Conical averagedness and convergence analysis of fixed point algorithms

TL;DR

This paper introduces conically averaged operators, explores their properties, and applies them to analyze the convergence of several fixed point algorithms, providing a unified framework that enhances understanding and results.

Contribution

It develops a conical extension of averaged operators, systematically investigates their properties, and applies these to improve convergence analysis of key algorithms.

Findings

Conically averaged operators are stable under relaxations, convex combinations, and compositions.

Properties of resolvents of generalized monotone operators are characterized as conically averaged.

The convergence of proximal point, forward-backward, and Douglas-Rachford algorithms is unified and improved.

Abstract

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.