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The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when…

Optimization and Control · Mathematics 2016-04-21 Heinz H. Bauschke , Walaa M. Moursi

The Douglas--Rachford algorithm is a classical and very successful splitting method for finding the zeros of the sums of monotone operators. When the underlying operators are normal cone operators, the algorithm solves a convex feasibility…

Optimization and Control · Mathematics 2015-04-16 Heinz H. Bauschke , Walaa M. Moursi

The Douglas-Rachford and Peaceman-Rachford algorithms have been successfully employed to solve convex optimization problems, or more generally find zeros of monotone inclusions. Recently, the behaviour of these methods in the inconsistent…

Optimization and Control · Mathematics 2024-05-28 Regina S. Burachik , Bethany I. Caldwell , C. Yalçın Kaya , Walaa M. Moursi , Matthew Saurette

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Weak convergence in this method to a solution of the underlying monotone inclusion problem in the general case remained an…

Optimization and Control · Mathematics 2018-09-10 Benar F. Svaiter

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we…

Optimization and Control · Mathematics 2025-11-07 Jan Harold Alcantara , Akiko Takeda

The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm is well…

Optimization and Control · Mathematics 2020-04-14 Minh N. Dao , Hung M. Phan

The Douglas-Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one…

Optimization and Control · Mathematics 2020-07-10 Heinz H. Bauschke , Walaa M. Moursi

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a…

Optimization and Control · Mathematics 2018-09-10 Benar F. Svaiter

The Douglas-Rachford algorithm is a simple yet effective method for solving convex feasibility problems. However, if the underlying constraints are inconsistent, then the convergence theory is incomplete. We provide convergence results when…

Optimization and Control · Mathematics 2020-04-14 Heinz H. Bauschke , Minh N. Dao , Walaa M. Moursi

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets.…

Optimization and Control · Mathematics 2013-12-24 Heinz H. Bauschke , J. Y. Bello Cruz , Tran T. A. Nghia , Hung M. Phan , Xianfu Wang

The Douglas--Rachford method is a splitting method frequently employed for finding zeroes of sums of maximally monotone operators. When the operators in question are normal cones operators, the iterated process may be used to solve…

Optimization and Control · Mathematics 2020-01-28 Scott B. Lindstrom , Brailey Sims

In recent years, a distributed Douglas-Rachford splitting method (DDRSM) has been proposed to tackle multi-block separable convex optimization problems. This algorithm offers relatively easier subproblems and greater efficiency for…

Optimization and Control · Mathematics 2024-11-19 Leyu Hu , Jiaxin Xie , Xingju Cai , Deren Han

The Douglas-Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper closed convex functions; more generally two maximally monotone operators. Recent results concerned with…

Optimization and Control · Mathematics 2018-05-25 Walaa M. Moursi , Lieven Vandenberghe

In this paper, we study a parameterized Douglas-Rachford splitting method for a class of nonconvex optimization problem. A new merit function is constructed to establish the convergence of the whole sequence generated by the parameterized…

Optimization and Control · Mathematics 2020-06-17 Fengmiao Bian , Xiaoqun Zhang

The Douglas-Rachford algorithm is widely used in sparse signal processing for minimizing a sum of two convex functions. In this paper, we consider the case where one of the functions is weakly convex but the other is strongly convex so that…

Optimization and Control · Mathematics 2015-11-13 İlker Bayram , Ivan W. Selesnick

The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for…

Optimization and Control · Mathematics 2022-06-16 Walaa M. Moursi

The Douglas-Rachford splitting method is a classical and widely used algorithm for solving monotone inclusions involving the sum of two maximally monotone operators. It was recently shown to be the unique frugal, no-lifting…

Optimization and Control · Mathematics 2025-12-12 Max Nilsson , Anton Åkerman , Pontus Giselsson

We study the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex, and inconsistent feasibility problems. This algorithm can be viewed as a convex relaxation between the cyclic Douglas-Rachford algorithm first introduced by…

Optimization and Control · Mathematics 2026-05-06 Thi Lan Dinh , G. S. Matthijs Jansen , D. Russell Luke

The Douglas-Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas-Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a…

Optimization and Control · Mathematics 2015-05-13 Heinz H. Bauschke , Walaa M. Moursi

The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular…

Optimization and Control · Mathematics 2014-08-01 Heinz H. Bauschke , Warren L. Hare , Walaa M. Moursi
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