A note on the equivalence of operator splitting methods

TL;DR

This paper surveys and simplifies the understanding of equivalences between various operator splitting methods, including ADMM, Douglas-Rachford, and Peaceman-Rachford, highlighting their interconnectedness and providing new proofs.

Contribution

It offers a comprehensive survey with simplified proofs of known equivalences and introduces new insights into the relationships among splitting methods.

Findings

Proved the equivalence of ADMM and Douglas-Rachford methods

Established the equivalence of ADMM with intermediate multiplier updates and Peaceman-Rachford method

Discussed equivalences among other splitting methods

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.