Essential Mathematics for Convex Optimization

In this chapter, we (a) outline the subject and the terminology of mathematical and convex programming, (b) introduce the Slater and relaxed Slater conditions and formulate the Convex Theorem on the Alternative -- the basis of Lagrange duality theory in convex programming, (c) introduce the notions of cone-convexity and of the convex programming problem in cone-constrained form, thus extending the standard mathematical programming setup of convex optimization, and (d) formulate and prove the Convex Theorem on the Alternative in cone-constrained form, justifying, as a byproduct, the standard Convex Theorem on the Alternative.