v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 101CXFigure 32: Convex set C ={X ∈ S × x∈ R | X ≽ xx T } drawn truncated.xe.g., Figure 32. Set C is the inverse image (2.1.9.0.1) of S n+1+ under theaffine mapping f . The set {X ∈ S n × x∈ R n | X ≼ xx T } is not convex, incontrast, having no Schur form. Yet for fixed x = x p , the set{X ∈ S n | X ≼ x p x T p } (169)is simply the negative semidefinite cone shifted to x p x T p .2.9.1.0.2 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 (170)X ∆ = {X | AX + B ≽ 0} ⊆ S N (171)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 (30) which is easier to visualize:vec g(X) = vec(AX) + vec B = (I ⊗A) vec X + vec B (172)