Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.6. EPIGRAPH, SUBLEVEL SET 239
3.6.2 semidefinite program via Schur
Schur complement (1484) 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 [180] (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
(558)
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
(559)
Next we use Schur complement [271,6.4.3] [243] and matrix vectorization
(2.2):
minimize
X∈ S M , t∈R
subject to
t
[
tI vec(A − X)
vec(A − X) T 1
S T XS ≽ 0
]
≽ 0
(560)
This semidefinite program (4) is an epigraph form in disguise, equivalent
to (558); it demonstrates how a quadratic objective or constraint can be
converted to a semidefinite constraint.
Were problem (558) instead equivalently expressed without the square
minimize ‖A − X‖ F
X∈ S M
subject to S T XS ≽ 0
(561)
then we get a subtle variation: