v2010.10.26 - Convex Optimization

168 CHAPTER 2. CONVEX GEOMETRY2.13.2 Abstractions of Farkas’ lemma2.13.2.0.1 Corollary. Generalized inequality and membership relation.[199,A.4.2] Let K be any closed convex cone and K ∗ its dual, and let xand y belong to a vector space R n . Theny ∈ K ∗ ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (319)which is, merely, a statement of fact by definition of dual cone (297). Byclosure we have conjugation: [307, thm.14.1]x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ (320)which may be regarded as a simple translation of Farkas’ lemma [133] asin [307,22] to the language of convex cones, and a generalization of thewell-known Cartesian cone factx ≽ 0 ⇔ 〈y , x〉 ≥ 0 for all y ≽ 0 (321)for which implicitly K = K ∗ = R n + the nonnegative orthant.Membership relation (320) is often written instead as dual generalizedinequalities, when K and K ∗ are pointed closed convex cones,x ≽K0 ⇔ 〈y , x〉 ≥ 0 for all y ≽ 0 (322)K ∗meaning, coordinates for biorthogonal expansion of x (2.13.7.1.2,2.13.8)[365] must be nonnegative when x belongs to K . Conjugating,y ≽K ∗0 ⇔ 〈y , x〉 ≥ 0 for all x ≽K0 (323)⋄When pointed closed convex cone K is not polyhedral, coordinate axesfor biorthogonal expansion asserted by the corollary are taken from extremedirections of K ; expansion is assured by Carathéodory’s theorem (E.6.4.1.1).We presume, throughout, the obvious:x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ (320)⇔x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ , ‖y‖= 1(324)