Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

Abstract

Given the success of Douglas–Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

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Acknowledgements

I would like to thank Wotao Yin for helpful comments and suggestions. I would also like to thank the anonymous associate editor and referees whose comments improved the paper significantly. In particular, the signal denoising numerical example was suggested by one of the anonymous reviewers. This work is supported in part by NSF Grant DMS-1720237 and ONR Grant N000141712162.

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Correspondence to Ernest K. Ryu.

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Ryu, E.K. Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting. Math. Program. 182, 233–273 (2020). https://doi.org/10.1007/s10107-019-01403-1

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Keywords

  • Douglas–Rachford splitting
  • Splitting methods
  • Maximal monotone operators
  • Lower bounds
  • First-order methods

Mathematics Subject Classification

  • 47H05
  • 47H09
  • 65K10
  • 90C25