v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 119is a diagonal projection matrix whose entries are either 1 or 0 (E.3). Wehave the complementary sumΦQ T x + (I − Φ)Q T x = Q T x (210)So, adding (I −Φ)Q T x to both sides of the membership within (208) admitsvec X = {x∈ R N2 | Q T x ∈ Λ † Q T (K − b) + (I − Φ)Q T x}= {x | Q T x ∈ Φ ( Λ † Q T (K − b) ) ⊕ (I − Φ)R N2 }= {x ∈ QΛ † Q T (K − b) ⊕ Q(I − Φ)R N2 }= (I ⊗A) † (K − b) ⊕ N(I ⊗A)(211)where we used the facts: linear function Q T x in x on R N2 is a bijection,and ΦΛ † = Λ † .vec X = (I ⊗A) † vec(S N + − B) ⊕ N(I ⊗A) (212)In words, set vec X is the vector sum of the translated PSD cone(linearly mapped onto the rowspace of I ⊗ A (E)) and the nullspace ofI ⊗ A (synthesis of fact fromA.6.3 andA.7.3.0.1). Should I ⊗A have nonullspace, then vec X =(I ⊗A) −1 vec(S N + − B) which is the expected result.2.9.2 Positive semidefinite cone boundaryFor any symmetric positive semidefinite matrix A of rank ρ , there mustexist a rank ρ matrix Y such that A be expressible as an outer productin Y ; [331,6.3]A = Y Y T ∈ S M + , rankA=ρ , Y ∈ R M×ρ (213)Then the boundary of the positive semidefinite cone may be expressed∂S M + = { A ∈ S M + | rankA