Convex Sets and Convex Functions

Abstract

This chapter explores sets that can be represented as intersections of (a possibly infinite number of) halfspaces of Rⁿ . As will be shown, these are exactly the closed convex subsets. Furthermore, convex functions are studied, which are closely connected to convex sets and provide a natural generalization of linear functions. Then convex minimization problems are considered. Quadratic programs are examined from a theoretical and practical point of view. We finally discuss the ellipsoid method as an algorithmic procedure to find feasible points in convex bodies and indicate how it can be used to design (theoretically) efficient algorithms for solving linear and more general convex programming problems.