Quantitative convergence analysis of iterated expansive, set-valued mappings

TL;DR

This paper introduces a new framework for analyzing the convergence of iterative algorithms involving expansive set-valued mappings, providing quantitative estimates and applying it to various nonconvex optimization methods.

Contribution

It generalizes existing convergence analysis tools to expansive, set-valued mappings and demonstrates their application to nonconvex algorithms.

Findings

Proves local linear convergence of cyclic projections for infeasible and feasible problems.

Establishes convergence of the forward-backward algorithm in nonconvex settings.

Shows local linear convergence of the Douglas--Rachford algorithm for structured nonconvex minimization.

Abstract

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive, set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity -- or inverse calmness -- of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural byproduct of the framework. To demonstrate the application of the theory, we prove for the first time a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas--Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.