , T],

Vt 0i the function hi(t) satisfies the differential equation (13) with initial conditions Mi

h,(0i) = h,_,(O,)4 2 l,k[f(x*(O,), Y,k, O,)--f(x*(0i), u*(O,), 0,)]. k-O

It is now possible to show that

h(t) = hrt,)(t) where r(t) was defined in (11). The function h(t) depends on {Yo}, {0i}, and {l~}.

2.5. A classical variation Suppose in addition to the above assumptions that the set U is convex and f is quasiditterentiable jointly in x and u, i.e.,

4f(x, U~ t ) lira l [ f ( x + a h , O[h, q] ~+o ct =

max [v,. v2] ~ o_.f~.,,(t )

u+aq, t ) - f ( x , u, t)]

[(v,, h)+(v2, q ) ] +

min [ wl. w2lc alT~.~(t )

[(wl, h) + (w2, q)].

V.F. Demyanov et aL / Optimal control problems

166

Now let

Au~(t)=e(u(t)--u*(t))=--eq(t),

u~ U.

Proceeding as above we find that h(t) satisfies the system of ordinary differential equations h(t) =

[(vb h(t))+(v2, q(t))]

max [vt,o2]~_~f,,w(O

+

min

[ ~,,,w2],- ~/,.~(t)

[(w~, h(t))+(w2, q(t))]

with initial condition h(0) =0. Here _0fx,~(t) c En+r and 0fx.~(t) c En+r are convex compact sets. Thus for all of the five control variations considered here we obtain

x,(t) = x*(t) + eh(t) + o(e) where h(t) satisfies a particular system of equations, depending on the control variation chosen.

3. Necessary optimality conditions Since ~b is quasidifferentiable and Lipschitzian we have

J(u~) = ok(x*( T)+ eh( T ) + o ( e ) ) = ~b(x*(T)) + e

Oek(x*( T)) ~o(~) Oh(T)

and therefore the following necessary condition holds:

Theorem 1. If u* ~ U is an optimal control then Ock(x*( T)) Oh(T)

max

~O~r

(v, h ( T ) ) +

min

~,~(.:~T))

(w, h(T))>~O

(16)

for all admissible variations of.trajectory h(T). It is possible to obtain different necessary conditions by considering different types of control variations. Suppose, for example, that f is smooth with respect to x and that we choose a needle variation. Then equation (8) becomes an ordinary system of variational equalities (see, e.g., [9]):

h ( t ) _ O f ( x * ' u * ' t ) h(t) ' Ox with initial condition (9).

t>~O,

V.F. Demyanov et al. / Optimal control problems

167

Applying the Cauchy formula, we obtain

h(T) = Y(T) Y-~(O)(f(x*(O), y, O) -f(x*(O), u*(0), 0)) = Y ( T ) Y '(O)Arf(X* , u*, 0), where Y(t) is the fundamental matrix of solutions to the system of variational equalities,

~try(x* , u*, O):f(x*(O), y, O)-f(x*(O), u*( O), 0)). Substituting the above expression for h(T) into (16) we obtain a~b(x*(T)) ah(T)

max ((Y(T) Y-'(o))rv,/trf(x*,y*,O)) ~O,~(x'(r)) +

min

wc~d~(x*(T))

(( Y(T) Y-'(0)) Tw,/tyf(x*, u*, 8)).

Let us introduce the following n-dimensional vector functions:

Ov(O)=(Y(T)Y-'(o))Tv,

veS_ck(x*(T)),

Ow(O)=(Y(T)Y-'(O))rw,

weOck(x*(T)).

It is not difficult to see that the function ~ ( 0 ) satisfies the following system of differential equations:

dO~(O) dO

Oft(x*, u*, 0)

- - =

Ox

0~(0),

0 ~< T,

veOqb(x*(T).

0v(T) =v,

(17)

Similarly, the function 0w(0) satisfies the system dOw(0)=

afT(x *, U*, 0)

dO

Ox

0w(T) = w,

0~(0),

0 ~< T, (18)

w e 0~b(x*(T)).

Using (16) we can deduce the following theorem: Theorem 2. For a control u* E U to be optimal it is necessary that min [ y~ v

max

L vc,),;t,(x*(T))

+

min

AyH(x*, u*, d/,,, O)

weSgO(x*( T) )

/tyH(X*,U*,Ow, o ) l = o 3

VOe(O,T)

where H(x, u, O, O) = (f(x(O), u(O), 0), qJ(O)),

AyH(x*, u*, ~, 0) = H(x*, y, ~b, O) - H(x*, u*, d/, 0).

(19)

V.F. Demyanov et al. / Optimal control problems

168

Condition (19) is a generalization of the Pontryagin maximum principle [9]. Functions O.(t) and ~bw(t) are referred to as conjugate functions and systems (17) and (18) as conjugate systems. Now let us consider a multiple needle variation (and f is again supposed to be smooth with respect to x). Making use of formula (12) and passing to the limit as r ~ +co it is possible to obtain the following 'integral' necessary optimality condition.

Theorem 3. For a control u* ~ U to be optimal it is necessary that inf

max

ueU I v~Ocb(x*(T))

Io

zl.H(x*, u*, ~ . r) dr

+ minwcs,(xO(r))fro AuH(x*'u*'O~'r) dr}=O"

(20)

In the case where the set U of admissible controls is convex and f(x, u, t) is smooth with respect to both x and u it is not difficult to deduce the following condition

Theorem 4. For a control u* ~ U to be optimal it is necessary that inf I

max

u e U [. vc_~q~(x*(T))

+

min

wc~r162

r (OH(x*'u*' \ ~u ~b'' r) , u(~) - u * ( r ) ) dr

I0( \

~u

'

u(r)-u*(r)

)}

dr =0.

(21)

If ~b is a smooth function then conditions (19) and (20) are equivalent. For nonsmooth problems condition (20) may happen to be 'stronger' than conditions (19) and (21). A more detailed comparison of different necessary conditions can be found in the paper by V.N. Nikulina and I.R. Shablinskaya (see [5, Chapter IV]).

Example 1. Consider the system of two equations .~(1) = U,

~c~2)= -x(')-i-U with the initial condition xr

xC2~(O)= 0. Let the functional be defined by

J(u) = d~(x(1, u)) = Isin xtl)(1, u ) l - Isin xr

u)l.

Find a quasidifferential of function ,fi at x = 0 = (0, 0): D(k(0) = [co{(cos 0; 0); ( - c o s 0;0)}; co{(0, cos 0); (0, - c o s 0)}]

= [co{(l, o), (-1, o)}, co{(O, 1), (o, -1)}].

V.F. Demyanov et al. / Optimal control problems

169

Let V = [ - 1 , 1]. Take i f ( t ) = 0 Vte[0, 1]. Then

s

Vt~[0,1]

and

J(i) =0. Functions q,o and ~,~ (defined by (17) and (18)) satisfy the same system

(~l')= 2x~'tq/2J'

451:~=0. Therefore 4,~(0) = (v ~'~, vt2~), ~w(0) = (w "~, w~2~) u

z[0, 1]

and

H(.~,i,g,~,O)=H(s

qJ~,O)=O V0c[0, 1],

H(.~, u, i/J~,0) = v(l)u+v(2~u 2,

(22)

H(:~, u, ~p~,0 ) = wtl)u+wt2~u 2. Let us first check condition (19): max (vtl~y+ v~2~y2)+ m_in (wr vc_~,b(O)

wt2~y2) = max{-y, y}+min{-y 2, y2}

wead~(O)

--ly[-y2>~O V y E [ - 1 , 1] r o e [ 0 , 1]; i.e., condition (19) is satisfied. Now let us verify condition (21). It follows from (22) that OH(~, i, ~ , 0) = v(,~'

OH(~, i, q'w, 0) _ w(,)

3u

3u

and

v~)u(O) dO+ m_in

max

v~_~(O)

w~a,b(0)

=max

{fo'

u(O)dO;-

fo

w~)u(O) dO

}Fro

u(O) dO =

I

u(O) dO >!0 Y u z U

i.e., condition (21) is also satisfied. But now we shall show that nevertheless condition (20) is not satisfied for i. To do this it is not necessary to find infimum in (20), it is enough to pick up a 'violator' of this condition. Take fi(t)=

1, -1,

tz[0, 89 tE[~,l].

V.F. Demyanov et al. / Optimal control problems

170

Then inf _ m a x uc u [woq,(o) + rain

we~q~(o)

!0 (25)

is satisfied for every y c V and for almost all 0 ~ [tk-1, tk] where k c I : ( N + 1). Here q,v(0) is the solution of the system of linear equations ~b(0)=--AT(0)qJ(0)

VO~[tk_,, tk]

(26)

with the terminal condition $o(tk) = RTv,

v60_Ob(x*(T)),

and Sw(0) is the solution of system (26) with the terminal condition ~bw(tk) = R~w,

w~qb(x*(T)).

Note that to check condition (16) it is necessary to solve the systems (8)-(9) for each y c V and every 0~[0, T]; but to check condition (25) it is enough to find functions ~0o(0) and ~0w(0) and then we can use them for all 0 and y.

Example 2. Let us. consider the following system of two equations .~(I) ----_Xf2),

(27) X'z'-- 21xI.

Recall that a finite function f defined on E, is called positively homogeneous if f(Ax) = Af(x) VA ~>O.

178

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

Let 12 be a set in E , , 0 9 int 12. The function

Ixl---Ixl,

= inf{)t > 0 ] x e )tO}

(1)

is called the gauge of set .(2 (or the Minkowski gauge function). If 12 is convex then the gauge coincides with the gauge function familiar from convex analysis; if 12 is a ball then the gauge is a n o r m corresponding to this ball.

Theorem 1. Let s be a functional defined on E,. The following propositions are then equivalent: (a) the functional s is positively homogeneous, nonnegative and continuous; (b) s coincides with the gauge o f a star-shaped set I2, where 12 = {xl s ( x ) O Is(x) O[xe AO} = U {A > Ol x ~ AO,},

(2)

i

{A > O l x e AO} = O {A > 0 l x e ) t O , } ,

(3)

i

which can be verified quite easily. Thus, the union and intersection of a finite number of star-shaped sets are themselves star-shaped sets. Furthermore, the union coincides with the infimum and the intersection with the supremum of these sets in lattice 5f.

Proposition 2. Let A be a set of indices and U~ be a star-shaped set with gauge ]. [,. I f the function Ix[ = i n f ~ a Ix[~ is continuous, then it is the gauge of the set cl U ~ u~. If the function ]xl = sup,,~a ]xl~ is finite and continuous, then it is the gauge of the set

A,,u~. We shall prove only the first part of the proposition. Since the function Ix I = inf~ca Ix]o is continuous it follows from Theorem 1 that this function is the gauge of some star-shaped set _O. It is now not difficult to check that

~=cJU u,,. Indeed, the continuity of functions I" [ and I" [,, implies that int _0 = {xllxl < 1} = {xlinf Ixl~ < Ct

1} = U t~

int U..

180

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

Therefore, taking into account the regularity of star-shaped sets we get O = cl int _/2 = cl U int u~ = cl I J u~. at

at

This proves the first part of the proposition.

3. Addition and multiplication The algebraic operations of addition and multiplication by a nonnegative number have been introduced within the family E{ of gauges of star-shaped sets in a natural way. We shall now introduce corresponding operations within the family ~ with the help of isomorphism ~,. Let ~ c 5e, h/>0. We shall describe the set h @ ~ with gauge ['1 = h ] ' l a , where [" It~ is the gauge of O, as the inverse product of set ~ and number h. The set ~ ( ~ ~2 with gauge l" I which satisfies the relation

1"l=l'l,+l'12, where [. I~ is the gauge of set 0~, is called the inverse sum of the shar-shaped sets ~21 and ~2. It follows from the definition that if h > 0 then

~o~ =!~. A If h = 0 then the set h O ~ coincides with the entire space E.. We shall now describe inverse summation. To do this we require the following elementary proposition.

Proposition 3. Let a , , . . . , am be nonnegative numbers. Then 1

al+'''+am=

min m a x i a i

(4)

(where it is assumed that 0 / 0 = 0). If a~ = 0 Vi then (4) is trivial. Otherwise, for any set {ct~} such that a~/> O, ~ a~ = 1 there exists an index j such that aj aJ ~ k=|

a k.

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

181

At the same time max i

-O1~i

ai = .

at,,

k=l

and this proves the proposition. Now let us consider star-shaped sets /21 and /22 with gauges l-In, and [.In2 respectively, and let 1. [ be the gauge of their inverse sum/21G 122. Then the following equality holds for every x: Ix[ = [xln, + ]xln: = min max

Ix[n,,

x],~2

0 ~ ar ~ I

min

Ixl,I-~

0~: tr ~ 1

O~t~'~

1

I 1.,

where l" Io is the gauge of set ct/2~ c~ (1 - ct).O2. (It is assumed that 0 9 = r'l~>o a/2.) Since the function l" I is continuous it follows from Proposition 2 that O1|

U

[~c~(1-a)/22].

0 ~ 1

Note that the role of zero (a neutral element) with respect to summation in a 'semilinear space' 5~ is played by the space E, (since the gauge of E, coincides with the identity zero). At the same time, E, is the smallest element of the ordered set ,5C We shall now give some computational examples.

Example 1. Consider the following rectangles in E~: U = [ - 1 , 1] x [ - 2 , 2],

V, = [-2A, 2A] • I-A, a].

Their inverse sum coincides with an octagon which is symmetric with respect to the coordinate axes. The intersection of this octagon with the first quadrant has the vertices: A

2A

2A

Rectangles U and V1 and their inverse sum are shown in Fig. 1. The set U | shown in Fig. 2.

V~o is

Example 2. Let U={(x,y)e E21y~~ @ v,,

(u,, e , ) - ( v , , v2) r

u,|189

u2|

We shall now factorize the set bv2 with respect to the equivalence relation ~. In other words, we shall consider the family T of all classes of equivalent pairs. Since the operators O and Q produce equivalent pairs when applied to equivalent pairs, the operations for inverse summation and inverse multiplication by a number can be introduced within T in q ~ t e a natural way. The order relation within T is derived naturally from 6e2. An element of T which contains a given pair (U~, /3"2) will be denoted by [ U], U2]. We shall identify an element U of the set 6e with the element [ U, E,] of the set T. The equality [ U,, U2] = [ U1, E,]@[En, /3"2]= [ U,, E . ] O [ U~, E~] (where srQ 77 = ~:@ ( - 1 ) Q 7/) then implies that every element of T can be represented as the difference of two elements of ~, i.e., T is the smallest vector-ordered space containing ~T. For this reason we shall call ~ the space of star-shaped sets (compare with the space of convex sets).

184

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

We shall associate with every pair ( U,, U2) e 5e2 a positively homogeneous function [. [, is the gauge of Ui. It is clear that two pairs generate the same function if and only if they are equivalent. Hence, the function f = l" h -1" 12 Co(E,) is associated with every element [ UI,/-/2] of the space T. Conversely, by representing a continuous positively homogeneous function f in various forms f = f l -f2 (where f E ~ ) , we conclude that every element of the space Co(E,) is associated with the class of equivalent pairs [U~, Uz], where Ui = {x[f(x) 0 1 - h Ilxll 01U = ;t Q B}. Let X be a star-shaped c o m p a c t set in E,, and ,~ be some subset of the family 5e(X) of all star-shaped subsets of X. Let U e , ~ , and I1 be the gauge of U. We shall consider the sets

~u = { x l l x l ~ = l } (the b o u n d a r y of U) and

a~, - - { x l l - e ~ 0 such that a~j + B~ c a~ V U c ~ where Ba =

{xlllxll < 81. Proof. Let us consider the set ~Ti.I of all functions from C ( S ~ ) - - t h i s represents a contraction (on S~) of the gauges of sets from ,~. The fact that the set 2~ is c o m p a c t is equivalent to the set E I I being compact. By the Arzeid-Ascoli t h e o r e m this property of E I I is equivalent to this set being b o u n d e d and equicontinuous. It is clear that condition (i) is satisfied if and only if-Vl. I is bounded.

186

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

We shall now show that c o n d i t i o n (ii) is e q u i v a l e n t to I; I.I b e i n g e q u i c o n t i n u o u s a s s u m i n g that (i) holds. Let c o n d i t i o n (ii) be satisfied a n d U ~ E. First o f all note that there exists an r > 0 such that for any U 9 I; a n d any x for which Ixl u = 1 the following i n e q u a l i t y holds: ]]xll > r. It follows i m m e d i a t e l y from c o n d i t i o n (i). Since X is c o m p a c t then there exists an R < oo such that

Ilxll 0 there exists a 8 > 0 such that relations Ilxll -- Ilyll = ] and [Ix - y l l < 8 imply

Ilxl.-lyl.[
~[U2, I/2] if U , - V2 m/-/2- VI. Let L be the subspace of the space Co(E,) which consists of functions which can be represented by the sum of a convex function and a concave function. The mapping qb : M ~ L defined by 4 ( [ U, V])(x) = max (u, x) + m i n (v, x) ut_ U

(9)

oC V

is an algebraic and ordering isomorphism (it is, of course, assumed that L is provided with natural algebraic operations and an order relation). The inverse mapping ~-~ associates an element [_0p,~q] from M with a function p + q ~ L (where _~p is the subdifferential of the sublinear functional p and ~q is the superdifferential of the superlinear functional q). Let us consider a subset U of the space En. Let U ~ denote its polar:

U~

Vu~ U}.

Here (and in (9)) 0', x) is the scalar product of y and x. Let us recall the main properties of the polar: (i) The set U ~ is convex and closed; 0 c U ~ (ii) I f U is convex and closed and 0 c U, then U ~176U. (iii) U is compact if and only if 0 c int U ~ (iv) Let U be a convex closed set, with 0 e U. Then the gauge function of U coincides with support function of the polar U ~ and the support function of U coincides with the gauge function of the polar. (v) Let UI and U2 be convex and closed and let 0 ~ U1, 0 ~ U2. Then the relations UI D U2, U ~c U ~ are equivalent and

(U1+U2)~176

~

(AU)~

h

~ ifh>0.

Now let us consider star-shaped convex sets U~ and U2. Since 0 ~ int U,, the polar U ~ is compact. Since the gauge [. [i of the set U, coincides with the support function of the polar U ~ the following relations holds: [ x [ , - [x[2 = max (1, x ) - m a x (1, x) 1~u ~ 1~tJ~ = max (l, x ) + l~u, ~

min i~t-u2 ~

(l, x).

(10)

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

189

Let ~b and 9 be mappings defined by formulas (5) and (9) respectively, ~t be an element of the space T containing the pair (U1, U2), and fl be an element of the space M containing the pair ( U ~ - U ~ From (10) it follows that

and hence = (~-1~)(,~).

The operator 7r = ~-~q/ defines the operation of taking the polar (or is the polar operator). It is defined on the subspace Tc of the space T which consists of elements a such that there exists a pair (U, V) c a, where U and V are convex sets. It is clear that Tc is a linear space (this follows from the equivalence of the convexity o f a star-shaped set and that of its gauge). The set of values of the operator Tr coincides with the space of convex sets. Indeed, for f l e M it is always possible to find a pair ( U, V) e fl such that 0 e U, 0 e V. Then U = (U~ ~ V = (V~ ~ so that fl = 7ra, where ~ = [ U ~ V~ e To. From the properties of the polar it follows that the operator r is linear and order-preserving.

6. Quasidifferentiability and a geometrical interpretation of directional derivatives

The space of star-shaped sets can be used to provide a geometrical intepretation of directional derivatives. Let f be a function defined on an open set f~ c En and suppose that at a point x ~ E, we can construct the directional derivative o f f : af(x) -f'x(g) =- lira 1 [ f ( x + c t g ) - f ( x ) ] , 8g , ~ o o~ where the function fy~(g) is continuous in g. Since the functional f~ is positively homogeneous, an element of the space T of star-shaped sets is associated with f~. In other words, a pair of star-shaped sets (U, V) exists such that

f'(g) = min{A > 0 1 g e A U } - m i n { A > 0 ] g ~ AV} or, equivalently, f ' ( g ) = min{A > 01ge AU}+max{A < 0 ] g c ( - A ) V}.

(11)

Note (from equation (11)) that the pairs (U, V) and (U~, V,) represent the derivative o f f if and only if they are equivalent. Let us denote the set U in (11) by Of(x) and the set V by df(x). Invoking the properties of the space T of star-shaped sets, it is possible to state rules for algebraic operations over functions and the

190

A.M. Rubinov and A.A. Yagubov / The .space of star-shaped sets and its applications

corresponding pairs:

d_(fl + f2)(x) = _dfl(x)(~ d_f2(x), a ( f , +f . ) ( x ) = a f , ( x ) |

d_(fl "f2)(x) =fl(x)Qd_f2(x)@f2(x)Qdfl(x), a ( f l 9f2)(x) = f l ( x ) Q a f 2 ( x ) O f 2 ( x ) Q a f , ( x ) . Using formulas (6) and (7) and the rules for ditterentiability of the m a x i m u m function it is easy to find _d(maxfi(x)),

a(maxfi(x)), i

d(minf~(x)), -

i

d(minfi(x)). i

It is clear that a function f is quasiditterentiable at x if and only if there exist convex sets df(x) and a f ( x ) . In this case

df(x) = [of(x)] ~

af(x) = [ - ~ f ( x ) ]

~

where 0f(x) and -gf(x) are a subdifferential and a superdifferential, respectively, of f at x. We shall now present a geometrical interpretation of necessary conditions for a minimum. It is based on the following lemma. Lemma 1. Let a functional f be directionalty differentiable at x c 17.,, the derivative f ' ( g ) be continuous in g and :K be a cone in E,. Then (i) The relation min f ' ( g ) = 0

is satisfied if and only if Of(x) ~ 5~c 8f(x). (ii) The relation max f'(g)=0 g t= :,Y

is satisfied if and only if d f ( x ) c~ :K c df(x). Proof. Let us write f ' ( g ) in the form

f'x(g) = Igl, -Ig[2, where l" I1 is the gauge of the set d_f(x) a n d [ . 12 is the gauge of the set df(x). Assume that minf'(g)=0

and

ged_f(x)c~ffL

Then Ig, I 0 such that lag[, = 1. Then Xg ~ d_f(x). But since hg c (If(x) we have the i n e q u a l i t y IAgl= 0. Thus, if [gh = 0 then [g[2 = 0 (since [g[2 0: y + a v c a ( x + ag) V a ~ (0, ao]}.

(18)

F ( x , y, g) = cl 7(x, y, g). We say (see [4]) that the m a p p i n g a:E~--> 2 ~,,, allows first-order a p p r o x i m a t i o n at x ~ En in the direction g ~ E, if for any numerical sequence {ak} such that a k --~ d-0 and any convergent sequence {Yk} such that YR E a ( x d- akg), Yk -~ Y, the representation YR = Y + Os -~- O( a k) holds, where Vk~I'(x,y,g),

CtkVk~O,

yea(x).

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

195

Assume also that a is a continuous m a p p i n g and that the t o p o l o g y of ~ is induced from the Banach space T. This is equivalent to saying that the m a p p i n g x-~ [. [~(x) = I t ( x , . ) is continuous. Fix an element yo~ Era, and for x ~ g2 take V ( x ) = [it(x, Yo), +oo). We shall now describe the set Fv(x, ", g) (the closure of set yv(X, ", g) constructed from formula (18)). Let A ~ V(x). The relation v ~ 7v(X, A, g) means that, for a sufficiently small, we have I t ( x + ag, Yo) It(x, Yo) then (19) is valid for every v (with a sufficiently small). IfA = It(x, Yo) then (19) can be rewritten in the form 1 v / > - - [ i t ( x + ag, Yo) - i t (x, Yo)]O~

N o w we have ( (-oo, +oo), Z'v(X, A, g) = {t [ ~ ' ( x , Yo, g), +co),

A > v-(x, Yo), A = It(x, Yo),

where /%'~(x, Yo, g) = lim 1 [ i t ( x + ag, Yo) - v- (x, Yo)]. c~ ~ t - 0 0 t '

Proposition 6. A mapping a is weakly star-shaped directionally differentiable at x if and only if the mapping V allows first-order approximation in every direction f o r all Yo ~ E,. Proof. 1. Let V be such that first-order a p p r o x i m a t i o n is allowed in a direction g, and

ot k -q. + 0 .

Then

I t ( x + akg , Yo) ~ It(x, Yo) and therefore It ( x + akg, Yo) = It (X, Yo) + akVk + O( ak ), where Vk ~ V-~(X, YO, g)" This leads to I t ' ( x , Yo, g) = lira __1 [v-(x + akg, Yo) -- It(x, Yo)] ~> ~ ' ( X , Yo, g). Otk

2. Let a be directionally differentiable. Then the derivative v-'~(x, Yo, g) exists for every y o g a ( x ) , g e En. Let Ak~A, Ak~ V ( X + a k g ) . Then Ak >1 V- ( X + akg, Y0) = It (X, Y0) + ak It'~ (X, Y0, g) + O( ak ). If A = It(X, Yo), set Vk = Otkv-t~(X,YO, g) and we have a representation which is used in the definition of the first-order a p p r o x i m a t i o n . If A > v-(x, Yo) then this representation is obvious, and the proposition is proved.

196

A . M . Rubinot~ and A.A. Yagubov / The space o f star-shaped sets and its applications

Remark. The gauge function can be viewed as a minimum function with dependent constraints ~t(x, y0)= min h, x~_ V(x)

and therefore its differentiability can be studied with the help of a theorem by Demyanov [2]. However, this theorem is proved under the assumption that V allows first-order appI'oximation. Proposition 6 shows that this assumption is absolutely essential in the case under consideration. It is clear that the inverse sum of weakly differentiable mappings is also weakly differentiable. If a is weakly differentiable,f(x)/> 0 a n d f is a directionally differentiable function, then the mapping b ( x ) = f ( x ) Q a ( x ) is also weakly differentiable. Let a~:12--> 6e (i ~ 1: N ) be a weakly directionally differentiable mapping. Then the union of these mappings _a(x)=LJ~l:N a~(x) and their intersection a ( x ) = ['-'),~l:n a~(x) are also weakly directionaily differentiable. I f / ~ is the gauge of the mapping ai then the derivatives of the gauge functions /2 and /z of the mappings a and a are described by the following equations: /2'~(x, y, g) = max

ic R(x, v)

/zi(x, y, g),

~',(x, y, g) = min /xi(x, y, g), i~. Q(x,.v)

where R ( x , y ) = { i e 1: Nlli(x,y)=/z,(x, y)}, Q(x,y)={i~ 1: Nl~_(x,y)=/xdx, y)}. We shall now consider some examples of weakly differentiable mappings. Example 4. Let l: En ~ Em be directionally differentiable and set

a(x) = {yi(l(x), y) -0, if (/(x), y) < 0,

[max{(l'x(g),y),O}

if(l(x),y)=O.

Thus the mapping a is at least weakly differentiable. The function y--, # ' ( x , y, g) may be discontinuous, and in this case the mapping a is not strongly differentiable. Example 5. Let a(x) = {y](li(x), y) OVi~

l:m}

198

A.M. Rubinov and A.A. Yagubov / The space of star-shaped sets and its applications

a n d a s s u m e that 12 is not empty. C o n s i d e r the m a p p i n g

a(x) = { y c E,,iy 1 i

and max gi(x) ~ O, i

where gi(x) = 1 -fi(x).

9. Trajectories of star-shaped mappings Let us now discuss the asymptotic behavior o f trajectories generated by a starshaped mapping. Problems of this type c o m m o n l y arise in mathematical economics, where they are studied u n d e r additional convexity assumptions. The same problems without the convexity assumptions have been discussed in [9]. Let ~ be a star-shaped c o m p a c t set in En. A Hausdorff continuous m a p p i n g a:~llst(~) defined on ,T is called a discrete dispersible dynamic system ( 0 3system). Here Hs,(~) is the family o f all star-shaped subsets o f set ~. A sequence {x~ I i ~ 0, 1 . . . . } o f elements o f ~ such that xi.lEa(xi) ,

ie0,1 .... ,

is calle&a trajectory o f the D3-system a. A n o n e m p t y subset H o f set ~ is called a semiinvariant set o f the D~-system a if a ( O ) c O. Take

Pa(~:) = cl U a'(~) t=l

for sc~ llst(~), where a ' + ' ( ~ ) = a(a'(~)). A point x e ~T is called a Poisson stable point o f D3-system a if x e Pa(x)= P~({x}). A set /~/~ IIs,(gT) is called a turn-pike set o f D3-system a if p(x,, 1(4)-~ 0 for any trajectory {x,} o f this system. Let M denote the intersection o f all turn-pike sets.

200

A.M. Rubinov and A.A. Yagubov / The space o f star-shaped sets and its applications

A functional h defined on eT is said to be in equilibrium if h is continuous, h(x) >~O V x e ~, and

h(y) 0 such that Ok + gk ~ ( l + "rf)C. Taking the limit as k-->oo we obtain v~ (1 + ' q ' ) C , which contradicts the inclusion v e C and thus proves the lemma.

T h e o r e m 4. I r a : ~ ~ lI~t( ~ ) is a quasihomogeneous mapping and a ( x ) c Z f o r every

x, then W = M = ~ where Y( is the f a m i l y o f all Poisson stable points.

Proof. It is necessary to check the inclusions ~ D W, M D ~. 1. We shall first verify ~ ~ W. If x ~ ~, then x ~ Pa(x). T h e set P~(x) is star-shaped (since a ( x ) c Z and Z is compact) and semi-invariant. Let h be the function defined in Proposition 7 with respect to set ~: = P~(x). Then h is in equilibrium and since x ~ P a ( x ) we have h ( x ) > 1. However, we also have a ( x ) e P a ( x ) and therefore ( h o a ) ( x ) = l . Thus x ~ Wh and hence x ~ W. 2. To verify M D ~, we first let x e ~ , i.e., x e P~(x). From L e m m a 2 it is clear that for every e e (0, l) there exists a n u m b e r t such that (1 - e)x e a'(x).

(20)

Consider a sequence o f positive numbers {ek} such that [ I k - i ( 1 - Ek) converges to some n u m b e r ~,e(0, 1). Using (20) and the q u a s i h o m o g e n e i t y o f a we can construct a trajectory X = {x,} starting from x and containing the subsequence {% =l-I~=~ (1--ek)X}. This means that ux is a limit point o f the trajectory x and therefore u x e M. Since u is an arbitrary n u m b e r we conclude that x e M. This completes the p r o o f o f the thereom.

References [1] J. Cassels, An introduction to the geometry of numbers (Springer-Verlag, Berlin, 1959). [2] V.F. Demyanov, Minimax: directional differentiability (in Russian) (Leningrad University Press, Leningrad, 1974). [3] V.F. Demyanov (Ed.), Nonsmooth problems in the theory of optimization and control (in Russian) (Leningrad University Press, Leningrad, 1982). [4] V.F. Demyanov and A.M. Rubinov, "On quasidifferentiable functionals', Doklady Akademii Nauk, SSSR 250 (1980) 21-25. (Translated in Soviet Mathematics Doklady 21 (1) (1980) 14-17.) [5] V.F. Demyanov and A.M. Rubinov, "On some approaches to the nonsmooth optimization problem" (in Russian), Ekonomika i Matematieheskie Metocly 17 (1981) 1153-1174.

202

A.M. Rubinov and A.A. Yagubot~ / The space of star-shaped sets and its applications

[6] V.F. Demyanov and A.M. Rubinov, "On quasidifferentiable mappings", Mathematische Operationsforschung und Statistik, Series Optimization 14 (l) (1983) 3-21. [7] B.N. Pschenichnyi, Convex analysis and extremal problems (in Russian) (Nauka, Moscow, 1980). [8] R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, 1970). [9] A.M. Rubinov, "Turn-pike sets in discrete dispersible dynamic systems'" (in Russian), Sibirskii Matematicheskii Zhurnal 21 (4) (1980) 136-145.

Mathematical Programming Study 29 (1986) 203-218 North-Holland

e-QUASIDIFFERENTIABILITY OF REAL-VALUED FUNCTIONS AND OPTIMALITY CONDITIONS IN E X T R E M A L P R O B L E M S V.V. G O R O K H O V I K Institute of Mathematics, Academy of Sciences of the Byelorussian SSR, Surganov St. 11, Minsk 220604, USSR Received 2 December 1983 Revised 6 June 1984

In this paper we demonstrate how the concept of quasidifferentiability introduced by Demyanov and Rubinov may be extended to the more general concepts of e-quasidifferentiability and approximate quasidifferentiability. We study the e-quasiditterentiability of composite functions and present some rules for e-quasidifferential calculus. The optimality conditions for some typical extremal problems are restated in terms of e-quasiditterentials.

Key words: Nonsmooth Analysis, e-Quasidifferential, Local Extremum, Inequality and Equality Constraints, Optimality Conditions.

1. Introduction

The main aim of this paper is to show how the concept of quasidifferentiability introduced into nonsmooth analysis by Demyanov and Rubinov [4] (see also [ 1, 2, 5, 6, 16]) may be extended to the concepts of e-quasidifferentiability and approximate quasidifferentiability. These concepts are more general than that of quasidifferentiability, every quasidifferentiable function being also approximately quasiditterentiable. In addition, we shall see that the class of approximately quasidifferentiable functions contains locally Lipschitzian functions that are also directionally differentiable. The concepts of e-quasidifferentiability and approximate quasidifferentiability (termed simply quasidifferentiability) were introduced in earlier papers by the present author [8, 9]. The structure of the paper is as follows. Section 2 provides a summary of the results and notation which are used in the body of the paper. In Section 3 we introduce the concepts of e-quasidifferentiability and approximate quasidifferentiability for real-valued functions. We also investigate the property of e-quasidifferentiability for compositions of e-quasidifferentiable functions, and present some rules for e-quasidifferential calculus. A number of typical extremal problems are considered in Section 4 and their optimality conditions are derived in terms of equasidifferentials. It should be noted that the choice of extremal problems considered in Section 4 and in fact the paper as a whole was greatly influenced by [1, 2, 4, 5, 6, 16]. Some familiarity with these papers would be useful to the reader. 203

V. V. Gorokhovik / e-Quasidifferentiability

204

2. Difference-snhlinear functions and their quasidifferentials Let ~ ( E . ) be the vector space of real-valued positively homogeneous ( h ( h x ) = hh(x), h > 0 , x ~ E . ) continuous functions defined on n-dimensional Euclidean space E., and l e t / q ( E . ) be a convex cone in Y((E.) consisting ofsublinear functions. We shall say that a function h from 9((E.) is difference-sublinear if h may be represented as the difference of two sublinear functions, i.e., if there exist sublinear functions h and k7 such that h ( x ) = h ( x ) - h ( x ) V x ~ E.. The collection of all difference-sublinear functions is denoted by H(E.). It is not hard to see that H(En) is the smallest vector subspace of Yg(E.) which contains /q(E.). Moreover, the vector subspace H ( E . ) is closed with respect to the operations of taking pointwise m a x i m u m and minimum on finite subsets of H(E.). To demonstrate this we consider difference-sublinear functions h, and h~ and assume that hi(x) = hi(x) - ~(x), x c E., i = 1, 2 where _hi,/~i, i = 1, 2, are sublinear functions. From the equalities max{hi(x), h2(x)} = max{_hl(x) +/~2(x), _h2(x)+/~,(x)} - (/~l(x) +/~2(x)) and min{ hi(x), h2(x)} = (_h,(x) + h2(x)) - max{/~,(x) + _h2(x),/~2(x) + _hi(x)} we can conclude that max{h,(x), h2(x)} and min{h,(x), h2(x)} are also differencesublinear functions. It is well-known that each sublinear function h e / ~ ( E . ) is uniquely associated with a convex compact subset ah(0) = {v c E, [(x, v) /0. A function f is said to be e-quasidifferentiable at a point x i f f is directionally ditierentiable at x and there exist convex compact sets O_~f(x) c En and -~ff(x)c En such that

If(g)-

max ( g , v ) + max v~-Orf(x)

wcS~f(x~

(g,w)[ e. In particular we can consider a quasidifferential Df(x) of a quasidifferentiable function f to be an e-quasiditterential of f for any e > 0. Hence quasiditierentiable functions are also e-quasidifferentiable for any positive e. However, this property is not limited to quasidifferentiable functions--it characterizes a much wider class of functions which can be defined as follows: Definition 2. A function f is said to be approximately quasidifferentiable at a point x i f f is e-quasidifferentiable at x for any positive e. The next theorem gives a criterion for approximate quasidifferentiability. Theorem 1. A function f: tJ ~ El is approximately quasidifferentiable at a point x if

and only if f is directionall): differentiable at x and its directional derivative f " : E, ~ E~ is continuous. Proof. This theorem follows immediately from the fact that the subspace H ( E , ) of difference-sublinear functions is dense in the space ~ ( E , ) of positively homogeneous continuous functions with respect to the norm II h II ~-: max I1~11~1Jh (x)[. From this theorem it follows that any uniformly directionally differentiable function is also approximately quasidifferentiable.

