v2006.03.09 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1572.13.2.1 Null certificate, Theorem of the alternativeIf in particular x p /∈ K a closed convex cone, then the construction inFigure 40(b) suggests there exists a hyperplane having inward-normalbelonging to dual cone K ∗ separating x p from K ; indeed, (260)x p /∈ K ⇔ ∃ y ∈ K ∗ 〈y , x p 〉 < 0 (275)The existence of any one such y is a certificate of null membership. From adifferent perspective,x p ∈ Kor in the alternative∃ y ∈ K ∗ 〈y , x p 〉 < 0(276)By alternative is meant: these two systems are incompatible; one system isfeasible while the other is not.2.13.2.1.1 Example. Theorem of the alternative for linear inequality.Myriad alternative systems of linear inequality can be explained in terms ofpointed closed convex cones and their duals.From membership relation (262) with affine transformation of dualvariable we write, for example,b − Ay ∈ K ∗ ⇔ x T (b − Ay)≥ 0 ∀x ∈ K (277)A T x=0, b − Ay ∈ K ∗ ⇒ x T b ≥ 0 ∀x ∈ K (278)where A∈ R n×m and b∈R n are given. Given membership relation (277),conversely, suppose we allow any y ∈ R m . Then because −x T Ay isunbounded below, x T (b −Ay)≥0 implies A T x=0: for y ∈ R mIn toto,A T x=0, b − Ay ∈ K ∗ ⇐ x T (b − Ay)≥ 0 ∀x ∈ K (279)b − Ay ∈ K ∗ ⇔ x T b ≥ 0, A T x=0 ∀x ∈ K (280)Vector x belongs to cone K but is constrained to lie in a subspaceof R n specified by an intersection of hyperplanes through the origin{x∈ R n |A T x=0}. From this, alternative systems of generalized inequalitywith respect to pointed closed convex cones K and K ∗