v2006.03.09 - Convex Optimization

364 CHAPTER 6. SEMIDEFINITE PROGRAMMINGSemidefinite 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[259,1.3.8] 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 6.6 to achieve 0 duality gap(primal-dual objective difference6.2.3). In our circumstance, assuminga nonempty intersection, a positive definite lemma is required to insure apoint of intersection closest to the origin is not at infinity; e.g., Figure 31.Then given A∈ R m×n(n+1)/2 having rank m , we wish to detect existence ofa nonempty relative interior of the primal feasible set; 6.7b ∈ int K ⇔ 〈y, b〉 > 0 for all y ∈ K ∗ , y ≠ 0 ⇔ A ∩ int S n + ≠ ∅(859)A positive definite Farkas’ lemma can easily be constructed from thismembership relation (267) and these proper convex cones K (301) andK ∗ (307):6.2.1.1.2 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} (848)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=16.6 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. [38,5.2.3] [259,1.3.8] [26, p.325]6.7 Detection of A ∩ int S n + by examining K interior is a trick need not be lost.i=1