v2010.10.26 - Convex Optimization

288 CHAPTER 4. SEMIDEFINITE PROGRAMMINGWhether vector b ∈ ∂K belongs to cone K boundary, that is a determinationwe can indeed make; one that is certainly expressible as a feasibility problem:Given linearly independent set 4.12 {A i ∈ S n , i=1... m} , for b ∈ K (662)find y ≠ 0subject to y T b = 0m∑y i A i ≽ 0i=1(671)Any such nonzero solution y certifies that affine subset A (652) intersects thepositive semidefinite cone S n + only on its boundary; in other words, nonemptyfeasible set A ∩ S n + belongs to the positive semidefinite cone boundary ∂S n + .4.2.2 DualsThe dual objective function from (649D) evaluated at any feasible solutionrepresents a lower bound on the primal optimal objective value from (649P).We can see this by direct substitution: Assume the feasible sets A ∩ S n + andD ∗ are nonempty. Then it is always true:〈 ∑i〈C , X〉 ≥ 〈b, y〉〉y i A i + S , X ≥ [ 〈A 1 , X〉 · · · 〈A m , X〉 ] y〈S , X〉 ≥ 0(672)The converse also follows becauseX ≽ 0, S ≽ 0 ⇒ 〈S , X〉 ≥ 0 (1485)Optimal value of the dual objective thus represents the greatest lower boundon the primal. This fact is known as the weak duality theorem for semidefiniteprogramming, [391,1.3.8] and can be used to detect convergence in anyprimal/dual numerical method of solution.4.12 From the results of Example 2.13.5.1.1, vector b on the boundary of K cannotbe detected simply by looking for 0 eigenvalues in matrix X . We do not consider askinny-or-square matrix A because then feasible set A ∩ S+ n is at most a single point.