v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 119given nonzero linearly independent A j ∈ S N and real b j . Define the affinesubsetA ∆ = {X | 〈A j , X〉=b j , j =1... m} ⊆ S N (230)If the intersection A ∩ S N + is nonempty, then there exists a matrixX ∈ A ∩ S N + such that given a number of equalities mrankX (rankX + 1)/2 ≤ m (231)whence the upper bound 2.36⌊√ ⌋ 8m + 1 − 1rankX ≤2or given desired rank instead, equivalently,(232)m < (rankX + 1)(rankX + 2)/2 (233)An extreme point of A ∩ S N + satisfies (232) and (233). (confer4.1.1.2)A matrix X =R ∆ T R is an extreme point if and only if the smallest face thatcontains X of A ∩ S N + has dimension 0 ; [174,2.4] id est, iff (138)dim F ( (A ∩ S N +)∋X )= rank(X)(rank(X) + 1)/2 − rank [ (234)svec RA 1 R T svec RA 2 R T · · · svec RA m R ] Tequals 0 in isomorphic R N(N+1)/2 .Now the intersection A ∩ S N + is assumed bounded: Assume a givennonzero upper bound ρ on rank, a number of equalitiesm=(ρ + 1)(ρ + 2)/2 (235)and matrix dimension N ≥ ρ + 2 ≥ 3. If the intersection is nonempty andbounded, then there exists a matrix X ∈ A ∩ S N + such thatrankX ≤ ρ (236)This represents a tightening of the upper bound; a reduction by exactly 1of the bound provided by (232) given the same specified number m (235) ofequalities; id est,rankX ≤√ 8m + 1 − 12− 1 (237)⋄2.364.1.1.2 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 + .