Random iteration and projection method

TL;DR

This paper analyzes how random iteration of nonexpansive maps can recover invariant sets and explains the behavior of the Kaczmarz--von Neumann projection algorithm in infeasible cases involving multiple affine varieties.

Contribution

It provides theoretical insights into the convergence of random orbits generated by nonexpansive maps and clarifies the algorithm's behavior in infeasible scenarios.

Findings

Random orbits recover invariant sets via omega-limit under certain conditions.

The analysis explains the behavior of the Kaczmarz--von Neumann algorithm in infeasible cases.

Conditions for convergence of random iteration in nonexpansive systems.

Abstract

We show that under suitable conditions a random orbit generated by a system of nonexpansive maps recovers an invariant set via its omega-limit. In particular, this explains what happens to the Kaczmarz--von Neumann projection algorithm in the infeasible case, that is, when one deals with more than two affine varieties having empty intersection.