v2009.01.01 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 237
Now we assume a twice differentiable function.
3.2.3.0.2 Definition. Differentiable convex matrix-valued function.
Matrix-valued function g(X) : R p×k →S M is convex in X iff domg is an
open convex set, and its second derivative g ′′ (X+ t Y ) : R→S M is positive
semidefinite on each point of intersection along every line {X+ t Y | t ∈ R}
that intersects domg ; id est, iff for each and every X, Y ∈ R p×k such that
X+ t Y ∈ domg over some open interval of t ∈ R
d 2
dt 2 g(X+ t Y ) ≽ S M +
0 (571)
Similarly, if
d 2
dt 2 g(X+ t Y ) ≻ S M +
0 (572)
then g is strictly convex; the converse is generally false. [53,3.1.4] 3.17 △
3.2.3.0.3 Example. Matrix inverse. (confer3.1.5)
The matrix-valued function X µ is convex on int S M + for −1≤µ≤0
or 1≤µ≤2 and concave for 0≤µ≤1. [53,3.6.2] In particular, the
function g(X) = X −1 is convex on int S M + . For each and every Y ∈ S M
(D.2.1,A.3.1.0.5)
d 2
dt 2 g(X+ t Y ) = 2(X+ t Y )−1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 ≽
S M +
0 (573)
on some open interval of t ∈ R such that X + t Y ≻0. Hence, g(X) is
convex in X . This result is extensible; 3.18 trX −1 is convex on that same
domain. [176,7.6, prob.2] [48,3.1, exer.25]
3.17 Quadratic forms constitute a notable exception where the strict-case converse is
reliably true.
3.18 d/dt trg(X+ tY ) = trd/dt g(X+ tY ). [177, p.491]