Iterative Optimization in Inverse Problems

30 Iterative Optimization in Inverse Problems

with interior U .Atthekth step of their method one minimizes a function

G

k

(x)=f(x)+d(x, x

k−1

) (2.9)

to get x

k

. Their distance d(x, y) is defined for x and y in U, and the gradient

with respect to the first variable, denoted ∇

1

d(x, y), is assumed to exist.

The distance d(x, y) is not assumed to be a Bregman distance. Instead, they

assume that the distance d has an associated induced proximal distance

H(a, b) ≥ 0, finite for a and b in U, with H(a, a)=0and

∇

1

d(b, a),c− b≤H(c, a) −H(c, b), (2.10)

for all c in U .

If d = D

h

,thatis,ifd is a Bregman distance, then from the equation

∇

1

d(b, a),c− b = D

h

(c, a) −