Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
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240 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS<br />
minimize t<br />
X∈ S M , t∈R<br />
subject to ‖A − X‖ F ≤ t<br />
S T XS ≽ 0<br />
(562)<br />
that leads to an equivalent for (561) (and for (558) by (491))<br />
minimize<br />
X∈ S M , t∈R<br />
subject to<br />
t<br />
[<br />
tI vec(A − X)<br />
vec(A − X) T t<br />
]<br />
≽ 0<br />
(563)<br />
S T XS ≽ 0<br />
3.6.2.0.1 Example. Schur anomaly.<br />
Consider a problem abstract in the <strong>convex</strong> constraint, given symmetric<br />
matrix A<br />
minimize ‖X‖ 2<br />
X∈ S M F − ‖A − X‖2 F<br />
(564)<br />
subject to X ∈ C<br />
the minimization <strong>of</strong> a difference <strong>of</strong> two quadratic <strong>functions</strong> each <strong>convex</strong> in<br />
matrix X . Observe equality<br />
‖X‖ 2 F − ‖A − X‖ 2 F = 2 tr(XA) − ‖A‖ 2 F (565)<br />
So problem (564) is equivalent to the <strong>convex</strong> optimization<br />
minimize tr(XA)<br />
X∈ S M<br />
subject to X ∈ C<br />
(566)<br />
But this problem (564) does not have Schur-form<br />
minimize t − α<br />
X∈ S M , α , t<br />
subject to X ∈ C<br />
‖X‖ 2 F ≤ t<br />
(567)<br />
‖A − X‖ 2 F ≥ α<br />
because the constraint in α is non<strong>convex</strong>. (2.9.1.0.1)