V.. V. Gorokhovik / e-Quasidifferentiability

207

Recall [3, 7, 11] that a f u n c t i o n f i s said to be uniformly directionally ditterentiable at a point x if it is directionally differentiable at x and for any g e E, and any A > 0 there exist ~ > 0 and 3' > 0 such that

It

' ( f ( x + tz) - f ( x ) ) - f ' ( g ) [ < A

for all t e (0, 3') and all z e Sa(g) = {y e E. [ Ily- gll ~ ~}. In particular, any function f that both satisfies the Lipschitz condition at the point x and is directionally differentiable at x is uniformly directionally differentiable at this point. Hence any locally Lipschitzian function that is also directionally differentiable must be a p p r o x i m a t e l y quasiditterentiable. Let a function f : U - E~ be a p p r o x i m a t e l y quasidifferentiable at a point x. We shall use ~ , f ( x ) to denote the set whose elements are e-quasiditterentials of the function f at x. From the definition of a p p r o x i m a t e quasiditterentiability it follows that the set @~f(x) is not e m p t y for any positive e and that ~ , f ( x ) c ~ , f ( x ) , where e~E

t.

The collection ~ f ( x ) = {~,f(x)l e > 0} is called an approximate quasidifferential of the function f at the point x. The intersection ("]~>o ~ , f ( x ) is not e m p t y if and only if the function f is quasiditterentiable at x, and in this case any element of("~,> o ~ f ( x ) is a quasiditterential o f f at x. R e m a r k 1. O u r a p p r o a c h to the notion of a p p r o x i m a t e quasidifferentiability is based on the uniform a p p r o x i m a t i o n of the directional derivative by ditterence-sublinear functions. In this sense it is similar to W a r g a ' s a p p r o a c h to the derivative container [18, 19], which is based on uniform a p p r o x i m a t i o n by C~-functions. As a first step towards a calculus for e-quasidifferentials we shall now derive a chain rule for the c o m p o s i t i o n of e-quasidifferentiable functions. Theorem 2. Let U be an open subset of En and V be an open subset of Era. Consider given functions f ~ : U ~ E l , i ~ l : m , u: V ~ E ~ and a point x ~ U such that y = (fl(x) . . . . ,f,,,(x))e V. Suppose furthermore that the functions .~, i e l : m , are e~quasidifferentiable at the point x and the function u is uniformly directionally differentiable at the point y. Let D , ~ ( x ) ; i e 1 : m, be any e~-quasidifferentials of the functions f~, i e l : m at the point x and D~u(y) be an ~-quasidifferential of the function u at the point y.

Then the composite functions S: x' ~ u ( f l ( x ' ) , . . . , f,, (x') ) is e-quasidifferentiable at the point x for e-= max

Ih,le,+ max

Ae_~;u(y)i=l

A ~ e u ( ) ' ) i=1

and if vectors p = ( v b . . . , v~ maxi~l:m f ( x ' ) is e-quasidifferentiable at the point x for e = maxi~i(x) el, l ( x ) = {i ~ 1: m[fi(x)=maxi~l:,.fi(x)}, and

i(:l:m

L

kiEl(x) \-

'

is an e-quasidifferential of the function at x.

je_l(x)

ie_l(x)

210

V. V. G o r o k h o v i k / e-Quasidifferentiability

(c) the function x ' ~ m i n ~ t : m f ( x ' ) is e-quasidifferentiable at the point x for e = max,~1(x) el, I ( x ) = {i c 1 : m l f ( x ) = m i n i ~ l : m f ( x ) } , and

": :

i~l(x)

li~:l(x)

j~-I(x) jr

is an e-quasidifferential o f the function at x. To prove Corollary 1 we make use of Theorem 2, defining the function u: Em -+ E1 as follows: (a) u(y)=Y~_~ AtYi; (b) u ( y ) = m a x i ~ : , , y ~ ; (c) u ( y ) = m i n ~ c l : , , y , We close this section with the following remark. Remark 2. In those cases where the function f : U ~ E~ is not directionally differenti-

able at the point x, the notions of e-quasidifferentiability and approximate quasidifferentiability may be defined on the basis of other positively homogeneous local approximations. Lower and upper Dini's derivatives [10, 15] are particularly suitable for this purpose. The approach based on Dini's derivatives is described in detail elsewhere [8, 9]. Here we note only that every locally Lipschitzian function is lower and upper approximately quasidifferentiable.

4. Conditions for a local extremum

In this section we shall present conditions for a local extremum expressed in terms of e-quasidifferentials. Let U be an open set in E. and f : U - E1 be a real-valued function with a finite value at a point x ~ U. Let the function f be e-quasidifferentiable at the point x for some e/> O. Theorem 3. l f a function f attains a local minimum ( m a x i m u m ) at a point x then for

any e-quasidifferential D ~ f ( x ) we have -O.f(x) = O_J'(x.) + eB.

(O_~f(x) c - ~ f ( x ) + eB.),

( 11 )

where B. is the closed unit ball in E.. Proof. Since [Igll = maxt,,n,. (g, v) we can rewrite relation (4) from Definition 1 in the following way: max (g, v ) v~_i_Jrf(x)

~O V g ~ En. Hence from the right-hand side o f the preceding inequality we have max (g, v)!O) of f a t x there exists a real ~ > e such that -O~f(x) + 8B, c O~f(x)

(~_,f(x) + 8B, ~ g,f(x)),

(12)

then the function f achieves a strict local minimum (maximum) at the point x. Proof. We shall confine ourselves to proving the sufficient condition for a strict local minimum. From condition (12) and the definition o f an e-quasidif[erential we obtain 0~(,5-e)llgll

VgE E,.

(13)

We shall n o w argue by contradiction. Suppose that f does not attain a strict local m i n i m u m at the point x. Then there exists a sequence xk ~ U, k =.1, 2 , . . . , tending to x and such that f(Xk)~O

VgcA(xtG),

(15)

where A ( x I G) is the cone of feasible directions for G at the point x [3]. Since the inclusions {g ~ E~ [ u ' ( g ) < 0} c A ( x I G) c {g E E. l u'~(g) N, which contradicts the fact that (17) is inconsistent. Hence, system (20) must be inconsistent.

