On compositions of special cases of Lipschitz continuous operators

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.