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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Unconstrained Optimization

Abstract
We devote this chapter primarily to optimality conditions and algorithms for solving unconstrained optimization problems of the form min x ∈ IR n f(x), where f(•) is a continuously differentiable cost function defined on IR n . When convenient, we will extend our results to problems of the form min x X f(x), where f (•) is a continuously differentiable cost function, defined on IR n , and X ⊂ IR n is an “unstructured”, convex, constraint set. We expect the reader to be familiar with the mathematical background contained in the first four sections of Chapter 5.
Elijah Polak

Chapter 2. Finite Min-Max and Constrained Optimization

Abstract
We devote this chapter to optimality conditions and algorithms for solving three classes of progressively more difficult optimization problems: min-max problems of the form {fy167|0a-1.
Elijah Polak

Chapter 3. Semi-Infinite Optimization

Abstract
We devote this chapter to optimality conditions and algorithms for solving semi-infinite optimization problems. In particular, we will consider semi-infinite min-max problems of the form.
Elijah Polak

Chapter 4. Optimal Control

Abstract
In this chapter we will develop optimality conditions and algorithms for several classes of optimal control problems that are extensions to a function space of the problems we have considered in the previous chapters. The study of optimal control requires an elementary knowledge of ordinary differential equation theory, and we therefore urge the reader to review the material presented in Section 5.6, the notation of which we will follow here as well.
Elijah Polak

Chapter 5. Mathematical Background

Abstract
The proofs presented in this book use results from functional analysis, convex analysis, the theory of set-valued maps, nonsmooth analysis, and minimax theory. Since there is no book available that contains all of these results, in this chapter we will summarize an essential collection of mathematical facts that are basic to the understanding of our presentation.
Elijah Polak

Backmatter

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