On the nonexpansive operators based on arbitrary metric: A degenerate analysis

TL;DR

This paper explores nonexpansive operators with arbitrary metrics, focusing on degenerate cases, and analyzes their convergence properties, thereby extending the understanding of operator splitting algorithms and generalized proximal point methods.

Contribution

It introduces a framework for analyzing nonexpansive operators under degenerate metrics, linking various operator properties and discussing convergence of related fixed-point iterations.

Findings

Established connections between firm nonexpansiveness, cocoerciveness, and averagedness.

Analyzed convergence of fixed-point iterations with degenerate metrics.

Provided insights for generalized proximal point algorithms with relaxation steps.

Abstract

We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is discussed with particular focus on the case of degenerate metric, since the degeneracy is often encountered when reformulating many existing first-order operator splitting algorithms as a metric resolvent. This work paves a way for analyzing the generalized proximal point algorithm with a non-trivial relaxation step and degenerate metric.