A first course in optimization theory

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Preface
This book developed out of a course I have taught since 1988 to first-year Ph.D.
students at the University of Rochester on the use of optimization techniques in
economic analysis. A detailed account of its contents is presented in Section 2.5 of
Chapter 2. The discussion below is aimed at providing a broad overview of the book,
as well as at emphasizing some of its special features.
An Overview of the Contents
The main body of this book may be divided into three parts. The first part,
encompassing Chapters 3 through 8, studies optimization in и-dimensional Euclidean space.
R". Several topics are covered in this span. These include—but are not limited to—
(i) the Weierstrass Theorem, and the existence of solutions to optimization problems:
(ii) the Theorem of Lagrange, and necessary conditions for optima in problems with
equality constraints; (Hi) the Theorem of Kuhn and Tucker, and necessary conditions
for optima in problems with inequality constraints; (iv) the role of convexity in
obtaining sufficient conditions for optima in constrained optimization problems; and
(v) the extent to which convexity can be replaced with q^asi-convexity, while still
obtaining sufficiency of the first-order conditions for global optima.
The second part of the book, comprised of Chapters 9 and 10, looks at the issue
of parametric variation in optimization problems, that is, at the manner in which
solutions to optimization problems respond to changes in the values of underlying
parameters. Chapter 9 begins this exercise with the question of parametric continuity:
under what conditions will solutions to optimization problems vary "continuously"
with changes in the underlying parameters? An answer is provided in the centerpiece
of this chapter, the Maximum Theorem. The strengthening of the Maximum Theorem
that is obtained by adding convexity restrictions to the problem is also examined,
Chapter 10 is concerned with parametric monotonicity, that is, with conditions undei
which increases in the values of parameters result in increases in the size of optimal

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