v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1532.13.4.2.1 Theorem. Discrete membership. (confer2.13.2.0.1)Given any set of generators (2.8.1.2) denoted by G(K) for closed convexcone K ⊆ R n and any set of generators denoted G(K ∗ ) for its dual, let xand y belong to vector space R n . Then discretization of the generalizedinequality and membership corollary is necessary and sufficient for certifyingmembership:x ∈ K ⇔ 〈γ ∗ , x〉 ≥ 0 for all γ ∗ ∈ G(K ∗ ) (317)y ∈ K ∗ ⇔ 〈γ , y〉 ≥ 0 for all γ ∈ G(K) (318)⋄2.13.4.2.2 Exercise. Test of discretized dual generalized inequalities.Test Theorem 2.13.4.2.1 on Figure 43(a) using the extreme directions asgenerators.From the discrete membership theorem we may further deduce a moresurgical description of dual cone that prescribes only a finite number ofhalfspaces for its construction when polyhedral: (Figure 42(a))K ∗ = {y ∈ R n | 〈γ , y〉 ≥ 0 for all γ ∈ G(K)} (319)2.13.4.2.3 Exercise. Comparison with respect to orthant.When comparison is with respect to the nonnegative orthant K = R n + , thenfrom the discrete membership theorem it directly follows:x ≼ z ⇔ x i ≤ z i ∀i (320)Generate simple counterexamples demonstrating that this equivalence withentrywise inequality holds only when the underlying cone inducing partialorder is the nonnegative orthant.2.13.5 Dual PSD cone and generalized inequalityThe dual positive semidefinite cone K ∗ is confined to S M by convention;S M ∗+∆= {Y ∈ S M | 〈Y , X〉 ≥ 0 for all X ∈ S M + } = S M + (321)The positive semidefinite cone is self-dual in the ambient space of symmetricmatrices [46, exmp.2.24] [28] [144,II]; K = K ∗ .