v2009.01.01 - Convex Optimization

4.1. CONIC PROBLEM 249
4.1.1.2 Reduced-rank solution
A simple rank reduction algorithm, for construction of a primal optimal
solution X ⋆ to (584P) satisfying an upper bound on rank governed by
Proposition 2.9.3.0.1, is presented in4.3. That proposition asserts existence
of feasible solutions with an upper bound on their rank; [25,II.13.1]
specifically, it asserts an extreme point (2.6.0.0.1) of the primal feasible
set A ∩ S n + satisfies upper bound
⌊√ ⌋ 8m + 1 − 1
rankX ≤
2
where, given A∈ R m×n(n+1)/2 and b∈ R m
(245)
A ∆ = {X ∈ S n | A svec X = b} (587)
is the affine subset from primal problem (584P).
4.1.1.3 Coexistence of low- and high-rank solutions; analogy
That low-rank and high-rank optimal solutions {X ⋆ } of (584P) coexist may
be grasped with the following analogy: We compare a proper polyhedral cone
S 3 + in R 3 (illustrated in Figure 70) to the positive semidefinite cone S 3 + in
isometrically isomorphic R 6 , difficult to visualize. The analogy is good:
int S 3 + is constituted by rank-3 matrices
int S 3 + has three dimensions
boundary ∂S 3 + contains rank-0, rank-1, and rank-2 matrices
boundary ∂S 3 + contains 0-, 1-, and 2-dimensional faces
the only rank-0 matrix resides in the vertex at the origin
Rank-1 matrices are in one-to-one correspondence with extreme
directions of S 3 + and S 3 + . The set of all rank-1 symmetric matrices in
this dimension
{
G ∈ S
3
+ | rankG=1 } (588)
is not a connected set.