Neustadt: Optimization A Theory Of Necessary Conditions (Princeton Legacy Library) - Hardcover

Neustadt

Hardcover

ISBN 10: 0691081417 ISBN 13: 9780691081410

Publisher: Princeton University Press, 1992

Synopsis

This book presents a comprehensive treatment of necessary conditions for general optimization problems. The presentation is carried out in the context of a general theory for extremal problems in a topological vector space setting.

Following a brief summary of the required background, generalized Lagrange multiplier rules are derived for optimization problems with equality and generalized "inequality" constraints. The treatment stresses the importance of the choice of the underlying set over which the optimization is to be performed, the delicate balance between differentiability-continuity requirements on the constraint functionals, and the manner in which the underlying set is approximated by a convex set. The generalized multiplier rules are used to derive abstract maximum principles for classes of optimization problems defined in terms of operator equations in a Banach space. It is shown that special cases include the usual maximum principles for general optimal control problems described in terms of diverse systems such as ordinary differential equations, functional differential equations, Volterra integral equations, and difference equations. Careful distinction is made throughout the analysis between "local" and "global" maximum principles.

Originally published in 1977.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

About this title

Excerpt. © Reprinted by permission. All rights reserved.

Optimization

A Theory of Necessary Conditions

By Lucien W. Neustadt

PRINCETON UNIVERSITY PRESS

Copyright © 1976 Princeton University Press
All rights reserved.

ISBN: 978-0-691-08141-0

Contents

Preface, ix, 
Summary of Notation, xiii, 
Chapter I: Mathematical Preliminaries, 3, 
Chapter II: A Basic Optimization Problem in Simplified Form, 63, 
Chapter III: A General Multiplier Rule, 96, 
Chapter IV: Optimization with Operator Equation Restrictions, 144, 
Chapter V: Optimal Control Problems with Ordinary Differential Equation Constraints, 211, 
Chapter VI: Optimal Control Problems with Parameters and Related Problems, 263, 
Chapter VII: Miscellaneous Optimal Control Problems, 315, 
Appendix: Volterra-Type Operators, 361, 
Notes and Historical Comments, 384, 
References, 413, 
Subject Index, 422, 

CHAPTER 1

Mathematical Preliminaries

In this chapter we shall present some mathematical background material which will be necessary in order to understand the remainder of this book. For the most part, we shall confine ourselves to stating definitions and to theorem statements. The proofs of many of the theorems are immediate. References will be given for those proofs which are less obvious but which are widely available in the literature. (In this regard, the reference numbers refer to books listed at the end of this chapter.) Proofs will be given for those lemmas and theorems which either are novel or are not widely known.

The material in this chapter is, of course, sketchy at best, and in many cases definitions and theorems are not presented in their full generality. Indeed, in those cases where full generality does not provide any additional insight and is not required in the remainder of this book, we shall confine ourselves to restrictive definitions and theorems.

Since linear topological spaces form the framework of much of what we have to say, most of this chapter deals with such spaces and with background material necessary to understand these spaces. The last section of this chapter, which contains the major part of the novel material in this chapter, is devoted to a study of a wide variety of differentials.

For the sake of completeness, we have included some quite elementary material, such as the properties of finite-dimensional Euclidean spaces. However, the reader is expected to be familiar with the fundamentals of analysis on the real line. In addition, in subsequent chapters — as well as in Section 6 of this chapter — we shall make use of a number of concepts and theorems from the theory of measure and integration on the real line (and, on a few occasions, on real finite-dimensional spaces). Because this material is so widely and easily accessible in many books, including, e.g., [4] and [7], we shall dispense with presenting any background material on it ourselves.

1. Linear Vector Spaces

A group is a nonempty collection Y of elements together with an operation called addition which assigns to each ordered pair (y1, y2) of elements from Y a new element y3 of Y — which will be written as y1 + y2 = y3 — subject to the following restrictions:

(i) y1 + (y2 + y3) = (y1 + y2) + y3 for all y1, y2, y3 in Y (the associative law);

(ii) there is in Y a unique 0 (which we shall call the identity) such that 0 + y = y + 0 = y for all y [member of] Y;

(iii) for ever

Edition details

• Publisher: Princeton University Press
• Publication date: 1992
• Language: English
• ISBN 10: 0691081417
• ISBN 13: 9780691081410
• Binding: Hardcover
• Number of pages: 440