v2010.10.26 - Convex Optimization

170 CHAPTER 2. CONVEX GEOMETRYThis implies{y | Ay ∈ K ∗ } = {A T x | x ∈ K} ∗ (333)When K is the selfdual nonnegative orthant (2.13.5.1), for example, thenand the dual relation{y | Ay ≽ 0} = {A T x | x ≽ 0} ∗ (334){y | Ay ≽ 0} ∗ = {A T x | x ≽ 0} (335)comes by a theorem of Weyl (p.148) that yields closedness for anyvertex-description of a polyhedral cone.2.13.2.1 Null certificate, Theorem of the alternativeIf in particular x p /∈ K a closed convex cone, then construction in Figure 55bsuggests there exists a supporting hyperplane (having inward-normalbelonging to dual cone K ∗ ) separating x p from K ; indeed, (320)x p /∈ K ⇔ ∃ y ∈ K ∗ 〈y , x p 〉 < 0 (336)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(337)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.Beginning from the simplest Cartesian dual generalized inequalities (321)(with respect to nonnegative orthant R m + ),y ≽ 0 ⇔ x T y ≥ 0 for all x ≽ 0 (338)