v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 219
Since optimal (x ⋆ , U ⋆ ) from problem (511) is feasible to problem (510), and
because their objectives are equivalent for projectors by (507), then converged
(x ⋆ , U ⋆ ) must also be optimal to (510) in the limit. Because problem (510)
is convex, this represents a globally optimal solution.
3.1.7.2 Semidefinite program via Schur
Schur complement (1410) can be used to convert a projection problem
to an optimization problem in epigraph form. Suppose, for example,
we are presented with the constrained projection problem studied by
Hayden & Wells in [158] (who provide analytical solution): Given A∈ R M×M
and some full-rank matrix S ∈ R M×L with L < M
minimize ‖A − X‖ 2
X∈ S M
F
subject to S T XS ≽ 0
(517)
Variable X is constrained to be positive semidefinite, but only on a subspace
determined by S . First we write the epigraph form:
minimize t
X∈ S M , t∈R
subject to ‖A − X‖ 2 F ≤ t
S T XS ≽ 0
(518)
Next we use Schur complement [239,6.4.3] [214] and matrix vectorization
(2.2):
minimize t
X∈ S M , t∈R
[
]
tI vec(A − X)
subject to
vec(A − X) T ≽ 0 (519)
1
S T XS ≽ 0
This semidefinite program (4) is an epigraph form in disguise, equivalent
to (517); it demonstrates how a quadratic objective or constraint can be
converted to a semidefinite constraint.
Were problem (517) instead equivalently expressed without the square
minimize ‖A − X‖ F
X∈ S M
subject to S T XS ≽ 0
(520)