v2009.01.01 - Convex Optimization

530 CHAPTER 7. PROXIMITY PROBLEMS
7.3.2 Minimization of affine dimension in Problem 3
When the desired affine dimension ρ is diminished, Problem 3 (1286) is
difficult to solve [159,3] because the feasible set in R N(N−1)/2 loses convexity.
By substituting rank envelope (1271) into Problem 3, then for any given H
we get a convex problem
minimize ‖D − H‖ 2 F
D
subject to − tr(V DV ) ≤ κρ
(1302)
D ∈ EDM N
where κ ∈ R + is a constant determined by cut-and-try. Given κ , problem
(1302) is a convex optimization having unique solution in any desired
affine dimension ρ ; an approximation to Euclidean projection on that
nonconvex subset of the EDM cone containing EDMs with corresponding
affine dimension no greater than ρ .
The SDP equivalent to (1302) does not move κ into the variables as on
page 520: for nonnegative symmetric input H and distance-square squared
variable ∂ as in (1290),
minimize − tr(V (∂ − 2H ◦D)V )
∂ , D
[ ]
∂ij d ij
subject to
≽ 0 , N ≥ j > i = 1... N −1
d ij 1
− tr(V DV ) ≤ κρ
D ∈ EDM N
(1303)
∂ ∈ S N h
That means we will not see equivalence of this cenv(rank)-minimization
problem to the non−rank-constrained problems (1289) and (1291) like we
saw for its counterpart (1273) in Problem 2.
Another approach to affine dimension minimization is to project instead
on the polar EDM cone; discussed in6.8.1.5.