Algorithms Based on Unions of Nonexpansive Maps

TL;DR

This paper introduces a framework for analyzing iterative algorithms based on unions of nonexpansive maps, demonstrating local convergence around strong fixed points, thus generalizing previous results in fixed point theory.

Contribution

It extends existing fixed point theorems to set-valued operators composed of unions of paracontracting maps, providing new insights into convergence behavior.

Findings

Proves local convergence of fixed point iterations for unions of paracontracting operators

Generalizes a theorem by Bauschke and Noll (2014)

Provides a structured approach to analyze set-valued operators in iterative algorithms

Abstract

In this note, we consider a framework for the analysis of iterative algorithms which can described in terms of a structured set-valued operator. More precisely, at each point in the ambient space, we assume that the value of operator can be expressed as a finite union of values of single-valued paracontracting operators. Our main result, which shows that the associated fixed point iteration is locally convergent around strong fixed points, generalises a theorem due to Bauschke and Noll (2014).