On compositions of special cases of Lipschitz continuous operators

TL;DR

This paper systematically studies the compositions of special Lipschitz continuous operators, such as firmly nonexpansive and averaged operators, to understand their properties and implications for convergence in optimization algorithms.

Contribution

It extends the analysis of compositions of Lipschitz operators to new special cases, enhancing understanding of their structure and applications in optimization.

Findings

Analyzed compositions of scaled conically nonexpansive mappings.

Applied results to Douglas--Rachford and forward-backward operators.

Provided examples illustrating the theoretical conclusions.

Abstract

Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas--Rachford and forward-backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.