v2007.09.17 - Convex Optimization

238 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.2.1.3 Boundary-membership criterion(confer (560) (561)) From boundary-membership relation (286) for propercones and from linear matrix inequality cones K (324) and K ∗ (330)b ∈ ∂K ⇔ ∃ y 〈y , b〉 = 0, y ∈ K ∗ , y ≠ 0, b ∈ K ⇔ ∂S n + ⊃ A ∩ S n + ≠ ∅(567)Whether vector b ∈ ∂K belongs to cone K boundary, that is adetermination we can indeed make; one that is certainly expressible as afeasibility problem: assuming b ∈ K (559) given linearly independent set{A i ∈ S n , i=1... m} 4.8 find y ≠ 0subject to y T b = 0m∑y i A i ≽ 0i=1(568)Any such feasible vector y ≠ 0 certifies that affine subset A (549) intersectsthe positive semidefinite cone S n + only on its boundary; in other words,nonempty feasible set A ∩ S n + belongs to the positive semidefinite coneboundary ∂S n + .4.2.2 DualsThe dual objective function evaluated at any feasible point represents a lowerbound on the primal optimal objective value. We can see this by directsubstitution: Assume the feasible sets A ∩ S n + and C ∗ are nonempty. Thenit is always true:〈 ∑i〈C , X〉 ≥ 〈b, y〉〉y i A i + S , X ≥ [ 〈A 1 , X〉 · · · 〈A m , X〉 ] y〈S , X〉 ≥ 0(569)The converse also follows becauseX ≽ 0, S ≽ 0 ⇒ 〈S,X〉 ≥ 0 (1285)4.8 From the results of Example 2.13.5.1.1, vector b on the boundary of K cannot bedetected simply by looking for 0 eigenvalues in matrix X .