v2007.09.13 - Convex Optimization

118 CHAPTER 2. CONVEX GEOMETRYThe 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} (228)ρ=kis convex for each k by Theorem 2.9.2.6.3.2.9.2.9 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.2.9.2.9.1 Exercise. Difference A − B .What about the difference of matrices A,B belonging to the positivesemidefinite cone? Show:The difference of any two points on the boundary belongs to theboundary or exterior.The difference A−B , where A belongs to the boundary while B isinterior, 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:2.9.3.0.1 Proposition. (Barvinok) Affine intersection with PSD cone.[20,II.13] [21,2.2] Consider finding a matrix X ∈ S N satisfyingX ≽ 0 , 〈A j , X〉 = b j , j =1... m (229)