Quasidifferential Calculus

229 Pages·1986·2.863 MB·English

by  V. F. Demyanov, L. N. Polyakova, A. M. Rubinov (auth.), V. F. Demyanov, L. C. W. Dixon (eds.)

MATHEMATICAL PROGRAMMING 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 Universi-
ty, 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.
M A T H E M A T I C A L
P R O G R A M M I N G
STUDY29
A PUBLICATION OF
THE MATHEMATICAL PROGRAMMING SOCIETY
Quasidifferential Calculus
Edited by V.F. DEMYANOV and L.C.W. DIXON
yaM 6891
NORTH-HOLLAND - AMSTERDAM
© The Mathematical Programming Society, Inc. - 1986
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or trans-
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Submission to this journal of a paper entails the author's irrevocable and exclusive authorization of the
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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 ,nrapBo
A no MHH bTtl~OX
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
iiv
iiiv ecaferP
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 and/or 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 vii
V.F. Demyanov, L.N. Polyakova and A.M. Rubinov, Nonsmoothness and
quasidifferentiability 1
V.F. Demyanov, Quasidifferentiable functions: Necessary conditions and
descent directions 02
L.N. Polyakova, On the minimization ofa quasidifferentiable function subject to
equality-type quasidifferentiable constraints 44
.A Shapiro, Quasidifferential calculus and first-order optimality conditions in
nonsmooth optimization 65
L.N. Polyakova, On minimizing the sum of a convex function and a concave
function 96
.F.V Demyanov, .S Gamidov and T.I. Sivelina, An algorithm for minimizing a
certain class of quasidifferentiable functions 47
K.C. Kiwiel, A linearization method for minimizing certain quasidifferentiable
functions 58
.A.V Demidova and V.F. Demyanov, A directional implicit function theorem for
quasidifferentiable functions 59
.F.V Demyanov and I.S. Zabrodin, Directional differentiability of a continual
maximum function of quasidifferentiable functions 801
.D Melzer, On the expressibility of piecewise-linear continuous functions as the
difference of two piecewise-linear convex functions 811
S.L. Pechersky, Positively homogeneous quasidifferentiable functions and their
application in cooperative game theory 531
N.A. Pecherskaya, Quasidifferentiable mappings and the differentiability of
maximum functions 541
V.F. Demyanov, V.N. Nikulina and I.R. Shablinskaya, Quasidifferentiable
functions in Optimal Control 061
A.M. Rubinov and A.A. Yagubov, The space of star-shaped sets and its
applications in nonsmooth optimization 671
V.V. Gorokhovik, e-Quasidifferentiability of real-valued functions and
optimality conditions in extremal problems 302
Appendix
A guide to the bibliography on quasidifferential calculus 912
Bibliography on quasidifferential calculus (January )5891 912
Mathematical Programming Study 29 (1986) 1-19
North-Holland
NONSMOOTHNESS AND QUASIDIFFERENTIABILITY
V.F. DEMYANOV
Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9,
Leningrad ,461991 USSR, and International Institute for Applied Systems Analysis,
A-2361 Laxenburg, Austria
L.N. POLYAKOVA
Department of Applied Mathematics, Leningrad State University, Universitetskaya nab. 7/9,
Leningrad ,461991 USSR
A.M. RUBINOV
Institute for Social and Economic Problems, USSR Academy of Sciences, .lu Voinova ,a-05
Leningrad ,510891 USSR
Received 9 April 4891
Revised manuscript received 51 November 4891
This paper si an introduction to the present volume. It si first shown that quasidifferentiable
functions form a very distinct class of nondifferentiable functions. This and other papers in this
volume demonstrate that ew 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
si 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, Sub-
differentials, Superdiilerentials, Optimization Problems, Directional Differentiability, Upper Con-
xev and Lower Concave Approximations, Clarke Subdifferential.
1. Introduction
This is not the place to go into the motivations and origins of nondifferentiability
(although these are very iml~rtant and interesting): for the purpose of this paper
it is only necessary to realize that although a nondifferentiable function can often
be approximated by a differentiable one, this substitution is usually unacceptable
from an optimization viewpoint since some very important properties of the function
are lost (see Example 2.1 below). We must therefore find some new analytical tool
to apply to the problem.
Define a finite-valued functionfon an open set D c E,. Iffunctionfis directionally
differentiable, i.e., if the following limit exists:
af(x) =
lim lf(x+ag)-f(x) VgcE,, (1.1)
Og ,,~+o to
1
2 V.F. Demyanov, L.N. Polyakova and A.M. Rubinov / Quasidifferentiability
then
f(x + ag) =f(x) + ~ of(x) + o(~).
ga
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)
ga - (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 off at .x
Diferentiable functions form a well-known and important class of functions.
The next cases that we shall consider are convex functions and maximum func-
tions. It turns out that for these functions the directional derivative has the form
Of(x)
max (v, g), (1.2)
gO )x(fOcv
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 many attempts to extend the concept of a
subdifferential to other classes of nondifferentiable functions (see, e.g., 1, 15, ,61
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 elbaitnereffidisauq
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)
max (v, g)+ re_in (w, g). (1.3)
gO o~f(x) )x(f~cew
The pair Df(x)= _0f(x), 0f(x) is called a quasidifferential off 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