Essential Mathematics for Convex Optimization

In this chapter, we (a) present an algebraic characterization of extreme points of polyhedral sets and extreme rays of polyhedral cones, (b) describe extreme points of several important polyhedral sets, including the Birkhoff--von-Neumann Theorem on extreme points of the polytope of doubly stochastic matrices, (c) establish the theorem on the structure of polyhedral sets stating that nonempty polyhedral sets are exactly the sets representable as sums of convex hulls of nonempty finite sets and conic hulls of finite sets, and vice versa, (d) extract from the latter theorem basic descriptive results of linear programming theory, and (e) present and justify the Majorization Principle.