v2009.01.01 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 511
where
R ∆ = Q T U ∈ R N−1×N−1 (1242)
in U on the set of orthogonal matrices is a linear bijection. We propose
solving (1241) by instead solving the problem sequence:
minimize ‖Υ − R T ΛR‖ 2 F
Υ
subject to rank Υ ≤ ρ
Υ ≽ 0
minimize ‖Υ ⋆ − R T ΛR‖ 2 F
R
subject to R −1 = R T
(a)
(b)
(1243)
Problem (1243a) is equivalent to: (1) orthogonal projection of R T ΛR
on an N − 1-dimensional subspace of isometrically isomorphic R N(N−1)/2
containing δ(Υ)∈ R N−1
+ , (2) nonincreasingly ordering the
[
result, (3) unique
R
ρ
]
+
minimum-distance projection of the ordered result on . (E.9.5)
0
Projection on that N−1-dimensional subspace amounts to zeroing R T ΛR at
all entries off the main diagonal; thus, the equivalent sequence leading with
a spectral projection:
minimize ‖δ(Υ) − π ( δ(R T ΛR) ) ‖ 2
Υ [ R
ρ
]
+
subject to δ(Υ) ∈
0
minimize ‖Υ ⋆ − R T ΛR‖ 2 F
R
subject to R −1 = R T
(a)
(b)
(1244)
Because any permutation matrix is an orthogonal matrix, it is always
feasible that δ(R T ΛR)∈ R N−1 be arranged in nonincreasing order; hence, the
permutation operator π . Unique minimum-distance projection of vector
π ( δ(R T ΛR) ) [ R
ρ
]
+
on the ρ-dimensional subset of nonnegative orthant
0