Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations

TL;DR

This paper introduces a family of distributed forward-backward splitting methods with variable stepsizes, based on relocated fixed-point iterations, extending existing approaches to broader classes of operators for solving structured monotone inclusion problems.

Contribution

It extends fixed-point iteration techniques to conically averaged operators, enabling variable stepsize methods that retain convergence and computational efficiency for structured problems.

Findings

Methods converge to fixed points of the iteration operator.

Shadow sequences converge to solutions of the problem.

Numerical experiments demonstrate effectiveness in sparse optimization.

Abstract

We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems.