v2010.10.26 - Convex Optimization

138 CHAPTER 2. CONVEX GEOMETRYhaving rank ρ successively 1 lower than M , appends a nonconvexconstituent to the cone boundary; but only in their union is the boundarycomplete: (confer2.9.2)∂S M + =⋃{Y ∈ S M + | rankY = ρ} (267)M−1ρ=0The composite sequence, the cone interior in union with each successiveconstituent, remains convex at each step; id est, for 0≤k ≤MM⋃{Y ∈ S M + | rankY = ρ} (268)ρ=kis convex for each k by Theorem 2.9.2.9.3.2.9.2.12 Peeling constituentsProceeding the other way: To peel constituents off the complete positivesemidefinite cone boundary, one starts by removing the origin; the onlyrank-0 positive semidefinite matrix. What remains is convex. Next, theextreme directions are removed because they constitute all the rank-1 positivesemidefinite matrices. What remains is again convex, and so on. Proceedingin this manner eventually removes the entire boundary leaving, at last, theconvex interior of the PSD cone; all the positive definite matrices.2.9.2.12.1 Exercise. Difference A − B .What about a difference of matrices A,B belonging to the PSD cone? Show:Difference of any two points on the boundary belongs to the boundaryor exterior.Difference A −B , where A belongs to the boundary while B is interior,belongs to the exterior.2.9.3 Barvinok’s propositionBarvinok posits existence and quantifies an upper bound on rank of a positivesemidefinite matrix belonging to the intersection of the PSD cone with anaffine subset: