v2009.01.01 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 433
where m=N(N ∆ −1)/2, we may rewrite (1054b) as an equivalent quadratic
program; a convex optimization problem [53,4] in terms of the
halfspace-description of K M+ :
minimize (σ − Πd) T (σ − Πd)
σ
subject to Y † σ ≽ 0
(1057)
This quadratic program can be converted to a semidefinite program via
Schur-form (3.1.7.2); we get the equivalent problem
minimize
t∈R , σ
subject to
t
[
tI σ − Πd
(σ − Πd) T 1
]
≽ 0
(1058)
Y † σ ≽ 0
5.13.2.3 Convergence
InE.10 we discuss convergence of alternating projection on intersecting
convex sets in a Euclidean vector space; convergence to a point in their
intersection. Here the situation is different for two reasons:
Firstly, sets of positive semidefinite matrices having an upper bound on
rank are generally not convex. Yet in7.1.4.0.1 we prove (1054a) is equivalent
to a projection of nonincreasingly ordered eigenvalues on a subset of the
nonnegative orthant:
minimize ‖−VN T(D − O)V N ‖ F minimize ‖Υ − Λ‖
D
F
subject to rankVN TDV Υ [ ]
N ≤ 3 ≡
R
3
+
D ∈ EDM N
subject to δ(Υ) ∈
0
(1059)
where −VN TDV ∆
N =UΥU T ∈ S N−1 and −VN TOV ∆
N =QΛQ T ∈ S N−1 are
ordered diagonalizations (A.5). It so happens: optimal orthogonal U ⋆
always equals Q given. Linear operator T(A) = U ⋆T AU ⋆ , acting on square
matrix A , is a bijective isometry because the Frobenius norm is orthogonally
invariant (41). This isometric isomorphism T thus maps a nonconvex
problem to a convex one that preserves distance.