Iterative Optimization in Inverse Problems

178 Iterative Optimization in Inverse Problems

≤ D

f

(p, q) −∇

2

D

f

(p

n

,q

n

),q−q

n

≤D

f

(p, q).

Therefore, we have

D

f

(p, p

n

)+D

f

(p, q) ≥ D

f

(p, q

n

).

This is the four-point property.

We now know that the alternating minimization method works for any

Bregman distance that is jointly convex. This includes the Euclidean and

the KL distances.

12.1.9 Minimizing a Proximity Function

We present now an example of alternating Bregman distance minimiza-

tion taken from [57]. The problem is the convex feasibility problem (CFP),

to find a member of the intersection C ⊆ R

J

of finitely many closed convex

sets C

i

,i=1, ..., I, or, failing that, to minimize the proximity function

F