Table Of ContentStahle Parametrie Programming
Applied Optimization
Volume 57
Series Editors:
Panos M. Pardalos
University of Florida, U.S.A.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
Stable Parametric
Programming
by
Sanjo Zlobec
McGill University, Montreal, Canada
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-4885-6 ISBN 978-1-4615-0011-7 (eBook)
DOI 10.1007/978-1-4615-0011-7
AII Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1s t edition
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 Sanro and Inti "Snicey"
my structural optima
Elements ofa rock painting
discovered in Spain
The range of the caveman 's arrow depends on the projection angle
(parameter). The problern of determining the angle that yields the maximal
rangt!, subject to the gravitational force, can be jormulated as a stable
parametric programming model. The angle of the two parallel arrows in the
painting appears to be "optimal". In a modern version of this problem, a
businessman is lookingfor a pricing policy that maximizes the profit subject to
economics constraints. ·T his book studies stability and optimality in linear and
convex parametric programming models. The basic knowledge of analysis,
calculus, and linear algebra is. assumed The last section requires some
familiarity withfonctional analysis.
Contents
General Preface, xiii
Preface, xvii
Acknowledgments, xxi
1 INTRODUCTION 1
1 Parametrie Programming in Ancient Times, 1
2 Motivation, 3
3 Stahle Linear Models, 4
4 Unstable Linear Models, 7
5 Idea of Input Optimization, 8
2 CLASSICAL OPTIMALITY CONDffiONS 11
1 Method ofLagrange, 12
2 Second-Order Optimality Conditions, 14
Examples and Exercises, 16
3 BASIC CONVEX PROGRAMMING 29
1 Convex Sets, 29
2 Convex Functions, 32
3 Systems of Convex Inequalities, 36
4 Optimality Conditions, 38
Examples and Exercises, 46
4 ASYMPTOTIC OPTIMALITY CONDITIONS 59
1 Convex LFS Functions, 59
2 Convex Programs with LFS Constraints, 61
3 General Convex Programs, 62
Examples and Exercises, 66
5 NON-SMOOTH PROGRAMS 73
1 Preliminaries, 73
2 Optimality for Non-Smooth Programs, 73
3 Non-Smooth LFS Functions, 75
4 An Equivalent Unconstrained Program, 79
Examples and Exercises, 83
X
6 MULTI-OBJECTIVE PROGRAMS 87
1 Preliminaries, 87
2 Pareto Optima for LFS Functions, 88
.3 Pareto Optima for Differentiable Functions, 90
4 Saddle-Point Characterization, 92
Examples and Exercises, 93
7 INTRODUCTION TO STABILITY 101
1 Preliminaries, 10 1
2 Point-to-Set Mappings, 102
3 Stahle Convex Models, 104
4 Regions ofStability, 108
Examples and Exercises, 112
8 LOCALLYOPTlldALP~ETERS 121
1 Characterizing Locally Optimal Parameters, 121
2 Input Constraint Qualifications, 124
3 Lagrange Point-to-Set Mappings, 126
Examples and Exercises, 128
9 GLOBALLYOPTIMALP~TERS 135
1 Characterizing Globally Optimal Parameters, 135
2 The Sandwich Condition, 137
3 Optimality in LFS Models, 139
4 Duality, 140
5 An Explicit Representation of Optimal Parameters, 145
Examples and Exercises, 147
10 OPTIMAL VA LUE FUNCTION 155
1 Marginal Value Formula, 155
2 Input Optimization, 162
3 Review of Minimum Principles, 164
4 Case Study: Restructuring in a Textile Mill, 166
5 Case Study: Planning of University Admission, 170
Examples and Exercises,173
11 PARTLYCONVEXPROGRAMMING 185
1 Sources of Partly Convex Programs, 186
2 Characterizations of Global and Local Optima, 191
3 Partly LFS Programs, 195
Examples and Exercises, 196
Xl
12 NUMERICAL METHODS IN PCP 203
1 Parametrie Steepest Descent Method, 203
2 Parametrie Quasi-Newton Methods, 205
3 Constrained Programs, 207
Examples and Exercises, 209
13 ZERMELO'S NAVIGATION PROBLEMS 213
1 Zennelo's Problem on the Water, 213
2 Solution by the Method ofLagrange, 215
3 Solution by Input Optimization, 216
4 Zennelo's Problem under the Water, 217
5 Dual Solutions: Interpretation, 219
Examples and Exercises, 220
14 EFFICIENCY TESTING IN DATA ENVELOPMENT
ANALYSIS 225
1 Chames-Cooper-Rhodes Tests, 225
2 Stability ofChames-Cooper-Rhodes Tests, 230
3 Stahle Post-Optimality Analysis, 231
4 Radius ofRigidity Method, 232
5 Case Study: Effieieney Evaluations ofUniversity Libraries, 236
Examples and Exercises, 239
15 ORIENTATION 243
1 Linear Parametrie Models, 243
2 Lexieographie Models, 248
3 Stahle Inverse Programming, 255
4 Semi-Abstract Parametrie Programming, 257
5 Abstract Parametrie Programming, 258
Examples and Exercises, 267
Appendix: Method ofWeierstrass 279
Glossary of Symbols 283
Referenees 285
Index 315
