v2009.01.01 - Convex Optimization

236 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.2.2.0.1 Example. Matrix fractional function redux.
Generalizing Example 3.1.7.0.4 consider a matrix-valued function of two
variables on domg = S N + ×R n×N for small positive constant ǫ (confer (1746))
g(A, X) = ǫX(A + ǫI) −1 X T (568)
where the inverse always exists by (1354). 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 (1413) fromA.4: for T ∈ S n
[ A + ǫI X
T
G(A, X, T ) =
X ǫ −1 T
⇔
T − ǫX(A + ǫI) −1 X T ≽ 0
A + ǫI ≻ 0
]
≽ 0
(569)
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
+
(570)
3.2.3 second-order condition, matrix function
The following line theorem is a potent tool for establishing convexity of a
multidimensional function. To understand it, what is meant by line must first
be solidified. Given a function g(X) : R p×k →S M and particular X, Y ∈ R p×k
not necessarily in that function’s domain, then we say a line {X+ t Y | t ∈ R}
passes through domg when X+ t Y ∈ domg over some interval of t ∈ R .
3.2.3.0.1 Theorem. Line theorem. [53,3.1.1]
Matrix-valued function g(X) : R p×k →S M is convex in X if and only if it
remains convex on the intersection of any line with its domain. ⋄