Iterative Optimization in Inverse Problems

238 Iterative Optimization in Inverse Problems

The affine function h(x)=a, x−γ is above the concave function g(x)

if h(x) − g(x) ≥ 0, for all x ∈ D;thatis,

a, x−γ − g(x) ≥ 0,

or

γ ≤a, x−g(x). (17.9)

The conjugate function associated with g is the function

g

∗

(a) = inf

x

(a, x−g(x)). (17.10)

For each fixed a, the value g

∗

(a) is the largest value of γ for which the

affine function h(x)=a, x−γ is above g(x).

It follows, using f(x)=−g(x), that

g

∗

(a) = inf

x

(a, x + f(x)) = −sup

x

(−a, x−f(x)) = −f

∗

(−a).

17.2 Fenchel’s Duality Theorem

Let f(x) be a proper convex function on C ⊆ R

J

and g(x)aproper

concave function on D ⊆ R

J

,whereC and D are closed convex sets ...