v2006.03.09 - Convex Optimization

156 CHAPTER 2. CONVEX GEOMETRYWhen pointed closed convex cone K(confer2.7.2.2)x ≽ 0 ⇔ x ∈ Kx ≻ 0 ⇔ x ∈ rel int Kis implicit from context:(266)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 K and its dual K ∗x ∈ int K ⇔ 〈y, x〉 > 0 for all y ∈ K ∗ , y ≠ 0 (267)x ∈ K , x ≠ 0 ⇔ 〈y, x〉 > 0 for all y ∈ int K ∗ (268)By conjugation, we also have the dual relations:y ∈ int K ∗ ⇔ 〈y, x〉 > 0 for all x ∈ K , x ≠ 0 (269)y ∈ K ∗ , y ≠ 0 ⇔ 〈y, x〉 > 0 for all x ∈ int K (270)Boundary-membership relations for proper cones are also useful:x ∈ ∂K ⇔ ∃ y 〈y, x〉 = 0, y ∈ K ∗ , y ≠ 0, x ∈ K (271)y ∈ ∂K ∗ ⇔ ∃ x 〈y, x〉 = 0, x ∈ K , x ≠ 0, y ∈ K ∗ (272)2.13.2.0.2 Example. Dual linear transformation. [217,4]Consider a given matrix A and closed convex cone K . By membershiprelation 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} ∗ (273)This implies{y | Ay ∈ K ∗ } = {A T x | x ∈ K} ∗ (274)If we regard A as a linear operator, then A T is its adjoint.