v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 109
is a diagonal projection matrix whose entries are either 1 or 0 (E.3). We
have the complementary sum
ΦQ T x + (I − Φ)Q T x = Q T x (192)
So, adding (I −Φ)Q T x to both sides of the membership within (190) admits
vec X = {x∈ R N 2 | 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 N 2 }
= {x ∈ QΛ † Q T (K − b) ⊕ Q(I − Φ)R N 2 }
= (I ⊗A) † (K − b) ⊕ N(I ⊗A)
(193)
where we used the facts: linear function Q T x in x on R N 2 is a bijection,
and ΦΛ † = Λ † .
vec X = (I ⊗A) † vec(S N + − B) ⊕ N(I ⊗A) (194)
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 of
I ⊗ A (synthesis of fact fromA.6.3 andA.7.3.0.1). Should I ⊗A have no
nullspace, then vec X =(I ⊗A) −1 vec(S N + − B) which is the expected result.
2.9.2 Positive semidefinite cone boundary
For any symmetric positive semidefinite matrix A of rank ρ , there must
exist a rank ρ matrix Y such that A be expressible as an outer product
in Y ; [287,6.3]
A = Y Y T ∈ S M + , rankA=ρ , Y ∈ R M×ρ (195)
Then the boundary of the positive semidefinite cone may be expressed
∂S M + = { A ∈ S M + | rankA