On the Stability of Krasnosel'ski\v{\i}-Mann Iterations

TL;DR

This paper extends convergence results of Krasnosel'ski-Mann iterations, including relaxed variants, and applies these findings to generalized proximal point algorithms for monotone inclusion problems.

Contribution

It introduces new convergence results for relaxed Krasnosel'ski-Mann iterations and their applications to proximal algorithms.

Findings

Proved weak and strong convergence of translated methods.

Extended convergence results to relaxed iterations.

Applied results to generalized proximal point algorithms.

Abstract

Firstly, we invoke the weak convergence (resp. strong convergence) of translated basic methods involving nonexpansive operators to establish the weak convergence (resp. strong convergence) of the associated method with both perturbation and approximation. Then we employ the technique obtaining the result above to extend convergence results from the classic Krasnosel'ski\v{\i}-Mann iterations to their relaxation variants for finding a common fixed point of associated nonexpansive operators. At last, we show applications on generalized proximal point algorithms for solving monotone inclusion problems.