Survey: sixty years of Douglas–Rachford

SB Lindstrom, B Sims - Journal of the Australian Mathematical …, 2021 - cambridge.org
The Douglas–Rachford method is a splitting method frequently employed for finding zeros of
sums of maximally monotone operators. When the operators in question are normal cone …

Adaptive Douglas--Rachford splitting algorithm for the sum of two operators

MN Dao, HM Phan - SIAM Journal on Optimization, 2019 - SIAM
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 …

Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems

MN Dao, HM Phan - Journal of Global Optimization, 2018 - Springer
In this paper, we study the generalized Douglas–Rachford algorithm and its cyclic variants
which include many projection-type methods such as the classical Douglas–Rachford …

A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting

MN Dao, MK Tam - Journal of Global Optimization, 2019 - Springer
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 …

Union averaged operators with applications to proximal algorithms for min-convex functions

MN Dao, MK Tam - Journal of Optimization Theory and Applications, 2019 - Springer
In this paper, we introduce and study a class of structured set-valued operators, which we
call union averaged nonexpansive. At each point in their domain, the value of such an …

Convergence analysis under consistent error bounds

T Liu, BF Lourenço - Foundations of Computational Mathematics, 2024 - Springer
We introduce the notion of consistent error bound functions which provides a unifying
framework for error bounds for multiple convex sets. This framework goes beyond the …

Primal necessary characterizations of transversality properties

ND Cuong, AY Kruger - Positivity, 2021 - Springer
This paper continues the study of general nonlinear transversality properties of collections of
sets and focuses on primal necessary (in some cases also sufficient) characterizations of the …

Regularity of Sets Under a Reformulation in a Product Space with Reduced Dimension

R Campoy - Set-Valued and Variational Analysis, 2023 - Springer
Different notions on regularity of sets and of collection of sets play an important role in the
analysis of the convergence of projection algorithms in nonconvex scenarios. While some …

Constraint reduction reformulations for projection algorithms with applications to wavelet construction

MN Dao, ND Dizon, JA Hogan, MK Tam - Journal of Optimization Theory …, 2021 - Springer
We introduce a reformulation technique that converts a many-set feasibility problem into an
equivalent two-set problem. This technique involves reformulating the original feasibility …

The Douglas–Rachford algorithm for a hyperplane and a doubleton

HH Bauschke, MN Dao, SB Lindstrom - Journal of Global Optimization, 2019 - Springer
Abstract The Douglas–Rachford algorithm is a popular algorithm for solving both convex
and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent …