v2007.11.26 - Convex Optimization

478 CHAPTER 7. PROXIMITY PROBLEMS7.2.2.2 Applying trace rank-heuristic to Problem 2Substituting rank envelope for rank function in Problem 2, for D ∈ EDM N(confer (876))cenv rank(−V T NDV N ) = cenv rank(−V DV ) ∝ − tr(V DV ) (1198)and for desired affine dimension ρ ≤ N −1 and nonnegative H [sic] we geta convex optimization problemminimize ‖ ◦√ D − H‖ 2 FDsubject to − tr(V DV ) ≤ κ ρ(1199)D ∈ EDM Nwhere κ ∈ R + is a constant determined by cut-and-try. The equivalentsemidefinite program makes κ variable: for nonnegative and symmetric HminimizeD , Y , κsubject toκ ρ + 2 tr(V Y V )[ ]dij y ij≽ 0 ,y ijh 2 ijj > i = 1... N −1(1200)− tr(V DV ) ≤ κ ρY ∈ S N hD ∈ EDM Nwhich is the same as (1191), the problem with no explicit constraint on affinedimension. As the present problem is stated, the desired affine dimension ρyields to the variable scale factor κ ; ρ is effectively ignored.Yet this result is an illuminant for problem (1191) and it equivalents(all the way back to (1184)): When the given measurement matrix His nonnegative and symmetric, finding the closest EDM D as in problem(1184), (1187), or (1191) implicitly entails minimization of affine dimension(confer5.8.4,5.14.4). Those non−rank-constrained problems are eachinherently equivalent to cenv(rank)-minimization problem (1200), in otherwords, and their optimal solutions are unique because of the strictly convexobjective function in (1184).7.2.2.3 Rank-heuristic insightMinimization of affine dimension by use of this trace rank-heuristic (1198)tends to find the list configuration of least energy; rather, it tends to optimize