Iterative Optimization in Inverse Problems

Operators 85

6.6 Firmly Nonexpansive Gradients

In this section we consider some useful properties of the gradient oper-

ator of a differentiable convex function.

It is convenient for us to consider functions on R

J

whose values may be

infinite. For example, we define the indicator function ι

C

of a set C ⊆ R

J

to have the value zero for x in C, and the value +∞ for x outside the set

C.

Definition 6.12 Afunctionf : R

J

→ [−∞, ∞] is proper if there is no x

for which f(x)=−∞ and some x for which f (x) < +∞.

All the functions we shall consider in this text will be proper.

Definition 6.13 Let f beaproperfunctiondefinedonR

J

. The subset of

R

J+1

defined by

epi(f)={(x, γ)|f (