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

258 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
on some open interval of t ∈ R such that X + t Y ≻0. Hence, g(X) is
convex in X . This result is extensible; 3.22 trX −1 is convex on that same
domain. [198,7.6, prob.2] [53,3.1, exer.25]
3.8.3.0.4 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 (614)
which is positive semidefinite when Y is symmetric because then Y 2 = Y T Y
(1433). 3.23
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.680), and strictly convex when A is positive definite
and X is skinny-or-square full-rank (Corollary A.3.1.0.5).
‖X‖ 2 sup ‖Xa‖ 2 = σ(X) 1 = √ λ(X T X) 1 = minimize
‖a‖=1
t∈R
subject to
t (615)
[ ] tI X
X T ≽ 0
tI
The matrix 2-norm (spectral norm) coincides with largest singular value.
This supremum of a family of convex functions in X must be convex because
it constitutes an intersection of epigraphs of convex functions.
3.8.3.0.5 Exercise. Squared maps.
Give seven examples of distinct polyhedra P for which the set
{X T X | X ∈ P} ⊆ S n + (616)
were convex. Is this set convex, in general, for any polyhedron P ?
(confer (1212) (1219)) Is the epigraph of function g(X)=X T X convex for
any polyhedral domain?
3.22 d/dt tr g(X+ tY ) = trd/dtg(X+ tY ). [199, p.491]
3.23 By (1451) inA.3.1, changing the domain instead to all symmetric and nonsymmetric
positive semidefinite matrices, for example, will not produce a convex function.