v2009.01.01 - Convex Optimization

520 CHAPTER 7. PROXIMITY PROBLEMS
7.2.2.2 Applying trace rank-heuristic to Problem 2
Substituting rank envelope for rank function in Problem 2, for D ∈ EDM N
(confer (947))
cenv rank(−V T NDV N ) = cenv rank(−V DV ) ∝ − tr(V DV ) (1271)
and for desired affine dimension ρ ≤ N −1 and nonnegative H [sic] we get
a convex optimization problem
minimize ‖ ◦√ D − H‖ 2 F
D
subject to − tr(V DV ) ≤ κρ
(1272)
D ∈ EDM N
where κ ∈ R + is a constant determined by cut-and-try. The equivalent
semidefinite program makes κ variable: for nonnegative and symmetric H
minimize
D , Y , κ
subject to
κρ + 2 tr(V Y V )
[ ]
dij y ij
≽ 0 ,
y ij
h 2 ij
N ≥ j > i = 1... N −1
(1273)
− tr(V DV ) ≤ κρ
Y ∈ S N h
D ∈ EDM N
which is the same as (1264), the problem with no explicit constraint on affine
dimension. As the present problem is stated, the desired affine dimension ρ
yields to the variable scale factor κ ; ρ is effectively ignored.
Yet this result is an illuminant for problem (1264) and it equivalents
(all the way back to (1257)): When the given measurement matrix H
is nonnegative and symmetric, finding the closest EDM D as in problem
(1257), (1260), or (1264) implicitly entails minimization of affine dimension
(confer5.8.4,5.14.4). Those non−rank-constrained problems are each
inherently equivalent to cenv(rank)-minimization problem (1273), in other
words, and their optimal solutions are unique because of the strictly convex
objective function in (1257).
7.2.2.3 Rank-heuristic insight
Minimization of affine dimension by use of this trace rank-heuristic (1271)
tends to find the list configuration of least energy; rather, it tends to optimize