# Shadow Douglas–Rachford Splitting for Monotone Inclusions

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## Abstract

In this work, we propose a new algorithm for finding a zero of the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.

## Keywords

Monotone operator Operator splitting Douglas–Rachford algorithm Dynamical systems## Mathematics Subject Classification

49M29 90C25 47H05 47J20 65K15## Notes

### Acknowledgements

The authors would like to thank the Erwin Sch\(\ddot{\mathrm{r}}\)odinger Institute for their support and hospitality during the thematic program “Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions”. The authors would also like to thank the two anonymous referees for their helpful comments as well as Sebastian Banert for sharing his nice counterexample that we mentioned in Remark 4.

### Funding

ERC was supported by Austrian Science Fund Project P 29809-N32. YM was supported by German Research Foundation Grant No. SFB755-A4.

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