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Springer
Series
in
Computational
Mathematics
3
Editorial Board
R.
L.
Graham, Murray Hill
J. Stoer, WOrzburg
R.
Varga,.Cleveland

N.Z.Shor
Minimization
Methods
for
Non-Differentiable
Functions
Translated from the Russian
by
K.
C.
Kiwiel and
A.
Ruszczynski
Springer-Verlag
Berlin Heidelberg New York Tokyo

Naum Zuselevich Shor
Institute
of
Cybernetics of the Academy
of
Sciences
of
the Ukrainian SSR, 142/144 4D-letiya Oktyabrya Avenue
25227,
Kiev-207, USSR
Krzysztof
C.
Kiwiel
Systems Research Institute, Polish Academy
of
Sciences,
ul. Newelska 6,01-447 Warsaw, Poland
Andrzej Ruszczynski
Institute
of
Automatic Control, Technical University
of
Warsaw,
ul. Nowowiejska 15/19, 00-665 Warsaw, Poland
Title
of
the Russian original edition:
Metody minimizatsii nedifferentsiruemykh funktsij
i ikh prilozheniya
Published by Naukova Dumka, Kiev 1979
AMS Subject Classifications: 26B25, 49007, 49027, 49037, 52A40,
65H10,
65
K05, 9OC06, 90C30
ISBN-13: 978-3-642-82120-2
001: 10.1007/978-3-642-82118-9
e-ISBN-13: 978-3-642-82118-9
Library of Congress Cataloging
in
Publication Data.
Shor, Naum Zuselevich. Minimization methods for non-differentiable functions.
(Springer series
in
computational mathematics;
3)
Translation of: Metody minimi-
zafuii nedifferentsiruemykh
funktsiT
i ikh prilozheniia. Bibliography:
p.
Includes in-
dex.
1.
Mathematical optimization.
2.
Nondifferentiable functions.
I.
Title.
II.
Series.
QA402.5.S5413
1985
519
84-23594
This work
is
subject to copyright. All rights are reserved, whether the whole
or
part
of the material
is
concerned, specifically those of translation, reprinting, re-use
of illustrations, broadcasting, reproduction by photocopying machine or similar
means, and storage
in
data banks. Under §
54
of the German Copyright Law
where copies are made for other than private
use,
a fee is payable to "Verwer-
tungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin, Heidelberg
1985
Softcover reprint of the hardcover 1 st edition 1985
Typesetting: Graphischer Betrieb Konrad Triltsch, WUrzburg
Bookbinding:
B.
Helm, Berlin
2141/~543210

Preface
In
recent years much attention has been given to the development
of
auto-
matic systems of planning, design and control
in
various branches
of
the
national economy. Quality
of
decisions
is
an issue which has come to the
forefront, increasing the significance
of
optimization algorithms in math-
ematical software packages for al,ltomatic systems
of
various levels and pur-
poses. Methods for minimizing functions with discontinuous gradients are
gaining in importance and the
~xperts
in the computational methods
of
mathematical programming tend to agree
that
progress in the development
of
algorithms for minimizing nonsmooth functions is the key to the con-
struction
of
efficient techniques for solving large scale problems.
This monograph summarizes to a certain extent fifteen years
of
the
author's work on developing generalized gradient methods for nonsmooth
minimization. This work started in the department
of
economic cybernetics
of the Institute of Cybernetics
of
the Ukrainian Academy
of
Sciences under
the supervision
of
V.S.
Mikhalevich, a member
of
the
Ukrainian
Academy
of Sciences, in connection with the need for solutions to important, practical
problems of optimal planning and design.
In
Chap. I
we
describe basic classes
of
nonsmooth functions that are dif-
ferentiable almost everywhere, and analyze various ways
of
defining
generalized gradient sets.
In
Chap. 2
we
study in detail various versions
of
the su bgradient method,
show their relation to the methods
of
Fejer-type approximations and briefly
present the fundamentals
of
e-subgradient methods.
In
Chap. 3
we
describe gradient-type algorithms with space dilation in
the direction
of
the gradient (the
SDG
algorithms) and in the direction
of
the difference of two successive gradients (the r-algorithms).
Great
attention
is
paid to establishing convergence and speed
of
convergence
of
these algo-
rithms. Extreme variants
of
the methods are also considered. We separately
analyze the applicability
of
the
SDG
algorithms to the solution
of
systems
of
nonlinear equations. The relations between cutting plane methods and
methods with space dilation are also discussed.
Chapter 4
is
devoted to the use
of
generalized gradient methods for im-
plementing decomposition schemes in numerous real-life problems
of
opera-

VI
Preface
tive and long-term planning, for minimax problems, for problems
of
inter-
preting gravimetric observations, etc.
The author expresses his sincere gratitude to
V.S.
Mihkalevich, L.A.
Galustova, G.I. Gorbach,
V.1.
Gershovich, N.G. Zhurbenko, T.V. Marchuk,
AI.
Momot, V.A Trubin, L.P. Shabashova, and G.N. Yun, who contributed
significantly
to
the elaboration and practical applications
of
the methods
of
nonsmooth optimization reflected in this monograph, and to V.N. Bog-
danyuk,
G.A
Verbe, T.G. Thrushko and
T.1.
Yatsenko for their assistance in
preparing the manuscript.
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