On the Douglas-Rachford and Peaceman-Rachford algorithms in the presence of uniform monotonicity and the absence of minimizers

TL;DR

This paper proves the convergence of Douglas-Rachford and Peaceman-Rachford algorithms' shadow sequences under uniform monotonicity and $3^*$ monotonicity, even without minimizers, supported by examples and numerical experiments.

Contribution

It establishes convergence results for these algorithms in the absence of solutions when one operator is uniformly monotone and $3^*$ monotone, extending previous knowledge.

Findings

Convergence of shadow sequences under new conditions

Examples illustrating the theoretical results

Numerical experiments confirming convergence behavior

Abstract

The Douglas-Rachford and Peaceman-Rachford algorithms have been successfully employed to solve convex optimization problems, or more generally find zeros of monotone inclusions. Recently, the behaviour of these methods in the inconsistent case, i.e., in the absence of solutions has triggered significant consideration. It has been shown that under mild assumptions the shadow sequence of the Douglas-Rachford algorithm converges weakly to a generalized solution when the underlying operators are subdifferentials of proper lower semicontinuous convex functions. However, no convergence behaviour has been proved in the case of Peaceman-Rachford algorithm. In this paper, we prove the convergence of the shadow sequences associated with the Douglas-Rachford algorithm and Peaceman-Rachford algorithm when one of the operators is uniformly monotone and $3^*$ monotone but not necessarily a subdifferential. Several examples illustrate and strengthen our conclusion. We carry out numerical experiments using example instances of optimization problems.