# A Lyapunov-type approach to convergence of the Douglas-Rachford algorithm

## TL;DR

This paper introduces a Lyapunov-type approach to prove convergence of the Douglas-Rachford algorithm in nonconvex settings, expanding its theoretical understanding and applicability.

## Contribution

It provides the first convergence proof for Douglas-Rachford in certain nonconvex problems using a Lyapunov functional approach.

## Key findings

- Convergence is established in nonconvex scenarios.
- The Lyapunov functional does not require convexity of the original sets.
- Examples demonstrate global convergence in nonconvex cases.

## Abstract

The Douglas-Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas-Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.