# Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections

## TL;DR

This paper establishes necessary conditions, including metric subregularity and subtransversality, for the linear convergence of fixed point iterations like alternating projections, especially when mappings are expansive or set-valued.

## Contribution

It introduces necessary conditions for linear convergence of expansive, set-valued mappings, extending the understanding of convergence behavior beyond nonexpansive cases, with applications to alternating projections.

## Key findings

- Metric subregularity is necessary for linear convergence.
- Subtransversality of sets is essential for the convergence of alternating projections.
- Various regularity conditions influence the necessity of these geometric properties.

## Abstract

We present necessary conditions for monotonicity, in one form or another, of fixed point iterations of mappings that violate the usual nonexpansive property. We show that most reasonable notions of linear-type monotonicity of fixed point sequences imply {\em metric subregularity}. This is specialized to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called {\em subtransversality}. Our more general results for fixed point iterations are specialized to establish the necessity of subtransversality for consistent feasibility with a number of reasonable types of sequential monotonicity, under varying degrees of assumptions on the regularity of the sets.