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Stable Parametric Programming : Applied Optimization - Sanjo Zlobec

Stable Parametric Programming

Applied Optimization

Hardcover Published: December 2009
ISBN: 9780792371397
Number Of Pages: 322

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Optimality and stability are two important notions in applied mathematics. This book is a study of these notions and their relationship in linear and convex parametric programming models. It begins with a survey of basic optimality conditions in nonlinear programming. Then new results in convex programming, using LFS functions, for single-objective, multi-objective, differentiable and non-smooth programs are introduced. Parametric programming models are studied using basic tools of point-to-set topology. Stability of the models is introduced, essentially, as continuity of the feasible set of decision variables under continuous perturbations of the parameters. Perturbations that preserve this continuity are regions of stability. It is shown how these regions can be identified. The main results on stability are characterizations of locally and globally optimal parameters for stable and also for unstable perturbations. The results are straightened for linear models and bi-level programs. Some of the results are extended to abstract spaces after considering parameters as controls'. Illustrations from diverse fields, such as data envelopment analysis, management, von Stackelberg games of market economy, and navigation problems are given and several case studies are solved by finding optimal parameters. The book has been written in an analytic spirit. Many results appear here for the first time in book form. Audience: The book is written at the level of a first-year graduate course in optimization for students with varied backgrounds interested in modeling of real-life problems. It is expected that the reader has been exposed to a prior elementary course in optimization, such as linear or non-linear programming. The last section of the book requires some knowledge of functional analysis.

Industry Reviews

'The book would be of great interest to both graduate students and researchers who are concerned with optimization problems.'
Zentalblatt MATH, 986 (2002)

General Prefacep. xiii
Prefacep. xvii
Acknowledgmentsp. xxi
Introductionp. 1
Parametric Programming in Ancient Timesp. 1
Motivationp. 3
Stable Linear Modelsp. 4
Unstable Linear Modelsp. 7
Idea of Input Optimizationp. 8
Classical Optimality Conditionsp. 11
Method of Lagrangep. 12
Second-Order Optimality Conditionsp. 14
Examples and Exercisesp. 16
Basic Convex Programmingp. 29
Convex Setsp. 29
Convex Functionsp. 32
Systems of Convex Inequalitiesp. 36
Optimality Conditionsp. 38
Examples and Exercisesp. 46
Asymptotic Optimality Conditionsp. 59
Convex LFS Functionsp. 59
Convex Programs with LFS Constraintsp. 61
General Convex Programsp. 62
Examples and Exercisesp. 66
Non-Smooth Programsp. 73
Preliminariesp. 73
Optimality for Non-Smooth Programsp. 73
Non-Smooth LFS Functionsp. 75
An Equivalent Unconstrained Programp. 79
Examples and Exercisesp. 83
Multi-Objective Programsp. 87
Preliminariesp. 87
Pareto Optima for LFS Functionsp. 88
Pareto Optima for Differentiable Functionsp. 90
Saddle-Point Characterizationp. 92
Examples and Exercisesp. 93
Introduction to Stabilityp. 101
Preliminariesp. 101
Point-to-Set Mappingsp. 102
Stable Convex Modelsp. 104
Regions of Stabilityp. 108
Examples and Exercisesp. 112
Locally Optimal Parametersp. 121
Characterizing Locally Optimal Parametersp. 121
Input Constraint Qualificationsp. 124
Lagrange Point-to-Set Mappingsp. 126
Examples and Exercisesp. 128
Globally Optimal Parametersp. 135
Characterizing Globally Optimal Parametersp. 135
The Sandwich Conditionp. 137
Optimality in LFS Modelsp. 139
Dualityp. 140
An Explicit Representation of Optimal Parametersp. 145
Examples and Exercisesp. 147
Optimal Value Functionp. 155
Marginal Value Formulap. 155
Input Optimizationp. 162
Review of Minimum Principlesp. 164
Case Study: Restructuring in a Textile Millp. 166
Case Study: Planning of University Admissionp. 170
Examples and Exercisesp. 173
Partly Convex Programmingp. 185
Sources of Partly Convex Programsp. 186
Characterizations of Global and Local Optimap. 191
Partly LFS Programsp. 195
Examples and Exercisesp. 196
Numerical Methods in PCPp. 203
Parametric Steepest Descent Methodp. 203
Parametric Quasi-Newton Methodsp. 205
Constrained Programsp. 207
Examples and Exercisesp. 209
Zermelo's Navigation Problemsp. 213
Zermelo's Problem on the Waterp. 213
Solution by the Method of Lagrangep. 215
Solution by Input Optimizationp. 216
Zermelo's Problem under the Waterp. 217
Dual Solutions: Interpretationp. 219
Examples and Exercisesp. 220
Efficiency Testing in Data Envelopment Analysisp. 225
Charnes-Cooper-Rhodes Testsp. 225
Stability of Charnes-Cooper-Rhodes Testsp. 230
Stable Post-Optimality Analysisp. 231
Radius of Rigidity Methodp. 232
Case Study: Efficiency Evaluations of University Librariesp. 236
Examples and Exercisesp. 239
Orientationp. 243
Linear Parametric Modelsp. 243
Lexicographic Modelsp. 248
Stable Inverse Programmingp. 255
Semi-Abstract Parametric Programmingp. 257
Abstract Parametric Programmingp. 258
Examples and Exercisesp. 267
Method of Weierstrassp. 279
Glossary of Symbolsp. 283
Referencesp. 285
Indexp. 315
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780792371397
ISBN-10: 0792371399
Series: Applied Optimization
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 322
Published: December 2009
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.39 x 15.6  x 2.06
Weight (kg): 0.66

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