v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 1392.9.3.0.1 Proposition. (Barvinok) Affine intersection with PSD cone.[26,II.13] [24,2.2] Consider finding a matrix X ∈ S N satisfyingX ≽ 0 , 〈A j , X〉 = b j , j =1... m (269)given nonzero linearly independent (vectorized) A j ∈ S N and real b j . Definethe affine subsetA {X | 〈A j , X〉=b j , j =1... m} ⊆ S N (270)If the intersection A ∩ S N + is nonempty given a number m of equalities,then there exists a matrix X ∈ A ∩ S N + such thatrankX (rankX + 1)/2 ≤ m (271)whence the upper bound 2.50⌊√ ⌋ 8m + 1 − 1rankX ≤2(272)Given desired rank instead, equivalently,m < (rankX + 1)(rankX + 2)/2 (273)An extreme point of A ∩ S N + satisfies (272) and (273). (confer4.1.2.2)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 ; [239,2.4] [240] id est, iff (171)dim F ( (A ∩ S N +)∋X )= rank(X)(rank(X) + 1)/2 − rank [ (274)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 (275)2.504.1.2.2 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 + .