v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 267G ≽ X T X (749)which is a convex relaxation of the desired equality constraint[ ] [ ]I X IX T =G X T [ I X ] (750)The rank constraint insures this equality holds thus restricting solution to R n .convex equivalent problem statementProblem statement (639) is nonconvex because of the rank constraint.We do not eliminate or ignore the rank constraint; rather, we find a convexway to enforce it: for 0 < n < Nminimize 〈Z , W 〉G∈S N , X∈R n×Nsubject to d ij ≤ 〈G , (e i − e j )(e i − e j ) T 〉 ≤ d ij ∀(i,j)∈ I〈G , e i e T i 〉 = ‖ˇx i ‖ 2 , i = N − m + 1... N〈G , (e i e T j + e j e T i )/2〉 = ˇx T i ˇx j , i < j , ∀(i,j)∈{N − m + 1... N}X(:, N − m + 1:N) = [ ˇx N−m+1 · · · ˇx N ][ ] I XZ =X T≽ 0 (640)GEach linear equality constraint in G∈ S N represents a hyperplane inisometrically isomorphic Euclidean vector space R N(N+1)/2 , while each linearinequality pair represents a convex Euclidean body known as slab (anintersection of two parallel but opposing halfspaces, Figure 9). In this convexoptimization problem (640), a semidefinite program, we substitute a vectorinner-product objective function for trace from nonconvex problem (639);〈Z , I 〉 = trZ ← 〈Z , W 〉 (641)a generalization of the known trace heuristic [90] for minimizing convexenvelope of rank, where W ∈ S N+n+ is constant with respect to (640).Matrix W is normal to a hyperplane minimized over a convex feasible setspecified by the constraints in (640). Matrix W is chosen so −W points inthe direction of a feasible rank-n Gram matrix. Thus the purpose of vectorinner-product objective (641) is to locate a feasible rank-n Gram matrixthat is presumed existent on the boundary of positive semidefinite cone S N + .