v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 147By conjugation, we also have the dual relations:y ∈ int K ∗ ⇔ 〈y , x〉 > 0 for all x ∈ K , x ≠ 0 (284)y ∈ K ∗ , y ≠ 0 ⇔ 〈y , x〉 > 0 for all x ∈ int K (285)Boundary-membership relations for proper cones are also useful:x ∈ ∂K ⇔ ∃ y 〈y , x〉 = 0, y ∈ K ∗ , y ≠ 0, x ∈ K (286)y ∈ ∂K ∗ ⇔ ∃ x 〈y , x〉 = 0, x ∈ K , x ≠ 0, y ∈ K ∗ (287)2.13.2.0.3 Example. Linear inequality. [252,4](confer2.13.5.1.1) Consider a given matrix A and closed convex cone K .By membership relation we haveThis impliesAy ∈ K ∗ ⇔ x T Ay≥0 ∀x ∈ K⇔ y T z ≥0 ∀z ∈ {A T x | x ∈ K}⇔ y ∈ {A T x | x ∈ K} ∗ (288){y | Ay ∈ K ∗ } = {A T x | x ∈ K} ∗ (289)If we regard A as a linear operator, then A T is its adjoint. When, forexample, K is the self-dual nonnegative orthant, (2.13.5.1) then{y | Ay ≽ 0} = {A T x | x ≽ 0} ∗ (290)2.13.2.1 Null certificate, Theorem of the alternativeIf in particular x p /∈ K a closed convex cone, then the constructionin Figure 42(b) suggests there exists a supporting hyperplane (havinginward-normal belonging to dual cone K ∗ ) separating x p from K ; indeed,(276)x p /∈ K ⇔ ∃ y ∈ K ∗ 〈y , x p 〉 < 0 (291)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(292)