Convex Analysis

Abstract

Convexity is an important part of optimization, and we devote several chapters to various aspects of it in this book. This chapter treats the most basic properties of convex sets and functions, while Chapter 5 is devoted to their deeper properties, such as the relative interior (both in the algebraic and topological senses) and boundary structure of convex sets, the continuity properties of convex functions, and homogenization of convex sets. Chapter 6 treats the separation properties of convex sets, a very important topic in optimization. The calculus of the relative interior of convex sets developed in Chapter 5 plays an important role here. Chapters 7 and 8 deal with two, related, special topics, the theory of convex polyhedra and the theory of linear programming, respectively. Both are important topics within optimization, and each has wide applicability within science, engineering, and technology. Many developments in optimization have in fact been inspired by linear programming. Finally, Chapter 13 investigates several special topics in convexity.