Stable Parametric Programming

By: S. Zlobec

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Hardcover | 1 December 2009

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348 Pages

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24.13 x 16.51 x 2.54

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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)

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Non-Fiction Mathematics Optimisation Linear Programming Medicine Medicine in General Engineering & Technology Energy Technology & Engineering Electrical Engineering Game Theory Applied Mathematics Business & Management

Operational Research Management & Management Techniques Economics Economic Theory & Philosophy

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Published: 26th November 2013

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General Preface	p. xiii
Preface	p. xvii
Acknowledgments	p. xxi
Introduction	p. 1
Parametric Programming in Ancient Times	p. 1
Motivation	p. 3
Stable Linear Models	p. 4
Unstable Linear Models	p. 7
Idea of Input Optimization	p. 8
Classical Optimality Conditions	p. 11
Method of Lagrange	p. 12
Second-Order Optimality Conditions	p. 14
Examples and Exercises	p. 16
Basic Convex Programming	p. 29
Convex Sets	p. 29
Convex Functions	p. 32
Systems of Convex Inequalities	p. 36
Optimality Conditions	p. 38
Examples and Exercises	p. 46
Asymptotic Optimality Conditions	p. 59
Convex LFS Functions	p. 59
Convex Programs with LFS Constraints	p. 61
General Convex Programs	p. 62
Examples and Exercises	p. 66
Non-Smooth Programs	p. 73
Preliminaries	p. 73
Optimality for Non-Smooth Programs	p. 73
Non-Smooth LFS Functions	p. 75
An Equivalent Unconstrained Program	p. 79
Examples and Exercises	p. 83
Multi-Objective Programs	p. 87
Preliminaries	p. 87
Pareto Optima for LFS Functions	p. 88
Pareto Optima for Differentiable Functions	p. 90
Saddle-Point Characterization	p. 92
Examples and Exercises	p. 93
Introduction to Stability	p. 101
Preliminaries	p. 101
Point-to-Set Mappings	p. 102
Stable Convex Models	p. 104
Regions of Stability	p. 108
Examples and Exercises	p. 112
Locally Optimal Parameters	p. 121
Characterizing Locally Optimal Parameters	p. 121
Input Constraint Qualifications	p. 124
Lagrange Point-to-Set Mappings	p. 126
Examples and Exercises	p. 128
Globally Optimal Parameters	p. 135
Characterizing Globally Optimal Parameters	p. 135
The Sandwich Condition	p. 137
Optimality in LFS Models	p. 139
Duality	p. 140
An Explicit Representation of Optimal Parameters	p. 145
Examples and Exercises	p. 147
Optimal Value Function	p. 155
Marginal Value Formula	p. 155
Input Optimization	p. 162
Review of Minimum Principles	p. 164
Case Study: Restructuring in a Textile Mill	p. 166
Case Study: Planning of University Admission	p. 170
Examples and Exercises	p. 173
Partly Convex Programming	p. 185
Sources of Partly Convex Programs	p. 186
Characterizations of Global and Local Optima	p. 191
Partly LFS Programs	p. 195
Examples and Exercises	p. 196
Numerical Methods in PCP	p. 203
Parametric Steepest Descent Method	p. 203
Parametric Quasi-Newton Methods	p. 205
Constrained Programs	p. 207
Examples and Exercises	p. 209
Zermelo's Navigation Problems	p. 213
Zermelo's Problem on the Water	p. 213
Solution by the Method of Lagrange	p. 215
Solution by Input Optimization	p. 216
Zermelo's Problem under the Water	p. 217
Dual Solutions: Interpretation	p. 219
Examples and Exercises	p. 220
Efficiency Testing in Data Envelopment Analysis	p. 225
Charnes-Cooper-Rhodes Tests	p. 225
Stability of Charnes-Cooper-Rhodes Tests	p. 230
Stable Post-Optimality Analysis	p. 231
Radius of Rigidity Method	p. 232
Case Study: Efficiency Evaluations of University Libraries	p. 236
Examples and Exercises	p. 239
Orientation	p. 243
Linear Parametric Models	p. 243
Lexicographic Models	p. 248
Stable Inverse Programming	p. 255
Semi-Abstract Parametric Programming	p. 257
Abstract Parametric Programming	p. 258
Examples and Exercises	p. 267
Method of Weierstrass	p. 279
Glossary of Symbols	p. 283
References	p. 285
Index	p. 315
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Stable Parametric Programming

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Industry Reviews

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