v2009.01.01 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 533
Our plan is to instead divide problem (1309) into two and then iterate
their solution:
∥
minimize ∥δ(Υ) − π ( δ(R T ΛR) )∥ ∥ 2
Υ ⎡ ⎤
subject to δ(Υ) ∈ ⎣ Rρ+1 +
(a)
0 ⎦ ∩ ∂H
R −
(1311)
minimize ‖R Υ ⋆ R T − Λ‖ 2 F
R
subject to δ(QR Υ ⋆ R T Q T ) = 0
R −1 = R T
(b)
Proof. We justify disappearance of the hollowness constraint in
convex optimization problem (1311a): From the arguments in7.1.3
with regard ⎡ to⎤π the permutation operator, cone membership constraint
δ(Υ) ∈⎣ Rρ+1 +
0 ⎦∩ ∂H from (1311a) is equivalent to
R −
⎡ ⎤
δ(Υ) ∈ ⎣ Rρ+1 +
0 ⎦ ∩ ∂H ∩ K M (1312)
R −
where K M is the monotone cone (2.13.9.4.3). Membership of δ(Υ) to the
polyhedral cone of majorization (Theorem A.1.2.0.1)
K ∗ λδ = ∂H ∩ K ∗ M+ (1329)
where K ∗ M+ is the dual monotone nonnegative cone (2.13.9.4.2), is a
condition (in absence of a hollowness [ constraint) ] that would insure existence
0 1
T
of a symmetric hollow matrix . Curiously, intersection of
1 −D
⎡ ⎤
this feasible superset ⎣ Rρ+1 +
0 ⎦∩ ∂H ∩ K M from (1312) with the cone of
majorization K ∗ λδ
R −
is a benign operation; id est,
∂H ∩ K ∗ M+ ∩ K M = ∂H ∩ K M (1313)