On the Bredies-Chenchene-Lorenz-Naldi algorithm

TL;DR

This paper extends the analysis of the Bredies-Chenchene-Lorenz-Naldi algorithm for monotone inclusion problems, establishing new convergence results and illustrating their applicability in various scenarios.

Contribution

It relates fixed point set projections to the algorithm's convergence and provides strong convergence results for linear relations, enhancing understanding of the algorithm's behavior.

Findings

Established relations between fixed point set projections and algorithm convergence.

Proved strong convergence results for linear relations.

Provided examples demonstrating applicability of the theoretical results.

Abstract

Monotone inclusion problems occur in many areas of optimization and variational analysis. Splitting methods, which utilize resolvents or proximal mappings of the underlying operators, are often applied to solve these problems. In 2022, Bredies, Chenchene, Lorenz, and Naldi introduced a new elegant algorithmic framework that encompasses various well known algorithms including Douglas-Rachford and Chambolle-Pock. They obtained powerful weak and strong convergence results, where the latter type relies on additional strong monotonicity assumptions. In this paper, we complement the analysis by Bredies et al. by relating the projections of the fixed point sets of the underlying operators that generate the (reduced and original) preconditioned proximal point sequences. We also obtain strong convergence results in the case of linear relations. Various examples are provided to illustrate the applicability of our results.