v2006.03.09 - Convex Optimization

6.2. FRAMEWORK 3636.2.1.1 A ∩ S n + emptiness determination via Farkas’ lemma6.2.1.1.1 Lemma. Semidefinite Farkas’ lemma.Given an arbitrary set {A i ∈ S n , i=1... m} and a vector b = [b i ]∈ R m ,define the affine subsetA = {X ∈ S n | 〈A i , X〉 = b i , i=1... m} (848)Primal feasible set A ∩ S n + is nonempty if and only if y T b ≥ 0 holds foreach and every vector y = [y i ]∈ R m ∑such that m y i A i ≽ 0.Equivalently, primal feasible set A ∩ S n + is nonempty if and onlyif y T b ≥ 0 holds for each and every norm-1 vector ‖y‖= 1 such thatm∑y i A i ≽ 0.⋄i=1Semidefinite Farkas’ lemma follows directly from a membership relation(2.13.2.0.1) and the closed convex cones from linear matrix inequalityexample 2.13.5.1.1; given convex cone K and its duali=1whereK = {A svec X | X ≽ 0} (301)m∑K ∗ = {y | y j A j ≽ 0} (307)⎡A = ⎣j=1⎤svec(A 1 ) T. ⎦ ∈ R m×n(n+1)/2 (846)svec(A m ) Tthen we have membership relationb ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ (260)and equivalentsb ∈ K ⇔ ∃X ≽ 0 A svec X = b ⇔ A ∩ S n + ≠ ∅ (858)