Linear convergence of relocated fixed-point iterations

TL;DR

This paper proves linear convergence of relocated fixed-point iterations under certain conditions, including contraction operators, and applies this to algorithms like Douglas--Rachford and resolvent splitting for monotone inclusions.

Contribution

It introduces a unified framework for establishing linear convergence of relocated fixed-point iterations, extending to various algorithms under monotonicity and Lipschitz conditions.

Findings

Linear convergence established for relocated fixed-point iterations.

Application to Douglas--Rachford algorithm with strong convergence guarantees.

Extension to variable stepsize resolvent splitting methods.

Abstract

We establish linear convergence of relocated fixed-point iterations as introduced by Atenas et al. (2025) assuming the algorithmic operator satisfies a linear error bound. In particular, this framework applies to the setting where the algorithmic operator is a contraction. As a key application of our framework, we obtain linear convergence of the relocated Douglas--Rachford algorithm for finding a zero in the sum of two monotone operators in a setting with Lipschitz continuity and strong monotonicity assumptions. We also apply the framework to deduce linear convergence of variable stepsize resolvent splitting algorithms for multioperator monotone inclusions.