Set Regularities and Feasibility Problems

TL;DR

This paper unifies and characterizes regularity notions in set collections, providing insights into convergence conditions for projection algorithms in feasibility problems.

Contribution

It introduces new characterizations of regularities, clarifying their relationships and suggesting necessary conditions for local linear convergence.

Findings

New characterizations of regularity notions

Relations between different regularity concepts clarified

Necessary conditions for convergence identified

Abstract

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms.