Topological Methods in Complementarity Theory

690 Pages·2000·23.59 MB·English

by  George Isac (auth.)

Topological Methods in Complementarity Theory
Nonconvex Optimization and Its Applications
Volume 41
Managing Editor:
Panos Pardalos
University 0/ Florida, USA
Advisory Board:
Ding-Zhu Du
University 0/ Minnesota, US.A.
C. A. Floudas
Princeton University, USA
J. Mockus
StanJord University, US.A.
H. D. Sherali
Virginia Polytechnic Institute and State University, USA
The tit/es puhlished in this series are listed at the end af this va/urne.
Topological Methods
•
In
Complementarity Theory
by
George Isac
Department of Mathematics
and Computer Science,
Royal Military Collef?e of Canada,
Kinf?ston, Ontario, Canada
Springer-Science+Business Media, B.V.
A C.LP. Catalogue re cord for this book is available from the Library of Congress.
ISBN 978-1-4419-4828-1 ISBN 978-1-4757-3141-5 (eBook)
DOI 10.1007/978-1-4757-3141-5
Printed on acidjree paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
OriginaIly published by Kluwer Acadernic Publishers in 2000.
Softcover reprint ofthe hardcover 1st edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To my parents Dumitru and Elena Isac
"Ne vous occupez pas des fautes
d'autrui, ni de leurs actes,
ni de leurs negligences. Soyez plutöt conscients de vos
propres actes et de vos propres negligences."
(Bouddha)
TOPOLOGICAL METHODS IN COMPLEMENT ARITY THEORY VII
CONTENTS
PREFACE IX
CHAPTER 1
CONVEX CONES
1.1 Preliminaries 1
1.2 Normal cones 13
1.3 Regular and completely regular cones 19
1. 4 Weil based cones 22
1.5 Polyhedral cones 27
1.6 Convex cones in Hilbert spaces 32
1.7 Galerkin cones 49
1.8 Tangent and normal cones 55
REFERENCES 57
CHAPTER2 61
COMPLEMENT ARITY PROBLEMS. ORIGIN AND
DEFINITIONS 61
2.1 Complementarity theory. Origin and necessity 62
2.2 Types of complementarity problems 66
2.3 Questions about complementarity problems 86
REFERENCES 89
CHAPTER3 101
COMPLEMENT ARITY PROBLEMS AS
101
MATHEMATICAL MODELS
3.1 Mathematical programming and complementarity 101
3.2 Game theory and complementarity 113
3.3 Variational inequalities and complementarity 118
3.4 Complementarity problems in economics 120
3.5 Complementarity and equilibrium oftraffic flows 140
3.6 Complementarity and maximizing oil production 143
3.7 Complementarity problems in mechanics and engineering 144
3.8 Other possible applications of complementarity problems 159
3.9 The order complementarity problem as mathematical model 163
REFERENCES 169
VIII CONTENTS
CHAPTER 4 181
EQUIVALENCES 181
REFERENCES 208
CHAPTER 5 213
TOPICS ON SOLVABILITY 213
5.1 Solvabi1ity and properties of solution set 213
5.2 Uniqueness 217
5.3 Global solvability 221
5.4 The basic theorem of complementarity 234
5.5 Feasibility and solvability 249
5.6 Some general existence theorems 272
5.7 Some existence theorems for the linear order complementarity
problems 287
5.8 The least element solution 293
5.9 Boundedness of solution set 297
5.10 Variational inequalities and existence theorems for complementarity
problems 309
5.11 Some special problems and results about complementarity problems 325
REFERENCES 357
CHAPTER 6 365
TOPOLOGICAL DEGREE AND COMPLEMENT ARITY 365
6.1 A minimal background on topological degree 365
6.2 Topological degree and linear complementarity 369
6.3 Topological degree and the generalized linear order complementarity
problem 383
6.4 Topological degree and the horizontal linear complementarity
problem 391
6.5 Topological degree and nonlinear complementarity problems 395
REFERENCES 401
CHAPTER 7 405
ZERO-EPI MAPPINGS AND COMPLEMENT ARITY 405
7.1 Zero-epi mappings on bounded sets 405
7.2 (O,k)-epi mappings 409
7.3 Applications to complementarity theory 414
7.4 (O,k)-epi mappings on cones 421
7.5 Applications to complementarity problems 424
TOPOLOGICAL METHODS IN COMPLEMENT ARITY THEORY IX
REFERENCES 427
CHAPTER8 431
EXCEPTIONAL F AMIL Y OF ELEMETNS AND
COMPLEMENT ARITY 431
8.1 Some topological preliminaries 432
8.2 Exceptional family of elements for single-valued mappings and com-
lementarity 434
8.3 Exceptional families of elements for set-valued mappings 450
