Optimization: Algorithms and Consistent Approximations

by Elijah Polak

Overview

This book deals with optimality conditions, algorithms, and discretization techniques for nonlinear programming, semi-infinite optimization, and optimal control 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 problems. 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 algorithms. 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 control 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 becomestransparent.

Product Details

ISBN-13:	9781461268611
Publisher:	Springer New York
Publication date:	12/21/2012
Series:	Applied Mathematical Sciences , #124
Edition description:	Softcover reprint of the original 1st ed. 1997
Pages:	782
Product dimensions:	6.10(w) x 9.25(h) x 0.06(d)

Table of Contents

Contents: Unconstrained Optimization.- Optimality Conditions.- Algorithm Models and Convergence Conditions I.- Gradient Methods.- Newton's Method.- Methods of Conjugate Directions.- Quasi-Newton Methods.- One Dimensional Optimization.- Newton's Method for Equations and Inequalities.- Finite Minimax and Constrained Optimization.- Optimality Conditions for Minimax.- Optimality Conditions for Constrained Optimization.- Algorithm Models and Convergence Conditions II.- First-Order Minimax Algorithms.- Newton's Method for Minimax Problems.- Phase I. Phase II Methods of Centers .- Penalty Function Algorithms.- An Augmented Lagrangian Method.