A First Course in Optimization Theory

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 n-dimensional Euclidean space, ℝ n . 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; (iii) 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 quasi-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.