v2007.09.13 - Convex Optimization

4.2. FRAMEWORK 235Equivalently, 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 dualwhereK = {A svec X | X ≽ 0} (324)m∑K ∗ = {y | y j A j ≽ 0} (330)⎡A = ⎣j=1then we have membership relationand equivalents⎤svec(A 1 ) T. ⎦ ∈ R m×n(n+1)/2 (547)svec(A m ) Tb ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ (276)b ∈ K ⇔ ∃X ≽ 0 A svec X = b ⇔ A ∩ S n + ≠ ∅ (559)b ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ ⇔ A ∩ S n + ≠ ∅ (560)Semidefinite Farkas’ lemma provides the conditions required for a setof hyperplanes to have a nonempty intersection A ∩ S n + with the positivesemidefinite cone. While the lemma as stated is correct, Ye points out[296,1.3.8] that a positive definite version of this lemma is required forsemidefinite programming because any feasible point in the relative interiorA ∩ int S n + is required by Slater’s condition 4.6 to achieve 0 duality gap(primal−dual objective difference4.2.3, Figure 45). In our circumstance,assuming a nonempty intersection, a positive definite lemma is requiredto insure a point of intersection closest to the origin is not at infinity;4.6 Slater’s sufficient condition is satisfied whenever any primal strictly feasible pointexists; id est, any point feasible with the affine equality (or affine inequality) constraintfunctions and relatively interior to convex cone K . If cone K is polyhedral, then Slater’scondition is satisfied when any feasible point exists relatively interior to K or on its relativeboundary. [46,5.2.3] [29, p.325]