v2007.09.13 - Convex Optimization

102 CHAPTER 2. CONVEX GEOMETRYwhereI ⊗A ∆ = QΛQ T ∈ S N 2 (173)is a block-diagonal matrix formed by Kronecker product (A.1 no.21,D.1.2.1). Assignx ∆ = vec X ∈ R N 2b ∆ = vec B ∈ R N 2 (174)then make the equivalent problem: Findwherevec X = {x∈ R N 2 | (I ⊗A)x + b ∈ K} (175)K ∆ = vec S N + (176)is a proper cone isometrically isomorphic with the positive semidefinite conein the subspace of symmetric matrices; the vectorization of every element ofS N + . Utilizing the diagonalization (173),vec X = {x | ΛQ T x ∈ Q T (K − b)}= {x | ΦQ T x ∈ Λ † Q T (K − b)} ⊆ R N 2 (177)where †denotes matrix pseudoinverse (E) andΦ ∆ = Λ † Λ (178)is 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 (179)So, adding (I −Φ)Q T x to both sides of the membership within (177) admitsvec 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)(180)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) (181)