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

256 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.8.2 epigraph of matrix-valued function, sublevel sets
We generalize epigraph to a continuous matrix-valued function [33, p.155]
g(X) : R p×k →S M :
epi g {(X , T )∈ R p×k × S M | X ∈ domg , g(X) ≼
T } (605)
from which it follows
g convex ⇔ epig convex (606)
Proof of necessity is similar to that in3.6 on page 233.
Sublevel sets of a matrix-valued convex function corresponding to each
and every S ∈ S M (confer (537))
S M +
L S
g {X ∈ domg | g(X) ≼
S } ⊆ R p×k (607)
are convex. There is no converse.
S M +
3.8.2.0.1 Example. Matrix fractional function redux. [33, p.155]
Generalizing Example 3.6.0.0.4 consider a matrix-valued function of two
variables on domg = S N + ×R n×N for small positive constant ǫ (confer (1830))
g(A, X) = ǫX(A + ǫI) −1 X T (608)
where the inverse always exists by (1427). This function is convex
simultaneously in both variables over the entire positive semidefinite cone S N +
and all X ∈ R n×N : Consider Schur-form (1487) fromA.4: for T ∈ S n
[ ]
A + ǫI X
T
G(A, X, T ) =
X ǫ −1 ≽ 0
T
⇔
T − ǫX(A + ǫI) −1 X T ≽ 0
A + ǫI ≻ 0
(609)
By Theorem 2.1.9.0.1, inverse image of the positive semidefinite cone S N+n
+
under affine mapping G(A, X, T ) is convex. Function g(A, X) is convex
on S N + ×R n×N because its epigraph is that inverse image:
epig(A, X) = { (A, X, T ) | A + ǫI ≻ 0, ǫX(A + ǫI) −1 X T ≼ T } = G −1( S N+n
+
(610)
)