v2007.09.17 - Convex Optimization

474 CHAPTER 7. PROXIMITY PROBLEMS7.3.2 Minimization of affine dimension in Problem 3When the desired affine dimension ρ is diminished, Problem 3 (1188) isdifficult to solve [134,3] because the feasible set in R N(N−1)/2 loses convexity.By substituting rank envelope (1172) into Problem 3, then for any given Hwe get a convex problemminimize ‖D − H‖ 2 FDsubject to − tr(V DV ) ≤ κ ρ(1204)D ∈ EDM Nwhere κ ∈ R + is a constant determined by cut-and-try. Given κ , problem(1204) is a convex optimization having unique solution in any desiredaffine dimension ρ ; an approximation to Euclidean projection on thatnonconvex subset of the EDM cone containing EDMs with correspondingaffine dimension no greater than ρ .The SDP equivalent to (1204) does not move κ into the variables as onpage 464: for nonnegative symmetric input H and distance-square squaredvariable ∂ as in (1192),minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1− tr(V DV ) ≤ κ ρD ∈ EDM N(1205)∂ ∈ S N hThat means we will not see equivalence of this cenv(rank)-minimizationproblem to the non−rank-constrained problems (1191) and (1193) like wesaw for its counterpart (1174) in Problem 2.Another approach to affine dimension minimization is to project insteadon the polar EDM cone; discussed in6.8.1.5.