v2009.01.01 - Convex Optimization

124 CHAPTER 2. CONVEX GEOMETRY
The composite sequence, the cone interior in union with each successive
constituent, remains convex at each step; id est, for 0≤k ≤M
M⋃
{Y ∈ S M + | rankY = ρ} (241)
ρ=k
is convex for each k by Theorem 2.9.2.6.3.
2.9.2.9 Peeling constituents
Proceeding the other way: To peel constituents off the complete positive
semidefinite cone boundary, one starts by removing the origin; the only
rank-0 positive semidefinite matrix. What remains is convex. Next, the
extreme directions are removed because they constitute all the rank-1 positive
semidefinite matrices. What remains is again convex, and so on. Proceeding
in this manner eventually removes the entire boundary leaving, at last, the
convex interior of the PSD cone; all the positive definite matrices.
2.9.2.9.1 Exercise. Difference A − B .
What about the difference of matrices A,B belonging to the positive
semidefinite cone? Show:
The difference of any two points on the boundary belongs to the
boundary or exterior.
The difference A−B , where A belongs to the boundary while B is
interior, belongs to the exterior.
2.9.3 Barvinok’s proposition
Barvinok posits existence and quantifies an upper bound on rank of a positive
semidefinite matrix belonging to the intersection of the PSD cone with an
affine subset:
2.9.3.0.1 Proposition. (Barvinok) Affine intersection with PSD cone.
[25,II.13] [23,2.2] Consider finding a matrix X ∈ S N satisfying
X ≽ 0 , 〈A j , X〉 = b j , j =1... m (242)