v2009.01.01 - Convex Optimization

4.1. CONIC PROBLEM 251
In any SDP feasibility problem, an SDP feasible solution with the lowest
rank must be an extreme point of the feasible set. Thus, there must exist
an SDP objective function such that this lowest-rank feasible solution
uniquely optimizes it. −Ye, 2006 [206,2.4] [207]
Rank of a sum of members F +G in Lemma 2.9.2.6.1 and location of
a difference F −G in2.9.2.9.1 similarly hold for S 3 + and S 3 + .
Euclidean distance from any particular rank-3 positive semidefinite
matrix (in the cone interior) to the closest rank-2 positive semidefinite
matrix (on the boundary) is generally less than the distance to the
closest rank-1 positive semidefinite matrix. (7.1.2)
distance from any point in ∂S 3 + to int S 3 + is infinitesimal (2.1.7.1.1)
distance from any point in ∂S 3 + to int S 3 + is infinitesimal
faces of S 3 + correspond to faces of S 3 + (confer Table 2.9.2.3.1)
k dim F(S+) 3 dim F(S 3 +) dim F(S 3 + ∋ rank-k matrix)
0 0 0 0
boundary 1 1 1 1
2 2 3 3
interior 3 3 6 6
Integer k indexes k-dimensional faces F of S 3 + . Positive semidefinite
cone S 3 + has four kinds of faces, including cone itself (k = 3,
boundary + interior), whose dimensions in isometrically isomorphic
R 6 are listed under dim F(S 3 +). Smallest face F(S 3 + ∋ rank-k matrix)
that contains a rank-k positive semidefinite matrix has dimension
k(k + 1)/2 by (204).
For A equal to intersection of m hyperplanes having independent
normals, and for X ∈ S 3 + ∩ A , we have rankX ≤ m ; the analogue
to (245).
Proof. With reference to Figure 70: Assume one (m = 1) hyperplane
A = ∂H intersects the polyhedral cone. Every intersecting plane
contains at least one matrix having rank less than or equal to 1 ; id est,
from all X ∈ ∂H ∩ S 3 + there exists an X such that rankX ≤ 1. Rank 1
is therefore an upper bound in this case.