v2010.10.26 - Convex Optimization

286 CHAPTER 4. SEMIDEFINITE PROGRAMMINGFigure 45. Then given A∈ R m×n(n+1)/2 having rank m , we wish to detectexistence of nonempty primal feasible set interior to the PSD cone; 4.11 (383)b ∈ int K ⇔ 〈y, b〉 > 0 ∀y ∈ K ∗ , y ≠ 0 ⇔ A ∩ int S n + ≠ ∅ (664)Positive definite Farkas’ lemma is made from proper cones, K (380) and K ∗(387), and membership relation (326) for which K closedness is unnecessary:4.2.1.1.2 Lemma. Positive definite Farkas’ lemma.Given l.i. set {A i ∈ S n , i=1... m} and vector b = [b i ]∈ R m , make affine setA = {X ∈ S n |〈A i , X〉=b i , i=1... m} (652)Primal feasible cone interior A ∩ int S n + is nonempty if and only if y T b > 0∑holds for each and every vector y = [y i ]≠ 0 such that m y i A i ≽ 0.Equivalently, primal feasible cone interior A ∩ int S n + is nonempty if and∑only if y T b > 0 holds for each and every vector ‖y‖= 1 m y i A i ≽ 0. ⋄4.2.1.1.3 Example. “New” Farkas’ lemma.Lasserre [232,III] presented an example in 1995, originally offered byBen-Israel in 1969 [32, p.378], to support closedness in semidefinite Farkas’Lemma 4.2.1.1.1:[ ] svec(A1 )A Tsvec(A 2 ) T =[ 0 1 00 0 1i=1i=1] [ 1, b =0Intersection A ∩ S n + is practically empty because the solution set{[ ] }α √2 1{X ≽ 0 | A svec X = b} =≽ 0 | α∈ R√120](665)(666)is positive semidefinite only asymptotically (α→∞). Yet the dual systemm∑y i A i ≽0 ⇒ y T b≥0 erroneously indicates nonempty intersection becausei=1K (380) violates a closedness condition of the lemma; videlicet, for ‖y‖= 1[ ]01 [ ][ ]√2 0 00y 1√1+ y20 2 ≽ 0 ⇔ y = ⇒ y T b = 0 (667)0 114.11 Detection of A ∩ int S n +≠ ∅ by examining int K instead is a trick need not be lost.