v2010.10.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 325which is a convex relaxation of the desired equality constraint[ ] [ ] I X I [I X ]X T =G X T (938)The rank constraint insures this equality holds, by Theorem A.4.0.1.3, thusrestricting solution to R n . Assuming full-rank solution (list) Xconvex equivalent problem statementrankZ = rankG = rankX (751)Problem statement (750) is nonconvex because of the rank constraint. Wedo not eliminate or ignore the rank constraint; rather, we find a convex wayto 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 (752)GConvex function trZ is a well-known heuristic whose sole purpose is torepresent convex envelope of rankZ . (7.2.2.1) In this convex optimizationproblem (752), a semidefinite program, we substitute a vector inner-productobjective function for trace;trZ = 〈Z , I 〉 ← 〈Z , W 〉 (753)a generalization of the trace heuristic for minimizing convex envelope of rank,where W ∈ S N+n+ is constant with respect to (752). Matrix W is normal toa hyperplane in S N+n minimized over a convex feasible set specified by theconstraints in (752). Matrix W is chosen so −W points in direction of rank-nfeasible solutions G . For properly chosen W , problem (752) becomes anequivalent to (750). Thus the purpose of vector inner-product objective (753)is to locate a rank-n feasible Gram matrix assumed existent on the boundaryof positive semidefinite cone S N + , as explained beginning in4.4.1; how tochoose direction vector W is explained there and in what follows: