v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 125
given nonzero linearly independent A j ∈ S N and real b j . Define the affine
subset
A ∆ = {X | 〈A j , X〉=b j , j =1... m} ⊆ S N (243)
If the intersection A ∩ S N + is nonempty given a number m of equalities,
then there exists a matrix X ∈ A ∩ S N + such that
rankX (rankX + 1)/2 ≤ m (244)
whence the upper bound 2.41
⌊√ ⌋ 8m + 1 − 1
rankX ≤
2
Given desired rank instead, equivalently,
(245)
m < (rankX + 1)(rankX + 2)/2 (246)
An extreme point of A ∩ S N + satisfies (245) and (246). (confer4.1.1.3)
A matrix X =R ∆ T R is an extreme point if and only if the smallest face that
contains X of A ∩ S N + has dimension 0 ; [206,2.4] [207] id est, iff (151)
dim F ( (A ∩ S N +)∋X )
= rank(X)(rank(X) + 1)/2 − rank [ (247)
svec RA 1 R T svec RA 2 R T · · · svec RA m R ] T
equals 0 in isomorphic R N(N+1)/2 .
Now the intersection A ∩ S N + is assumed bounded: Assume a given
nonzero upper bound ρ on rank, a number of equalities
m=(ρ + 1)(ρ + 2)/2 (248)
and matrix dimension N ≥ ρ + 2 ≥ 3. If the intersection is nonempty and
bounded, then there exists a matrix X ∈ A ∩ S N + such that
rankX ≤ ρ (249)
This represents a tightening of the upper bound; a reduction by exactly 1
of the bound provided by (245) given the same specified number m (248) of
equalities; id est,
rankX ≤
√ 8m + 1 − 1
2
− 1 (250)
⋄
2.414.1.1.3 contains an intuitive explanation. This bound is itself limited above, of course,
by N ; a tight limit corresponding to an interior point of S N + .