# The Douglas-Rachford Algorithm for Convex and Nonconvex Feasibility Problems

## TL;DR

This paper reviews the Douglas-Rachford algorithm's application to convex and nonconvex feasibility problems, providing convergence theory, formulation strategies, and demonstrating its use on combinatorial and moment problems.

## Contribution

It develops the convergence theory for projection algorithms, explains how to formulate feasibility problems for the Douglas-Rachford method, and illustrates its application on specific combinatorial and moment problems.

## Key findings

- Convergence theory for the Douglas-Rachford algorithm is established.
- Feasibility problem formulations can effectively utilize the Douglas-Rachford method.
- Successful application demonstrated on the (m,n)-queens problem and moment construction.

## Abstract

The Douglas-Rachford method, a projection algorithm designed to solve continuous optimization problems, forms the basis of a useful heuristic for solving combinatorial optimization problems. In order to successfully use the method, it is necessary to formulate the problem at hand as a feasibility problem with constraint sets having efficiently computable nearest points. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas-Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the "eight queens puzzle" known as the "(m,n)-queens problem", and the problem of constructing a probability distribution with prescribed moments.