v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 287On the other hand, positive definite Farkas’ Lemma 4.2.1.1.2 certifies thatA ∩ int S n + is empty; what we need to know for semidefinite programming.Lasserre suggested addition of another condition to semidefinite Farkas’lemma (4.2.1.1.1) to make a new lemma having no closedness condition.But positive definite Farkas’ lemma (4.2.1.1.2) is simpler and obviates theadditional condition proposed.4.2.1.2 Theorem of the alternative for semidefinite programmingBecause these Farkas’ lemmas follow from membership relations, we mayconstruct alternative systems from them. Applying the method of2.13.2.1.1,then from positive definite Farkas’ lemma we getA ∩ int S n + ≠ ∅or in the alternativem∑y T b ≤ 0, y i A i ≽ 0, y ≠ 0i=1(668)Any single vector y satisfying the alternative certifies A ∩ int S n + is empty.Such a vector can be found as a solution to another semidefinite program:for linearly independent (vectorized) set {A i ∈ S n , i=1... m}minimizeysubject toy T bm∑y i A i ≽ 0i=1‖y‖ 2 ≤ 1(669)If an optimal vector y ⋆ ≠ 0 can be found such that y ⋆T b ≤ 0, then primalfeasible cone interior A ∩ int S n + is empty.4.2.1.3 Boundary-membership criterion(confer (663) (664)) From boundary-membership relation (330) for linearmatrix inequality proper cones K (380) and K ∗ (387)b ∈ ∂K ⇔ ∃ y ≠ 0 〈y , b〉 = 0, y ∈ K ∗ , b ∈ K ⇔ ∂S n + ⊃ A ∩ S n + ≠ ∅(670)