v2009.01.01 - Convex Optimization

532 CHAPTER 7. PROXIMITY PROBLEMS
dimension not in excess of that ρ desired; id est, spectral projection on
⎡ ⎤
⎣ Rρ+1 +
0 ⎦ ∩ ∂H ⊂ R N+1 (1306)
R −
where
∂H = {λ ∈ R N+1 | 1 T λ = 0} (1017)
is a hyperplane through the origin. This pointed polyhedral cone (1306), to
which membership subsumes the rank constraint, has empty interior.
Given desired affine dimension 0 ≤ ρ ≤ N −1 and diagonalization (A.5)
of unknown EDM D
[
0 1
T
1 −D
]
∆
= UΥU T ∈ S N+1
h
(1307)
and given symmetric H in diagonalization
[ ] 0 1
T
∆
= QΛQ T ∈ S N+1 (1308)
1 −H
having eigenvalues arranged in nonincreasing order, then by (1030) problem
(1304) is equivalent to
∥
minimize ∥δ(Υ) − π ( δ(R T ΛR) )∥ ∥ 2
Υ , R
⎡ ⎤
subject to δ(Υ) ∈ ⎣ Rρ+1 +
0 ⎦ ∩ ∂H
(1309)
R −
δ(QRΥR T Q T ) = 0
R −1 = R T
where π is the permutation operator from7.1.3 arranging its vector
argument in nonincreasing order, 7.18 where
R ∆ = Q T U ∈ R N+1×N+1 (1310)
in U on the set of orthogonal matrices is a bijection, and where ∂H insures
one negative eigenvalue. Hollowness constraint δ(QRΥR T Q T ) = 0 makes
problem (1309) difficult by making the two variables dependent.
7.18 Recall, any permutation matrix is an orthogonal matrix.