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Optimization : Algorithms and Consistent Approximations - Elijah Polak

Optimization

Algorithms and Consistent Approximations

By: Elijah Polak (Editor)

Hardcover

Published: 20th June 1997
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This book deals with optimality conditions, algorithms, and discretization tech­ niques for nonlinear programming, semi-infinite optimization, and optimal con­ trol problems. The unifying thread in the presentation consists of an abstract theory, within which optimality conditions are expressed in the form of zeros of optimality junctions, algorithms are characterized by point-to-set iteration maps, and all the numerical approximations required in the solution of semi-infinite optimization and optimal control problems are treated within the context of con­ sistent approximations and algorithm implementation techniques. Traditionally, necessary optimality conditions for optimization problems are presented in Lagrange, F. John, or Karush-Kuhn-Tucker multiplier forms, with gradients used for smooth problems and subgradients for nonsmooth prob­ lems. We present these classical optimality conditions and show that they are satisfied at a point if and only if this point is a zero of an upper semicontinuous optimality junction. The use of optimality functions has several advantages. First, optimality functions can be used in an abstract study of optimization algo­ rithms. Second, many optimization algorithms can be shown to use search directions that are obtained in evaluating optimality functions, thus establishing a clear relationship between optimality conditions and algorithms. Third, estab­ lishing optimality conditions for highly complex problems, such as optimal con­ trol problems with control and trajectory constraints, is much easier in terms of optimality functions than in the classical manner. In addition, the relationship between optimality conditions for finite-dimensional problems and semi-infinite optimization and optimal control problems becomes transparent.

Preface
Conventions and Symbols
Unconstrained Optimizationp. 1
Optimality Conditionsp. 1
Algorithm Models and Convergence Conditions Ip. 15
Gradient Methodsp. 56
Newton's Methodp. 70
Methods of Conjugate Directionsp. 87
Quasi-Newton Methodsp. 104
One-Dimensional Optimizationp. 138
Newton's Method for Equations and Inequalitiesp. 157
Finite Min-Max and Constrained Optimizationp. 167
Optimality Conditions for Min-Maxp. 168
Optimality Conditions for Constrained Optimizationp. 185
Algorithm Models and Convergence Conditions IIp. 215
First-Order Min-Max Algorithmsp. 222
Newton's Method for Min-Max Problemsp. 250
Phase I - Phase II Methods of Centersp. 259
Penalty Function Algorithmsp. 280
Augmented Lagrangian Methodsp. 315
Sequential Quadratic Programmingp. 333
Semi-Infinite Optimizationp. 368
Optimality Conditions for Semi-Infinite Min-Maxp. 369
Optimality Conditions for Constrained Semi-Infinite Optimizationp. 378
Theory of Consistent Approximationsp. 389
Semi-Infinite Min-Max Algorithmsp. 418
Algorithms for Inequality-Constrained Semi-Infinite Optimizationp. 445
Algorithms for Semi-Infinite Optimization with Mixed Constraintsp. 466
Optimal Controlp. 482
Canonical Forms of Optimal Control Problemsp. 482
Optimality Conditions for Optimal Controlp. 495
Algorithms for Unconstrained Optimal Controlp. 534
Min-Max Algorithms for Optimal Controlp. 562
Algorithms for Problems with State Constraints I: Inequality Constraintsp. 589
Algorithms for Problems with State Constraints II: Equality Constraintsp. 609
Algorithms for Problems with State Constraints III: Equality and Inequality Constraintsp. 630
Mathematical Backgroundp. 646
Results from Functional Analysisp. 646
Convex Sets and Convex Functionsp. 665
Properties of Set-Valued Functionsp. 676
Properties of Max Functionsp. 682
Minimax Theoremsp. 696
Differential Equationsp. 709
Bibliographyp. 743
Indexp. 773
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387949710
ISBN-10: 0387949712
Series: Applied Mathematical Sciences
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 782
Published: 20th June 1997
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.81
Weight (kg): 2.87