V.V. Gorokhovik / e-Quasidifferentiability

214

It follows from the inconsistency of (20) that

ch(1)~,f(x) )(g) + el I[gll ~>0 for all g satisfying the inequality

d~(D~2u(x))(g) + ~211gll ~ 0. Continuing the argument as in [1, 2, 5] leads to (19), thus proving the theorem. Remark 3. The necessary condition presented in [1,2, 5] for this problem with quasidiiterentiable functions f and u is a consequence of Theorem 6. Remark 4. Condition (14) is a consequence of condition (19). Indeed, noting that the pairs [~,f(x)+-O~:u(x), O~,f(x)+O~u(x)] and [O~:u(x)+-O~,f(x), -O~u(x)+ 0~,f(x)] are respectively el- and e2-quasidifierentials of the functions f and u at x, we can rewrite condition (19) in the following way:

"O.,f(x) + "O~u(x) ~ O~,f(x) + -O~u(x) + e, B. + cone(_0.,u(x) + -O~,f(x) + e2B. s { w})

(21)

for all we'O~,f(x)+O~u(x). Let v ~ -g.,f(x) and w e O.~u(x) and let us consider two possible cases: (a) v+w~O~u(x)+-a.,f(x)+ezB., (b) v + w c O~2u(x)+'~,f(x)+ezB,,. Since for any convex compact set M satisfying 0~ M we have cone M = cone M then for case (a) it follows from (21) that

v+ w e O_~,f(x)+'O~u(x)+ e,B. +cone(O~u(x)+-O~,f(x)+ e2B. - { v + w}). Hence there exist vte O~,f(x) +-O~u(x) + e 1Bn, ~)2e 0e2U(X ) "~'0~,f(x) + e2B n and a real number X >/0 such that v+

w =

v, + X ( v ~ - ( v + w)).

From this we have

v+W=l+AVi+

1)2

and therefore

v+weco((O_~,f(x)+'O~u(x)+e,B.)u('O.,f(x)+O_.~u(x)+e2B.)).

(22)

For case (b) the inclusion (22) is trivial. Since we are considering an arbitrary v9 and an arbitrary w 9 inclusion (22) implies that inclusion (14) is true. We shall now compare the necessary conditions (14) and (19) with other known necessary conditions by considering a particular case.

V. V. Gorokhovik / e-Quasidifferentiability

215

Example 1. Let us suppose that the function f is locally convex at the point x and let u(x')=maxi~l:mui(x'), where u, is also locally convex at x for i e l ( x ) = {i 9 1 : ml u,(x) = 0}. T h e n f and u are quasidifferentiable at x and their quasidifferentials are

Df(x) =

[ 0 f ( x ) , {0}],

Du(x) =

[co{Ou,(x), i 9 l(x)}, {0}],

where Of(x) and Ou~(x), i e l ( x ) , are respectively the subdifferentials o f f " (u~)', i 9 I ( x ) , at zero. In this special case condition (14) has the form

and

0 e co{0f(x), 0ui(x), i 9 which is equivalent to the existence of vectors v 9 0f(x), v~ 9 Ou.(x), i 9 l ( x ) , and real numbers Ao~>O, ;q>~O, i 9 l ( x ) , A o + ~ i ~ x ) A~= 1, such that AoV+ Y~ h~v~=0. ie I ( x )

Since u ' ( g ) = m a x ~ . ~ ) ( u ~ ) ' ~ ( g ) , g 9 E~, satisfies condition (a) then for the set G = { x ' 9 E~ Imax,~,:m u,(x')~< 0} to satisfy the regularity condition at x we require the existence of a vector ~ e E. such that ( u ~ ) ' ( ~ ) < 0 for i 9 I(x). For the example under consideration the necessary condition (19) may be stated as follows:

0 9 Of(x) + cone(co{Ou~(x), i 9 l(x)}). This is equivalent to the existence of vectors v 9 0f(x), v~9 Ou~(x), i e I ( x ) , and real numbers Ai ~>0, i 9 l ( x ) , such that v+

~

A~v~=0.

icl(x)

Thus we see that necessary conditions (14) and (19) generalize the Fritz John multiplier rule [12] and the K u h n - T u c k e r multiplier rule [13], respectively.

4.2. Minimization under an equality constraint We shall now consider the problem of minimizing the real-valued function f : En -~ El subject to the equality constraint G = {x'e E. [ u(x') = 0}, where u: E. ~ El is a real-valued function~defined on E.. The set G = { x ' e E n l u ( x ' ) = 0 } is said to satisfy the regularity condition for an equality constraint at a point x if (i) the function u: E.-~ E~ is uniformly directionally differentiable at the point x. and (ii) the function U'x:En ~ E~ does not achieve a local extremum (maximum or minimum) at any point. Using the same arguments as in the case of the regularity condition under an equality constraint, we can see that condition (ii) is equivalent to

V. E Gorokhovik / e-Quasidifferentiability

216

(ii') for any g e E. satisfying u'(g) = 0 and any real y > 0, there exist ~ , .~z e E. such that u ' ( g t ) < 0 , I1~,-gll < T and u'(g2) > 0, II~=-gll < T. (A discussion of condition (ii') is given in [3].) Theorem 7. Suppose that a function f is e~-quasidifferentiable at a point x and the constraint G = { x ' e E. [u(x') = 0} satisfies the regularity condition for an equality

constraint at x. I f the function f achieves a local minimum on the set G at the point x then for every el-quasidifferential D~,f(x) and for every e2-quasidifferential (e2 > O) D~u(x) of the function u at x, the following inclusion holds: {_a~,f(x) + e, B. + cone[(_a~ u(x) + e2B,, - { w})

~9.,f(x) c w e,7,: u ( x )

vca_,~u(x)

u (~2u(x) + e2B, - {v})]}.

(23)

Proof. U n d e r the regularity condition we have (see [3])

f'(g)/> 0

for all g e E, such that u ' ( g ) = 0,

(24)

where the function f achieves a local m i n i m u m on G at the point x. We shall now show that for every w e ~ , f ( x ) either the system max ( g , v ) - ( g , w ) + e , l l g l l < o , v~a_~f(x)

u'(g)