v2010.10.26 - Convex Optimization

118 CHAPTER 2. CONVEX GEOMETRY2.9.1.0.2 Example. Inverse image of positive semidefinite cone.Now consider finding the set of all matrices X ∈ S N satisfyinggiven A,B∈ S N . Define the setAX + B ≽ 0 (201)X {X | AX + B ≽ 0} ⊆ S N (202)which is the inverse image of the positive semidefinite cone under affinetransformation g(X)AX+B . Set X must therefore be convex byTheorem 2.1.9.0.1.Yet we would like a less amorphous characterization of this set, so insteadwe consider its vectorization (37) which is easier to visualize:wherevec g(X) = vec(AX) + vec B = (I ⊗A) vec X + vec B (203)I ⊗A QΛQ T ∈ S N2 (204)is block-diagonal formed by Kronecker product (A.1.1 no.31,D.1.2.1).Assignx vec X ∈ R N2(205)N2b vec B ∈ Rthen make the equivalent problem: Findwherevec X = {x∈ R N2 | (I ⊗A)x + b ∈ K} (206)K vec S N + (207)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 (204),vec X = {x | ΛQ T x ∈ Q T (K − b)}= {x | ΦQ T x ∈ Λ † Q T N2(208)(K − b)} ⊆ Rwhere †denotes matrix pseudoinverse (E) andΦ Λ † Λ (209)