Convexity and Optimization
Optimization is a central theme of applied mathematics that involves minimizing or maximizing various quantities. This is an important application of the derivative tests in calculus. In addition to the first and second derivative tests of one-variable calculus, there is the powerful technique of Lagrange multipliers in several variables. This chapter is concerned with analogues of these tests that are applicable to functions that are not differentiable. Of course, some different hypothesis must replace differentiability, and this is the notion of convexity. It turns out that many applications in economics, business, and related areas involve convex functions. As in other chapters of this book, we concentrate on the theoretical underpinnings of the subject. The important aspect of constructing algorithms to carry out our program is not addressed. However, the reader will be well placed to read that material. Results from both linear algebra and calculus appear regularly.
The study of convex sets and convex functions is a comparatively recent development. Although convexity appears implicitly much earlier (going back to work of Archimedes, in fact), the first papers on convex sets appeared at the end of the nineteenth century. The main theorems of this chapter, characterizations of solutions of optimization problems, first appeared around the middle of the twentieth century. Starting in the 1970s, there has been considerable work on extending these methods to nonconvex functions.
KeywordsSaddle Point Convex Function Extreme Point Convex Subset Directional Derivative
Unable to display preview. Download preview PDF.