8.4 Exceptional family of elements and feasibility 453
8.5 Exceptional family of elements and continuous path of E-solutions 464
8.6 Exceptional family of elements and the relational complementarity
problem 469
8.7 Exceptional family of elements and variational inequalities 474
8.8 Function without exceptional family of elements and existence theo-
rems 479
REFERENCES 493
CHAPTER 9 497
CONDITIONS (S)+ AND (S)l+: APPLICATIONS TO COM-
PLEMENT ARITY THEORY 497
9.1 Conditions (Sh and (S)l + 497
9.2 Applications to complementarity problems 507
9.3 Applications to complementarity problems for set-valued mappings 522
9.4 Condition (S)+, PM-mappings and complementarity 525
REFERENCES 529
CHAPTER 10 531
FIXED POINTS, COINCIDENCE EQUATIONS ON
CONES AND COMPLEMENT ARITY 531
10.1 Fixed points, coincidence equations and complementarity 531
10.2 Fixed, points, isotone projection cones and complementarity 556
10.3 Fixed points, coincidence equations and implicit complementarity 567
10.4 Fixed points, heterotonic operators and iterative methods for the
generalized order complementarity problem 578
10.5 Applications of complementarity to fixed point theory 584
10.6 An application to economics 594
REFERENCES 599
x
CONTENTS
CHAPTER 11 603
OTHER TOPOLOGICAL RESUL TS IN COMPLEMENT A-
RITY THEORY 603
11.1 A topological index on cones and the fold complementarity problem 603
11.2 Leray-Schauder alternative and the implicit complementarity 612
problem
11.3 Complementarity problems. Connectedness of solution set 616
11.4 The Mountain Pass Theorem and complementarity 628
REFERENCES 631
BIBLIOGRAPHY (Complementarity problems) 635
GLOSSARY OF NOTATION 677
INDEX 679
TOPOLOGICAL METHODS IN COMPLEMENT ARITY THEORY XI
PREFACE
This book is intended for mathematicians, engineers, economists, and specialists
working in operations research or in optimization and for anybody interested in
applied mathematics or in mathematical modelling.
The Complementarity Theory, is a new domain in applied mathematics and its
subject is the study of complementarity problems. Complementarity problems
represent a wide class of mathematical models related to optimization,game theory,
economics, engineering, mechanics, elasticity, fluid mechanics, stochastic optimal
control et cetern.
The complementarity condition is a kind of general equilibrium containing the
equilibrium in the physical sense and in the economical sense.
The concept of equilibrium is central to the understanding of many problems in
physics, engineering, economics, and other fields. Equilibrium is frequently used in
the study of competitive systems arising in different disciplines.
Particularly, in economics, examples of equilibrium problems include: markets in
which finns compete to detennine their profit-maximizing production outputs,
general economic equilibrium problems in which all the commodity prices are to be
detennined, congested urban transportation systems in which users seek to determine
their cost -minimizing routes of travel et cetera.
An interesting characteristic of Complementarity Theory is the fact that it has
multiple connections with other domains including: Linear Algebra, Functional
Analysis, Topology, the Fixed Point Theory, the Theory of Variational Inequalities,
the Topological Degree and Numerical Analysis. Because the diversity of its
application, the Complementarity Theory is a good stimulant for research in
fundamental mathematics.
This book is especially dedicated to the study of nonlinear complementarity
problems in infinite dimensional spaces.
The literature on this subject is very large. Concerning the subjects presented in this
book, we selected only results related to some topological methods and susceptible to
new developments. The numerical methods are not considered, as this subject may be
developed in a future book.
The structure of this book is as folIows:
In the first Chapter we present the necessary background on topological vector
spaces and especially on convex cones in topological vector spaces. Several classes
of cones used currently in Complementarity Theory are studied.