Unrestricted Douglas-Rachford algorithms for solving convex feasibility problems in Hilbert space

TL;DR

This paper introduces a generalized unrestricted Douglas-Rachford algorithm for convex feasibility problems in Hilbert space, expanding its framework with new iterative methods involving strongly nonexpansive operators.

Contribution

The paper develops a novel unrestricted Douglas-Rachford framework that incorporates products of strongly nonexpansive operators, broadening the algorithm's applicability.

Findings

Provides a new iterative method for convex feasibility problems

Extends Douglas-Rachford algorithm with unrestricted operator products

Offers a flexible framework for future algorithmic developments

Abstract

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted and extended the algorithm with the aim of solving CFPs and even more general problems, such as finding zeros of the sum of two maximally monotone operators. Many developments which implement various concepts concerning this algorithm have occurred during the last decade. We introduce an unrestricted DR algorithm, which provides a general framework for such concepts. Using unrestricted products of a finite number of strongly nonexpansive operators, we apply this framework to provide new iterative methods, where, \textit{inter alia}, such operators may be interlaced between the operators used in the scheme of our \ unrestricted \color DR algorithm.