E. Polak

Optimization : Algorithms and Consistent Approximations

(Applied Mathematical Sciences Vol. 124)

Springer, New York, 1997 VI, 779 pp. 32 figs. Hardcover $69.95 ISBN 0-387-94971-2

This book covers algorithms and discretization procedures for the solution of nonlinear programming, semi-infinite optimization and optimal control problems. Readers will find of particular interest the exhaustive modern treatment of optimality conditions and algorithms for min-max problems, as well as the newly developed theory of consistent approximations and the treatment of semi-infinite optimization and optimal control problems in this framework. This book presents the first treatment of optimization algorithms for optimal control problems with state-trajectory and control constraints, and fully accounts for all the approximations that one must make in their solution. It is also the first to make use of the concepts of epi-convergence and optimality functions in the construction of consistent approximations to infinite dimensional problems.

Contents:

1. 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.
2. 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 - Decomposition of Problems Using Penalty Functions.- An Augmented Lagrangian Method.- Sequential Quadratic Programming.
3. Semi-Infinite Optimization: Optimality Conditions for Semi-Infinite Minimax.- Optimality Conditions for Constrained Semi-Infinite Optimization.- Theory of Consistent Approximations.- Semi-Infinite Minimax Algorithms.- Algorithms for Inequality Constrained Semi-Infinite Optimization.- Algorithms for Semi-Infinite Optimization with Mixed Constraints.
4. Optimal Control: Canonical Forms of Optimal Control Problems.- Optimality Conditions for Optimal Control.- Algorithms for Unconstrained Optimal Control.- Minimax Algorithms for Optimal Control.- Algorithms for Problems with State Constraints: Inequality Constraints.- Algorithms for Problems with State Constraints: Equality Constraints.- Algorithms for Problems with State Constraints: Equality and Inequality Constraints.
5. Mathematical Background: Results from Functional Analysis.- Convex Sets and Convex Functions.- Properties of Set-Valued Functions.- Properties of Max Functions.- Minimax Theorems.- Differential Equations.

Mathematics, Engineering

For researchers, graduate students

Level: Monograph