Distributed Solutions for Loosely Coupled Feasibility Problems Using Proximal Splitting Methods

TL;DR

This paper introduces distributed algorithms based on proximal splitting methods to solve convex feasibility problems with loosely coupled sets, providing convergence guarantees and feasibility tests.

Contribution

It develops novel distributed algorithms for convex feasibility problems using proximal splitting, along with convergence tests and rate analysis for both feasible and infeasible cases.

Findings

Algorithms converge under bounded linear regularity

Distributed feasibility/infeasibility tests are effective

Convergence rates are similar to classical projection methods

Abstract

In this paper, we consider convex feasibility problems where the underlying sets are loosely coupled, and we propose several algorithms to solve such problems in a distributed manner. These algorithms are obtained by applying proximal splitting methods to convex minimization reformulations of convex feasibility problems. We also put forth distributed convergence tests which enable us to establish feasibility or infeasibility of the problem distributedly, and we provide convergence rate results. Under the assumption that the problem is feasible and boundedly linearly regular, these convergence results are given in terms of the distance of the iterates to the feasible set, which are similar to those of classical projection methods. In case the feasibility problem is infeasible, we provide convergence rate results that concern the convergence of certain error-bounds.