v2009.01.01 - Convex Optimization

220 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
then we get a subtle variation:
minimize t
X∈ S M , t∈R
subject to ‖A − X‖ F ≤ t
S T XS ≽ 0
(521)
that leads to an equivalent for (520) (and for (517) by (455))
minimize
X∈ S M , t∈R
subject to
t
[
tI vec(A − X)
vec(A − X) T t
]
≽ 0
(522)
S T XS ≽ 0
3.1.7.2.1 Example. Schur anomaly.
Consider a problem abstract in the convex constraint, given symmetric
matrix A
minimize ‖X‖ 2
X∈ S M F − ‖A − X‖2 F
(523)
subject to X ∈ C
the minimization of a difference of two quadratic functions each convex in
matrix X . Observe equality
‖X‖ 2 F − ‖A − X‖ 2 F = 2 tr(XA) − ‖A‖ 2 F (524)
So problem (523) is equivalent to the convex optimization
minimize tr(XA)
X∈ S M
subject to X ∈ C
(525)
But this problem (523) does not have Schur-form
minimize t − α
X∈ S M , α , t
subject to X ∈ C
‖X‖ 2 F ≤ t
(526)
‖A − X‖ 2 F ≥ α
because the constraint in α is nonconvex. (2.9.1.0.1)