v2009.01.01 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 235
→Y −X
where dg(X) is the directional derivative (D.1.4) of function g at X in
direction Y −X . By discretized dual generalized inequalities, (2.13.5)
g(Y ) − g(X) −
→Y −X
dg(X) ≽
0 ⇔
〈
g(Y ) − g(X) −
→Y −X
〉
dg(X) , ww T ≥ 0 ∀ww T (≽ 0)
S M +
For each and every X,Y ∈ domg (confer (551))
(563)
S M +
g(Y ) ≽
g(X) +
→Y −X
dg(X) (564)
S M +
must therefore be necessary and sufficient for convexity of a matrix-valued
function of matrix variable on open convex domain.
3.2.2 epigraph of matrix-valued function, sublevel sets
We generalize the epigraph to a continuous matrix-valued function
g(X) : R p×k →S M :
epig ∆ = {(X , T )∈ R p×k × S M | X ∈ domg , g(X) ≼
T } (565)
S M +
from which it follows
g convex ⇔ epi g convex (566)
Proof of necessity is similar to that in3.1.7 on page 214.
Sublevel sets of a matrix-valued convex function corresponding to each
and every S ∈ S M (confer (496))
L S
g ∆ = {X ∈ dom g | g(X) ≼
S } ⊆ R p×k (567)
S M +
are convex. There is no converse.