Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting
Felipe Atenas, Heinz H. Bauschke, Minh N. Dao, Matthew K. Tam
TL;DR
The paper develops a relocated fixed-point framework to enable convergence analysis of resolvent-based splitting methods with variable stepsizes, without assuming a common fixed-point across parameters. It introduces fixed-point relocators that map fixed-points across parameter changes and proves a parametric demiclosedness principle to support convergence proofs. The framework is applied to graph-based Douglas–Rachford for sums of $N\ge2$ maximally monotone operators and to variable-stepsize Malitsky–Tam type splits, establishing weak convergence to solutions under mild step-size conditions. This approach unifies nonstationary and graph-based DR variants and provides a path to rate and extension analyses for broader monotone inclusion problems.
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.