v2010.10.26 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 583substituting rank envelope (1368) into Problem 3, then for any given H weget a convex problemminimize ‖D − H‖ 2 FDsubject to − tr(V DV ) ≤ κρ(1399)D ∈ EDM Nwhere κ ∈ R + is a constant determined by cut-and-try. Given κ , problem(1399) 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 (1399) does not move κ into the variables as onpage 573: for nonnegative symmetric input H and distance-square squaredvariable ∂ as in (1387),minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , N ≥ j > i = 1... N −1d ij 1− tr(V DV ) ≤ κρD ∈ EDM N(1400)∂ ∈ S N hThat means we will not see equivalence of this cenv(rank)-minimizationproblem to the non−rank-constrained problems (1386) and (1388) like wesaw for its counterpart (1370) in Problem 2.Another approach to affine dimension minimization is to project insteadon the polar EDM cone; discussed in6.8.1.5.7.3.3 Constrained affine dimension, Problem 3When one desires affine dimension diminished further below what can beachieved via cenv(rank)-minimization as in (1400), spectral projection can beconsidered a natural means in light of its successful application to projectionon a rank ρ subset of the positive semidefinite cone in7.1.4.Yet it is wrong here to zero eigenvalues of −V DV or −V GV or a variantto reduce affine dimension, because that particular method comes from