v2007.09.13 - Convex Optimization

146 CHAPTER 2. CONVEX GEOMETRYfor which implicitly K = K ∗ = R n + the nonnegative orthant.Membership relation (276) 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 (278)K ∗meaning, coordinates for biorthogonal expansion of x (2.13.8) [273] mustbe nonnegative when x belongs to K . By conjugation [228, thm.14.1]y ≽K ∗0 ⇔ 〈y , x〉 ≥ 0 for all x ≽K0 (279)⋄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 ∗ (276)⇔x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ , ‖y‖= 1(280)2.13.2.0.2 Exercise. Test of dual generalized inequalities.Test Corollary 2.13.2.0.1 and (280) graphically on the two-dimensionalpolyhedral cone and its dual in Figure 43.When pointed closed convex cone K(confer2.7.2.2)x ≽ 0 ⇔ x ∈ Kx ≻ 0 ⇔ x ∈ rel int Kis implicit from context:(281)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 (282)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ (283)