v2007.09.17 - Convex Optimization

4.2. FRAMEWORK 237is positive semidefinite only asymptotically (α→∞). Yet the dual systemm∑y i A i ≽0 ⇒ y T b≥0 indicates nonempty intersection; videlicet, for ‖y‖= 1i=1y 1[01 √21 √20]+ y 2[ 0 00 1][ ] 0≽ 0 ⇔ y =1⇒ y T b = 0 (564)On the other hand, positive definite Farkas’ Lemma 4.2.1.1.2 showsA ∩ int S n + is empty; what we need to know for semidefinite programming.Based on Ben-Israel’s example, Lasserre suggested addition of anothercondition to semidefinite Farkas’ Lemma 4.2.1.1.1 to make a “new” lemma.Ye recommends positive definite Farkas’ Lemma 4.2.1.1.2 instead; which issimpler and obviates Lasserre’s proposed additional condition. 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, for example, we getA ∩ int S n + ≠ ∅or in the alternativem∑y T b ≤ 0, y i A i ≽ 0, y ≠ 0i=1(565)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 set {A i ∈ S n , i=1... m}minimizeysubject toy T bm∑y i A i ≽ 0i=1‖y‖ 2 ≤ 1(566)If an optimal vector y ⋆ ≠ 0 can be found such that y ⋆T b ≤ 0, then relativeinterior of the primal feasible set A ∩ int S n + from (558) is empty.