Optimization: A Theory of Necessary Conditions

Description

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. A brief summary of the mathematical background nec­essary to understand the material in the text is presented in Chapter I. On the assumption that the reader is familiar with the fundamentals of real analysis and basic elements of measure theory and integration, per­tinent definitions and results needed for the subsequent analysis in the linear topological space setting are given. Included also in Chapter I is a rather complete discussion of various types of differentials for functions from a linear (vector) space into a topological vector space. In Chapter II is found a generalized Lagrange multiplier rule for abstract optimization problems with a finite number of equality and inequality constraints. It is shown that application of this multiplier rule to a particular class of optimization problems defined in terms of operator equations in a Banach space yields a maximum principle which solutions of the problems must satisfy. Sufficiency of these conditions is discussed under certain convexity hypotheses on the problem data. Chapter III is devoted to a development of an extremal theory that leads to a generalization of the multiplier rule given in Chapter II. These generalizations involve a weakening of the hypoth­eses on the underlying set on which the optimization is carried out and a relaxation on the allowable constraints to permit a considerably more general type of "inequality" constraint. In Chapter IV the fundamental multiplier rules developed earlier are used to treat the general optimization problem: Given a family W of continuously differentiable operators T:A -> SC, where A is an open subset of the Banach space 3C, choose xe A satisfying (i) Tx = χ for some Τ e iV, (ii) certain equality and generalized "inequality" constraints, and which is in some sense optimal. The formulation here is such that not only are the usual necessary conditions for restricted phase coordinate optimal control problems with ordinary differential equation restrictions obtained as special cases (the subject matter of Chapter V), but many other general optimal control problems can also be easily treated as special cases. This is discussed in Chapter VI, where results are given for control problems with parameters and control problems with mixed control-phase inequality constraints. In Chapter VII necessary conditions using the framework of Chapter IV are obtained for control problems governed by such diverse systems as functional differential equations (differential-difference equations being a special case), Volterra integral equations, and difference equations. An appendix contains fundamental results (existence, continuation, uniqueness, continuous dependence) for equations defined in terms of the Volterra-type operators used in the formulation of certain of the problems discussed in Chapter IV. A concluding chapter (Notes and Historical Comments) comprises an extensive literature survey in which the development of necessary conditions and sufficient conditions in modern optimization theory is outlined and comments are made on the relationship between the differing approaches of various contri­butors to the literature.