In this chapter, we (a) introduce the notion of a convex problem in cone-constrained form, (b) present the Lagrange function of a cone-constrained convex problem, (c) prove the convex programming Duality Theorem in cone-constrained form, and (d) discuss conic programming and conic duality, and present the conic programming Duality Theorem.

Given the linear dynamic system

In this chapter we present convex programming optimality conditions in both sadde point form and Karush--Kuhn--Tucker form for mathematical programming, and also optimality conditions for cone-constrained convex programs and for conic problems. We conclude the chapter by revisiting linear programming duality as a special case of conic duality and reproducing the classical results on the dual of a linearly constrained convex quadratic minimization problem.

In this chapter, we (a) outline operations preserving convexity of functions, (b) present differential criteria for convexity, (c) establish convexity of several important multivariate functioins, (d) present the gradient inequality, and (e) establish local boundedness and Lipschitz continuity of convex functions.

In this chapter, we (a) present the notion of a polyhedral representation and illustrate its importance, (b) demonstrate via Fourier--Motzkin eliminaton that every polyhedrally representable set is polyhedral, and (c) outline the calculus of polyhedral representations. As an immediate application, we demonstrate that a bounded and feasible LP problem is solvable.

In this chapter, we (a) present epigraph characterization of cone-convexity, (b) introduce cone-monotonicity, and describe differential criteria of cone-convexity and cone-monotonicity, (c) present instructive examples of cone-convex and cone-monotone functions, (d) outline basic operations preserving cone-convexity and cone-monotonicity. Taken together, (b)--(d) provide simple and powerful tools allowing one to detect and utilize cone-convexity and cone-monotonicity.

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.

In this chapter, we present preliminaries on convex functions -- definitions via convexity inequality and via the convexity of the epigraph, basic examples, Jensen’s inequality, convexity of sublevel sets; we introduce the notion of the domain of a convex function and its representation as a function taking values in the extended real axis and introduce the concept of a proper convex function.

In this chapter, we present and illustrate Caratheodory’s Theorem (in plain and conic forms), Radon’s Theorem, and Helly’s Theorem (for finite and for infinite families of convex sets).