v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 107
C
X
Figure 38: Convex set C ={X ∈ S × x∈ R | X ≽ xx T } drawn truncated.
x
id est, for A 1 , A 2 ≽ 0 and each and every ζ 1 , ζ 2 ≥ 0
ζ 1 x T A 1 x + ζ 2 x T A 2 x ≥ 0 for each and every normalized x ∈ R M (179)
The convex cone S M + is more easily visualized in the isomorphic vector
space R M(M+1)/2 whose dimension is the number of free variables in a
symmetric M ×M matrix. When M = 2 the PSD cone is semi-infinite in
expanse in R 3 , having boundary illustrated in Figure 37. When M = 3 the
PSD cone is six-dimensional, and so on.
2.9.1.0.1 Example. Sets from maps of positive semidefinite cone.
The set
C = {X ∈ S n × x∈ R n | X ≽ xx T } (180)
is convex because it has Schur-form; (A.4)
X − xx T ≽ 0 ⇔ f(X , x) ∆ =
[ X x
x T 1
]
≽ 0 (181)
e.g., Figure 38. Set C is the inverse image (2.1.9.0.1) of S n+1
+ under the
affine mapping 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 } (182)
is simply the negative semidefinite cone shifted to x p x T p .