v2009.01.01 - Convex Optimization

4.3. RANK REDUCTION 269
whose rank does not satisfy upper bound (245), we posit existence of a set
of perturbations
{t j B j | t j ∈ R , B j ∈ S n , j =1... n} (638)
such that, for some 0≤i≤n and scalars {t j , j =1... i} ,
X ⋆ +
i∑
t j B j (639)
j=1
becomes an extreme point of A ∩ S n + and remains an optimal solution of
(584P). Membership of (639) to affine subset A is secured for the i th
perturbation by demanding
〈B i , A j 〉 = 0, j =1... m (640)
while membership to the positive semidefinite cone S n + is insured by small
perturbation (649). In this manner feasibility is insured. Optimality is proved
in4.3.3.
The following simple algorithm has very low computational intensity and
locates an optimal extreme point, assuming a nontrivial solution:
4.3.1.0.1 Procedure. Rank reduction. (Wıκımization)
initialize: B i = 0 ∀i
for iteration i=1...n
{
1. compute a nonzero perturbation matrix B i of X ⋆ + i−1 ∑
j=1
t j B j
2. maximize t i
subject to X ⋆ + i ∑
j=1
t j B j ∈ S n +
}