v2009.01.01 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 233
3.2 Matrix-valued convex function
We need different tools for matrix argument: We are primarily interested in
continuous matrix-valued functions g(X). We choose symmetric g(X)∈ S M
because matrix-valued functions are most often compared (557) with respect
to the positive semidefinite cone S M + in the ambient space of symmetric
matrices. 3.16
3.2.0.0.1 Definition. Convex matrix-valued function:
1) Matrix-definition.
A function g(X) : R p×k →S M is convex in X iff domg is a convex set and,
for each and every Y,Z ∈domg and all 0≤µ≤1 [186,2.3.7]
g(µY + (1 − µ)Z) ≼
µg(Y ) + (1 − µ)g(Z) (557)
S M +
Reversing sense of the inequality flips this definition to concavity. Strict
convexity is defined less a stroke of the pen in (557) similarly to (446).
2) Scalar-definition.
It follows that g(X) : R p×k →S M is convex in X iff w T g(X)w : R p×k →R is
convex in X for each and every ‖w‖= 1; shown by substituting the defining
inequality (557). By dual generalized inequalities we have the equivalent but
more broad criterion, (2.13.5)
g convex ⇔ 〈W , g〉 convex
∀W ≽
S M +
0 (558)
Strict convexity on both sides requires caveat W ≠ 0. Because the set of
all extreme directions for the positive semidefinite cone (2.9.2.4) comprises
a minimal set of generators for that cone, discretization (2.13.4.2.1) allows
replacement of matrix W with symmetric dyad ww T as proposed. △
3.16 Function symmetry is not a necessary requirement for convexity; indeed, for A∈R m×p
and B ∈R m×k , g(X) = AX + B is a convex (affine) function in X on domain R p×k with
respect to the nonnegative orthant R m×k
+ . Symmetric convex functions share the same
benefits as symmetric matrices. Horn & Johnson [176,7.7] liken symmetric matrices to
real numbers, and (symmetric) positive definite matrices to positive real numbers.