v2009.01.01 - Convex Optimization

258 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Any single vector y satisfying the alternative certifies A ∩ int S n + is empty.
Such a vector can be found as a solution to another semidefinite program:
for linearly independent set {A i ∈ S n , i=1... m}
minimize
y
subject to
y T b
m∑
y i A i ≽ 0
i=1
‖y‖ 2 ≤ 1
(604)
If an optimal vector y ⋆ ≠ 0 can be found such that y ⋆T b ≤ 0, then relative
interior of the primal feasible set A ∩ int S n + from (596) is empty.
4.2.1.3 Boundary-membership criterion
(confer (598) (599)) From boundary-membership relation (298) for proper
cones and from linear matrix inequality cones K (337) and K ∗ (343)
b ∈ ∂K ⇔ ∃ y ≠ 0 〈y , b〉 = 0, y ∈ K ∗ , b ∈ K ⇔ ∂S n + ⊃ A ∩ S n + ≠ ∅
(605)
Whether vector b ∈ ∂K belongs to cone K boundary, that is a
determination we can indeed make; one that is certainly expressible as a
feasibility problem: assuming b ∈ K (597) given linearly independent set
{A i ∈ S n , i=1... m} 4.9 find y ≠ 0
subject to y T b = 0
m∑
y i A i ≽ 0
i=1
(606)
Any such feasible vector y ≠ 0 certifies that affine subset A (587) intersects
the positive semidefinite cone S n + only on its boundary; in other words,
nonempty feasible set A ∩ S n + belongs to the positive semidefinite cone
boundary ∂S n + .
4.9 From the results of Example 2.13.5.1.1, vector b on the boundary of K cannot be
detected simply by looking for 0 eigenvalues in matrix X .