A note on the equivalence of operator splitting methods

2018/06/08 by Walaa M. Moursi, Moursi, Walaa M., Yuriy Zinchenko +1
Computer Science · Mathematics · #47H09 #49M27 #49N15 #90C25 #Differential Equations and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods for differential equations #Optimization and Control (math.OC) #Primary 47H05 #Secondary 49M29

paper · pdf · doi:10.48550/arxiv.1806.03353

openalex publication_date 2018/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper provides a comprehensive discussion of the equivalences between splitting methods. These equivalences have been studied over the past few decades and, in fact, have proven to be very useful. In this paper, we survey known results and also present new ones. In particular, we provide simplified proofs of the equivalence of the ADMM and the Douglas-Rachford method and the equivalence of the ADMM with intermediate update of multipliers and the Peaceman-Rachford method. Other splitting methods are also considered.

Related

• Linear Convergence and Metric Selection for Douglas-Rachford Splitting\n and ADMM
• A Three-Operator Splitting Scheme Derived from Three-Block ADMM
• Relative-error inertial-relaxed inexact versions of Douglas-Rachford and\n ADMM splitting algorithms
• A Three-Operator Splitting Scheme and its Optimization Applications
• A quick probability-oriented introduction to operator splitting methods