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.
Author(s): Lucien W. Neustadt
Publisher: Princeton University Press
Year: 1976
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.
