Bi-Level Strategies in Semi-Infinite Programming

219 Pages·2003·17.091 MB·English

by  Oliver Stein (auth.)

Bi-Ievel Strategies
in Semi-infinite Programming
Nonconvex Optimization and Its Applications
Volume 71
Managing Editor:
Panos Pardalos
University 0/ Florida, U.SA.
Advisory Board:
J. R. Birge
University a/Michigan, U.SA.
Ding-ZhuDu
University a/Minnesota, U.S.A.
C. A. Floudas
Princeton University, U.SA.
J. Mockus
Lithuanian Academy o/Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.SA.
G. Stavroulakis
Technical University Braunschweig, Germany
BI-LEVEL STRATEGIES
IN SEMI-INFINITE PROGRAMMING
OLIVER STEIN
Department of Mathematics
Aachen University
Germany
~.
"
Springer Science+Business Media, LLC
tt
Electronic Services <http://www.wkap.nl>
Library or Congress Cataloging-in-Publication
Stein, Oliver
Bi-Ievel Strategies in Semi-Infinite Programming
ISBN 978-1-4613-4817-7 ISBN 978-1-4419-9164-5 (eBook)
DOI 10.1007/978-1-4419-9164-5
Copyright © 2003 by Springer Science+Business Media New York
Originally published by KIuwer Academic Publishers in 2003
Softcover reprint of the hardcover 1s t edition 2003
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
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Printed on acid-free paper.
Dedicated to the memory
of Rainer Hettich
Contents
List of Symbols xi
List of Figures xv
List of Tables xix
Preface xxi
Acknowledgments xxvii
1. INTRODUCTION 1
1.1 Standard semi-infinite programming 2
1.2 General semi-infinite programming 4
1.3 The misconception about the generality of GSIP 6
1.4 Development to a field of active research 8
2. EXAMPLES AND APPLICATIONS 11
2.1 Chebyshev and reverse Chebyshev approximation 12
2.2 Minimax problems 14
2.3 Robust optimization 15
2.4 Design centering 18
2.5 Defect minimization for operator equations 20
2.6 Disjunctive programming 22
viii BI-LEVEL STRATEGIES IN SEMI-INFINITE PROGRAMMING
3. TOPOLOGICAL STRUCTURE OF THE FEASIBLE SET 25
3.1 Abstract index set mappings 25
3.1.1 A projection formula 27
3.1.2 A bi-Ievel formula and semi-continuity properties 31
3.1.3 A set-valued mapping formula 41
3.1.4 The local structure of M 42
3.1.5 The completely convex case 44
3.2 Index set mappings with functional constraints 46
3.2.1 The convex case 46
3.2.2 The linear case 47
3.2.3 The C1 case 60
3.2.4 The C2 case 62
3.2.5 Genericity results 66
4. OPTIMALITY CONDITIONS 85
4.1 Abstract primal optimality conditions 85
4.2 First order approximations of the feasible set 90
4.2.1 General constraint qualifications 91
4.2.2 Descriptions of the linearization cones 96
4.2.3 Degenerate index sets 108
4.3 Dual first order optimality conditions 116
4.3.1 The standard semi-infinite case 118
4.3.2 The completely convex case 120
4.3.3 The convex case 123
4.3.4 The C2 case with Reduction Ansatz 126
4.3.5 The C1 case 128
4.4 Second order optimality conditions 142
Contents ix
5. BI-LEVEL METHODS FOR GSIP 145
5.1 Reformulations of GSIP 146
5.1.1 The Stackelberg game reformulation of GSIP 146
5.1.2 The MPEC refonnulation of GSIP 148
5.1.3 A regularization of MPEC by NCP functions 149
5.1.4 The regularized Stackelberg game 152
5.2 Convergence results for a bi-Ievel method 154
5.2.1 A parametric reduction lemma 155
5.2.2 Convergence of global solutions 157
5.2.3 Convergence of Fritz John points 158
5.2.4 Quadratic convergence of the optimal values 162
5.2.5 An outer approximation property 163
5.3 Other bi-Ievel approaches and generalizations 167
6. COMPUTATIONAL RESULTS 171
6.1 Design centering in two dimensions 172
6.2 Design centering in higher dimensions 177
6.3 Robust optimization 178
6.4 Optimal error bounds for an elliptic operator equation 181
7. FINAL REMARKS 187
References 191
Index 201
List of Symbols
A set complement of the set A
C
8A topological boundary of the set A
A topological closure of the set A
int(A) topological interior of the set A
B(x, <5) closed ball about x with radius <5
codim V codimension of the manifold V
cony ( S) convex hull of the set S
Cr;o Whitney topology
Cd Cd (lRn+m, IRP) x £Bd
d(x, A) distance of x from the set A
diag(a) diagonal matrix with a as vector of diagonal entries
dom Y domain of the set-valued mapping Y
D aF gradient of F with respect to a
D~ F Hessian of F with respect to a
e all-ones vector of appropriate dimension
epi f epigraph of f
epi< f strict epigraph of f