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

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. 2018;168:55–61] 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 analysing many operator splitting algorithms.

Keywords:

• Douglas-Rachford splitting (DRS)
• degenerate proximal point algorithm
• proximal mapping
• maximal monotonicity
• cyclic monotonicity

AMS subject classifications:

• 47H09
• 47H05
• 90C25
• 68Q25

Acknowledgements

I am gratefully indebted to the anonymous reviewer for helpful discussions, particularly related to the proofs of Lemma 2.7 and Proposition 2.8.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

There is no associated data with this manuscript.

Notes

1 (4(4) $\lfloor \begin{array}{rl}{u}^{k+1}& :=J_{\frac{1}{\tau }{B}^{-1}}\left(\frac{1}{\tau }{z}^k\right),\\ s^{k+1}& :=J_{\frac{1}{\tau }{A}^{-1}}(\frac{1}{\tau }{z}^k-2u^{k+1}),\\ z^{k+1}& :=z^k-\tau (s^{k+1}+u^{k+1}).\end{array}$(4) ) only increases the apparent size of (2(2) $z^{k+1}:=z^k-J_{\mathrm{\tau B}}(z^k)+J_{\mathrm{\tau A}}(2J_{\mathrm{\tau B}}(z^k)-z^k).$(2) ), while keeping the actual dimension unchanged. Hence, we use ‘size expansion’ rather than dimension expansion.

2 The word ‘general’ means that there exists an exceptional case of $\mathcal{H}=\mathbb{R}$.