v2009.01.01 - Convex Optimization

250 CHAPTER 4. SEMIDEFINITE PROGRAMMING
C
0
P
Γ 1
Γ 2
S+
3
A=∂H
Figure 70: Visualizing positive semidefinite cone in high dimension: Proper
polyhedral cone S 3 + ⊂ R 3 representing positive semidefinite cone S 3 + ⊂ S 3 ;
analogizing its intersection with hyperplane S 3 + ∩ ∂H . Number of facets
is arbitrary (analogy is not inspired by eigen decomposition). The rank-0
positive semidefinite matrix corresponds to the origin in R 3 , rank-1 positive
semidefinite matrices correspond to the edges of the polyhedral cone, rank-2
to the facet relative interiors, and rank-3 to the polyhedral cone interior.
Vertices Γ 1 and Γ 2 are extreme points of polyhedron P =∂H ∩ S 3 + , and
extreme directions of S 3 + . A given vector C is normal to another hyperplane
(not illustrated but independent w.r.t ∂H) containing line segment Γ 1 Γ 2
minimizing real linear function 〈C, X〉 on P . (confer Figure 22, Figure 26)