# Quantitative convergence analysis of iterated expansive, set-valued mappings

arXiv: https://arxiv.org/abs/1605.05725 · HTML: https://mlsanity.com/paper/1605.05725/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2016.

## Built on by (most cited first, 2 of 2)

- [Provable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2210.09248/) (2022)
- [Inexact subgradient methods for semialgebraic functions](https://mlsanity.com/paper/2404.19517/) (2024)

## Abstract (arXiv)

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.