v2009.01.01 - Convex Optimization

216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1.7.0.4 Example. Matrix fractional function. (confer3.2.2.0.1)
Continuing Example 3.1.7.0.2, now consider a real function of two variables
on domf = S n +×R n for small positive constant ǫ (confer (1746))
f(A, x) = ǫx T (A + ǫI) −1 x (503)
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 : Consider Schur-form (1413) fromA.4: for t ∈ R
[ ] A + ǫI z
G(A, z , t) =
z T ǫ −1 ≽ 0
t
⇔
t − ǫz T (A + ǫI) −1 z ≥ 0
A + ǫI ≻ 0
(504)
Inverse image of the positive semidefinite cone S n+1
+ under affine mapping
G(A, z , t) is convex by Theorem 2.1.9.0.1. Function f(A, z) is convex on
S+×R n n because its epigraph is that inverse image:
epif(A, z) = { (A, z , t) | A + ǫI ≻ 0, ǫz T (A + ǫI) −1 z ≤ t } = G −1( S n+1
+
(505)
3.1.7.1 matrix fractional projector function
Consider nonlinear function f having orthogonal projector W as argument:
f(W , x) = ǫx T (W + ǫI) −1 x (506)
Projection matrix W has property W † = W T = W ≽ 0 (1791). Any
orthogonal projector can be decomposed into an outer product of
orthonormal matrices W = UU T where U T U = I as explained in
E.3.2. From (1746) for any ǫ > 0 and idempotent symmetric W ,
ǫ(W + ǫI) −1 = I − (1 + ǫ) −1 W from which
Therefore
f(W , x) = ǫx T (W + ǫI) −1 x = x T( I − (1 + ǫ) −1 W ) x (507)
lim
ǫ→0 +f(W
, x) = lim (W + ǫI) −1 x = x T (I − W )x (508)
ǫ→0 +ǫxT
where I − W is also an orthogonal projector (E.2).
)