Iterative Optimization in Inverse Problems

218 Iterative Optimization in Inverse Problems

15.4.2 The General Case

Now we have x

k+1

= Tx

k

where S = γG and T = P

C

(I −SP

C

(I −S)).

We assume that Tz = z,sothatz solves VIP(G, C).

The key Proposition now is the following.

Proposition 15.1 Let G : C → R

J

be pseudo-monotone and L-Lipschitz,

let σ = γL < 1,andletS = γG. For any k let y

k

= P

C

(I −S)x

k

.Then

z − x

k



2

−z − x

k+1



2

≥ (1 −σ

2

)y

k

− x

k



2

. (15.13)

The proof of Proposition 15.1 follows that in [120]. The inequality in

(15.13) emerges as a consequence of a sequence of inequalities and equa-

tions. We list these results first, and then discuss their proofs. For conve-

nience, we let w

k

= x

k

− Sx

k

.

1. Sy

k

,y

k