Iterative Optimization in Inverse Problems

Variational Inequality Problems and Algorithms 223

ProofoftheTheorem: Assume that z is a solution of the SVIP. The

operator γA

T

(I − T )A is

1

2γφL

-ism. The operator V will be averaged if

γφL < 1, or

γ<

1

φL

=

2

(δ +1)L

. (15.17)

If γ ≤

1

L

, then the inequality (15.17) holds for all choices of λ<2α.

In similar iterative algorithms, such as the CQ algorithm and the

Landweber algorithm (see [58, 59]), the upper bound on γ is

2

L

.Wecan

allow γ to approach

2

L

here, but only by making δ approach zero, that is,

only by taking λ near zero.

Since U is also averaged, the operator S is averaged. Since the inter-

section of Fix(U)andFix(V ) is not empty, this intersection equals ...