# On the Bredies-Chenchene-Lorenz-Naldi algorithm

arXiv: https://arxiv.org/abs/2307.09747 · HTML: https://mlsanity.com/paper/2307.09747/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration](https://mlsanity.com/paper/2407.13120/) (2024)

## Abstract (arXiv)

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.