v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1692.13.2.0.2 Exercise. Test of dual generalized inequalities.Test Corollary 2.13.2.0.1 and (324) graphically on the two-dimensionalpolyhedral cone and its dual in Figure 56.(confer2.7.2.2) When pointed closed convex cone K is implicit from context:x ≽ 0 ⇔ x ∈ Kx ≻ 0 ⇔ x ∈ rel int K(325)Strict inequality x ≻ 0 means coordinates for biorthogonal expansion of xmust be positive when x belongs to rel int K . Strict membership relationsare useful; e.g., for any proper cone 2.63 K and its dual K ∗x ∈ int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ , y ≠ 0 (326)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ (327)Conjugating, we get the dual relations:y ∈ int K ∗ ⇔ 〈y , x〉 > 0 for all x ∈ K , x ≠ 0 (328)y ∈ K ∗ , y ≠ 0 ⇔ 〈y , x〉 > 0 for all x ∈ int K (329)Boundary-membership relations for proper cones are also useful:x ∈ ∂K ⇔ ∃ y ≠ 0 〈y , x〉 = 0, y ∈ K ∗ , x ∈ K (330)y ∈ ∂K ∗ ⇔ ∃ x ≠ 0 〈y , x〉 = 0, x ∈ K , y ∈ K ∗ (331)which are consistent; e.g., x∈∂K ⇔ x /∈int K and x∈ K .2.13.2.0.3 Example. Linear inequality. [337,4](confer2.13.5.1.1) Consider a given matrix A and closed convex cone K .By membership relation we haveAy ∈ K ∗ ⇔ x T Ay≥0 ∀x ∈ K⇔ y T z ≥0 ∀z ∈ {A T x | x ∈ K}⇔ y ∈ {A T x | x ∈ K} ∗ (332)2.63 An open cone K is admitted to (326) and (329) by (19).