v2007.09.17 - Convex Optimization

100 CHAPTER 2. CONVEX GEOMETRY2.9.0.1.1 Example. Equality constraints in semidefinite program (546).Employing properties of partial ordering (2.7.2.2) for the pointed closedconvex positive semidefinite cone, it is easy to show, given A + S = CS ≽ 0 ⇔ A ≼ C (163)2.9.1 Positive semidefinite cone is convexThe set of all positive semidefinite matrices forms a convex cone in theambient space of symmetric matrices because any pair satisfies definition(144); [150,7.1] videlicet, for all ζ 1 , ζ 2 ≥ 0 and each and every A 1 , A 2 ∈ S Mζ 1 A 1 + ζ 2 A 2 ≽ 0 ⇐ A 1 ≽ 0, A 2 ≽ 0 (164)a fact easily verified by the definitive test for positive semidefiniteness of asymmetric matrix (A):A ≽ 0 ⇔ x T Ax ≥ 0 for each and every ‖x‖ = 1 (165)id 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 (166)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 semi-infinite inexpanse in R 3 , having boundary illustrated in Figure 31. 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 } (167)is convex because it has Schur-form; (A.4)X − xx T ≽ 0 ⇔ f(X , x) ∆ =[ X xx T 1]≽ 0 (168)