v2009.01.01 - Convex Optimization

108 CHAPTER 2. CONVEX GEOMETRY
2.9.1.0.2 Example. Inverse image of positive semidefinite cone.
Now consider finding the set of all matrices X ∈ S N satisfying
given A,B∈ S N . Define the set
AX + B ≽ 0 (183)
X ∆ = {X | AX + B ≽ 0} ⊆ S N (184)
which is the inverse image of the positive semidefinite cone under affine
transformation g(X) =AX+B ∆ . Set X must therefore be convex by
Theorem 2.1.9.0.1.
Yet we would like a less amorphous characterization of this set, so instead
we consider its vectorization (30) which is easier to visualize:
where
vec g(X) = vec(AX) + vec B = (I ⊗A) vec X + vec B (185)
I ⊗A ∆ = QΛQ T ∈ S N 2 (186)
is a block-diagonal matrix formed by Kronecker product (A.1.1 no.25,
D.1.2.1). Assign
x ∆ = vec X ∈ R N 2
b ∆ = vec B ∈ R N 2 (187)
then make the equivalent problem: Find
where
vec X = {x∈ R N 2 | (I ⊗A)x + b ∈ K} (188)
K ∆ = vec S N + (189)
is a proper cone isometrically isomorphic with the positive semidefinite cone
in the subspace of symmetric matrices; the vectorization of every element of
S N + . Utilizing the diagonalization (186),
vec X = {x | ΛQ T x ∈ Q T (K − b)}
= {x | ΦQ T x ∈ Λ † Q T (K − b)} ⊆ R N 2 (190)
where †
denotes matrix pseudoinverse (E) and
Φ ∆ = Λ † Λ (191)