On Douglas-Rachford splitting that generally fails to be a proximal mapping: A degenerate proximal point analysis

TL;DR

This paper investigates the Douglas-Rachford splitting method, revealing that it often cannot be considered a proximal mapping, and extends recent findings to broader contexts using degenerate proximal point analysis.

Contribution

It demonstrates that Douglas-Rachford splitting reduces to a resolvent but generally isn't a proximal mapping, broadening the understanding of its properties through degenerate proximal point analysis.

Findings

Douglas-Rachford splitting reduces to a resolvent

Generally fails to be a proximal mapping

Implications for operator splitting algorithms

Abstract

Based on a degenerate proximal point analysis, we show that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but generally fails to be a proximal mapping. This extends the recent result of [Bauschke, Schaad and Wang. Math. Program., 168 (2018), pp. 55-61] to more general setting. The related concepts and consequences are also discussed. In particular, the results regarding the maximal and cyclic monotonicity are instrumental for analyzing many operator splitting algorithms.