v2009.01.01 - Convex Optimization

238 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.2.3.0.4 Example. Matrix squared.
Iconic real function f(x)= x 2 is strictly convex on R . The matrix-valued
function g(X)=X 2 is convex on the domain of symmetric matrices; for
X, Y ∈ S M and any open interval of t ∈ R (D.2.1)
d 2
d2
g(X+ t Y ) =
dt2 dt 2(X+ t Y )2 = 2Y 2 (574)
which is positive semidefinite when Y is symmetric because then Y 2 = Y T Y
(1360). 3.19
A more appropriate matrix-valued counterpart for f is g(X)=X T X
which is a convex function on domain {X ∈ R m×n } , and strictly convex
whenever X is skinny-or-square full-rank. This matrix-valued function can
be generalized to g(X)=X T AX which is convex whenever matrix A is
positive semidefinite (p.632), and strictly convex when A is positive definite
and X is skinny-or-square full-rank (Corollary A.3.1.0.5).
3.2.3.0.5 Exercise. Squared maps.
Give seven examples of distinct polyhedra P for which the set
{X T X | X ∈ P} ⊆ S n + (575)
were convex. Is this set convex, in general, for any polyhedron P ?
(confer (1141) (1148)) Is the epigraph of function g(X)=X T X convex for
any polyhedral domain?
3.2.3.0.6 Example. Matrix exponential.
The matrix-valued function g(X) = e X : S M → S M is convex on the subspace
of circulant [142] symmetric matrices. Applying the line theorem, for all t∈R
and circulant X, Y ∈ S M , from Table D.2.7 we have
d 2
dt 2eX+ t Y = Y e X+ t Y Y ≽
S M +
0 , (XY ) T = XY (576)
because all circulant matrices are commutative and, for symmetric matrices,
XY = Y X ⇔ (XY ) T = XY (1378). Given symmetric argument, the matrix
3.19 By (1379) inA.3.1, changing the domain instead to all symmetric and nonsymmetric
positive semidefinite matrices, for example, will not produce a convex function.