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).

In this chapter, we (a) present the definition and game theory interpretation of saddle points, (b) describe primal and dual optimization problems induced by an antagonistic game, (c) provide a characterization of saddle points in terms of optimal solutions and optimal values of primal and dual problems induced by the game, and (d) formulate and prove he Minimax Lemma and the Sion--Kakutani Theorem on existence of saddle points in convex--concave antagonistic games.

In this chapter, we (a) demonstrate that every local minimizer of a convex function is its global minimizer, (b) show that the Fermat rule provides a necessary and sufficient condition for an interior point of the domain of a convex function to be a global minimizer of the function, provided the function is differentiable at the point, (c) introduce radial and normal cones and express in terms of these cones the necessary and sufficient condition for a point to be the minimizer of a convex function whenever the function is differentiable at the point, (d) introduce the symmetry principle and (e) provide basic information on maximizers of convex functions.

Which of the following functions are convex on the indicated domains

In this chapter, we (a) present the definition and basic examples (linear and affine subspace, polyhedron, norm ball, simplex, cone) of convex sets, (b) introduce and study the notions of convex combination and convex hull, (c) outline basic convexity-preserving operations with sets, (d) establish nonemptiness of the relative interior of a nonempty convex set and the fact that the relative interior of a convex sets is dense in its closure, and (e) introduce the notions of conic and perspective transforms of a convex set.

In this chapter, we (a) introduce the Lagrange function of an inequality-constrained mathematical programming problem, (b) formulate and prove convex programming Lagrange Duality Theorem, and (c) establish the connection between Lagrange duality and saddle points of the Lagrange function.

In this chapter, we prove the Separation Theorem for convex sets and extract from it basic results on the geometry of closed convex sets, specifically (a) supporting hyperplanes, extreme points, and the (finite-dimensional) Krein--Milman Theorem, (b) recessive directions and recessive cone of a convex set, (c) the definition and basic properties of the dual cone, (d) the (finite-dimensional) Dubovitski--Milutin Lemma, (e) existence of bases and extreme rays of nontrivial closed pointed cones and their relation to extreme points of the cone’s base, (f)the (finite-dimensional) Krein--Milman Theorem in conic form, and (g) polarity.

>Mark in the following list the sets which are convex

Mark with Y those of the cases listed below where the linear form f⊤x separates the sets S and T

In this chapter, we demonstrate that (a) substituting the vector of eigenvalues of a symmetric n x n matrix into a convex permutation symmetric function of n real variables results in a convex function of the matrix, and (b) that if g is a convex function on the real axis, and G is the set of symmetric matrices of a given size with spectrum in the domain of g, then G is a convex set, and when X is a matrix from G, the trace of the matrix g(X), is a convex function of X; here g(X) is the matrix acting at a spectral subspace of X associated with eigenvalue v as multiplication by g(v); both these facts will be heavily used when speaking about cone-convexity is chapter 21.

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.

In this chapter, we derive the standard first- and second-order necessary/sufficient conditions for local optimality of a feasible solution to a (possibly nonconvex) mathematical programming problem. We conclude the chapter by illustrating these on the S-Lemma.

In this chapter, we (a) discuss the notion of lower semicontinuity of a function and demonstrate that functions with this property have closed epigraphs, (b) show that the pointwise supremum of a family of lower semicontinous functions is lower discontinuous, (c) demonstrate that a proper lower semiconscious convex function is the pointwise supremum of the affine minorants of the function, (d) introduce the notion of a subgradient and the subdifferential of a convex function at a point and demonstrate existence of subgradients at points from the relative interior of the function’s domain, (e) outline elementary rules of subdifferential calculus, and (f) establish basic properties of the directional derivatives of convex functions and the connection between directional derivatives and subdifferentials.

In this chapter, we extract from the results of Chapter 3 the basic theory of finite systems of linear inequalities - Farkas’ Lemmas, General Theorem on the Alternative, certificates for feasibility/infeasibility of polyhedral sets, and linear programming Duality Theorem.