v2009.01.01 - Convex Optimization

272 CHAPTER 4. SEMIDEFINITE PROGRAMMING
(t ⋆ i) −1 = max {ψ(Z i )λ(Z i ) j , j =1... ρ} (649)
When Z i is indefinite, the direction of perturbation (determined by ψ(Z i )) is
arbitrary. We may take an early exit from the Procedure were Z i to become
0 or were
rank [ svec R T iA 1 R i svec R T iA 2 R i · · · svec R T iA m R i
]
= ρ(ρ + 1)/2 (650)
which characterizes the rank ρ of any [sic] extreme point in A ∩ S n + .
[206,2.4] [207]
Proof. Assuming the form of every perturbation matrix is indeed (643),
then by (646)
svec Z i ⊥ [ svec(R T iA 1 R i ) svec(R T iA 2 R i ) · · · svec(R T iA m R i ) ] (651)
By orthogonal complement we have
rank [ svec(R T iA 1 R i ) · · · svec(R T iA m R i ) ] ⊥
+ rank [ svec(R T iA 1 R i ) · · · svec(R T iA m R i ) ] = ρ(ρ + 1)/2
(652)
When Z i can only be 0, then the perturbation is null because an extreme
point has been found; thus
[
svec(R
T
i A 1 R i ) · · · svec(R T iA m R i ) ] ⊥
= 0 (653)
from which the stated result (650) directly follows.