v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 117CXFigure 44: Convex set C ={X ∈ S × x∈ R | X ≽ xx T } drawn truncated.xid est, for A 1 , A 2 ≽ 0 and each and every ζ 1 , ζ 2 ≥ 0ζ 1 x T A 1 x + ζ 2 x T A 2 x ≥ 0 for each and every normalized x ∈ R M (197)The convex cone S M + is more easily visualized in the isomorphic vectorspace R M(M+1)/2 whose dimension is the number of free variables in asymmetric M ×M matrix. When M = 2 the PSD cone is semiinfinite inexpanse in R 3 , having boundary illustrated in Figure 43. When M = 3 thePSD cone is six-dimensional, and so on.2.9.1.0.1 Example. Sets from maps of positive semidefinite cone.The setC = {X ∈ S n × x∈ R n | X ≽ xx T } (198)is convex because it has Schur-form; (A.4)X − xx T ≽ 0 ⇔f(X , x) [ X xx T 1]≽ 0 (199)e.g., Figure 44. Set C is the inverse image (2.1.9.0.1) of S n+1+ under affinemapping f . The set {X ∈ S n × x∈ R n | X ≼ xx T } is not convex, in contrast,having no Schur-form. Yet for fixed x = x p , the set{X ∈ S n | X ≼ x p x T p } (200)is simply the negative semidefinite cone shifted to x p x T p .