v2007.09.13 - Convex Optimization

236 CHAPTER 4. SEMIDEFINITE PROGRAMMINGe.g., Figure 33. Then given A∈ R m×n(n+1)/2 having rank m , we wish todetect existence of a nonempty relative interior of the primal feasible set; 4.7b ∈ int K ⇔ 〈y, b〉 > 0 ∀y ∈ K ∗ , y ≠ 0 ⇔ A∩int S n + ≠ ∅ (561)A positive definite Farkas’ lemma can easily be constructed from thismembership relation (282) and these proper convex cones K (324) andK ∗ (330):4.2.1.1.2 Lemma. Positive definite Farkas’ lemma.Given a linearly independent set {A i ∈ S n , i=1... m}b = [b i ]∈ R m , define the affine subsetand a vectorA = {X ∈ S n | 〈A i , X〉 = b i , i=1... m} (549)Primal feasible set relative 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 set relative interior A ∩ int S n + is nonemptyif and only if y T b > 0 holds for each and every norm-1 vector ‖y‖= 1 such∑that m y i A i ≽ 0.⋄i=1i=14.2.1.1.3 Example. “New” Farkas’ lemma.In 1995, Lasserre [166,III] presented an example originally offered byBen-Israel in 1969 [26, p.378] as evidence of failure in semidefinite Farkas’Lemma 4.2.1.1.1:[ ]A =∆ svec(A1 ) Tsvec(A 2 ) T =[ 0 1 00 0 1] [ 1, b =0](562)The intersection A ∩ S n + is practically empty because the solution set{X ≽ 0 | A svec X = b} ={[α1 √2√120]≽ 0 | α∈ R}(563)4.7 Detection of A ∩ int S n + by examining K interior is a trick need not be lost.