v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 285Semidefinite Farkas’ lemma provides necessary and sufficient conditionsfor a set of hyperplanes to have nonempty intersection A ∩ S n + with thepositive semidefinite cone. Given⎡ ⎤svec(A 1 ) TA = ⎣ . ⎦ ∈ R m×n(n+1)/2 (650)svec(A m ) Tsemidefinite Farkas’ lemma assumes that a convex coneK = {A svec X | X ≽ 0} (380)is closed per membership relation (320) from which the lemma springs:[232,I] K closure is attained when matrix A satisfies the cone closednessinvariance corollary (p.183). Given closed convex cone K and its dual fromExample 2.13.5.1.1m∑K ∗ = {y | y j A j ≽ 0} (387)j=1then we can apply membership relationto obtain the lemmab ∈ K ⇔ 〈y , b〉 ≥ 0 ∀y ∈ K ∗ (320)b ∈ K ⇔ ∃X ≽ 0 A svec X = b ⇔ A ∩ S n + ≠ ∅ (662)b ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ ⇔ A ∩ S n + ≠ ∅ (663)The final equivalence synopsizes semidefinite Farkas’ lemma.While the lemma is correct as stated, a positive definite version is requiredfor semidefinite programming [391,1.3.8] because existence of a feasiblesolution in the cone interior A ∩ int S n + is required by Slater’s condition 4.10to achieve 0 duality gap (optimal primal−dual objective difference4.2.3,Figure 58). Geometrically, a positive definite lemma is required to insurethat a point of intersection closest to the origin is not at infinity; e.g.,4.10 Slater’s sufficient constraint qualification is satisfied whenever any primal or dualstrictly feasible solution exists; id est, any point satisfying the respective affine constraintsand relatively interior to the convex cone. [330,6.6] [41, p.325] If the cone were polyhedral,then Slater’s constraint qualification is satisfied when any feasible solution exists (relativelyinterior to the cone or on its relative boundary). [61,5.2.3]