Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting

TL;DR

This paper introduces a new convergence framework for iterative algorithms with variable fixed-point operators, enabling more flexible and adaptive splitting methods without requiring common fixed points.

Contribution

It develops a novel convergence analysis for nonexpansive operator families with relocated fixed points, extending existing methods to variable stepsize resolvent splitting.

Findings

Established a parametric demiclosedness principle for nonexpansive operators.

Proposed a variable stepsize Douglas--Rachford algorithm for monotone inclusions.

Proved convergence without common fixed-point assumptions.

Abstract

In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed-points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed-point, as is common in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations.