Skip to main content

Shadow Douglas–Rachford Splitting for Monotone Inclusions

Applied Mathematics & Optimization volume 80, pages 665–678 (2019)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Abbas, B., Attouch, H., Svaiter, B.F.: Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces. J. Optim. Theory Appl. 161(2), 331–360 (2014)

    Article  MathSciNet  Google Scholar 

  2. Al’ber, Y.I.: Continuous regularization of linear operator equations in a Hilbert space. Math. Notes 4(5), 793–797 (1968)

    Article  MathSciNet  Google Scholar 

  3. Antipin, A.S.: Minimization of convex functions on convex sets by means of differential equations. Diff. Equ. 30(9), 1365–1375 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Arrow, K., Hurwicz, L.: Gradient methods for constrained maxima. Op. Res. 5(2), 258–265 (1957)

    Article  MathSciNet  Google Scholar 

  5. Attouch, H., Cabot, A.: Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09584-z

  6. Banert, S., Boţ, R.I.: A forward-backward-forward differential equation and its asymptotic properties. J. Convex. Anal. 25(2), 371–388 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 1st edn. Springer, New York (2011)

    Book  Google Scholar 

  8. Bello Cruz, J., Díaz Millán, R.: A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy. Optimization 64(7), 1471–1486 (2015)

    Article  MathSciNet  Google Scholar 

  9. Bolte, J.: Continuous gradient projection method in Hilbert spaces. J. Optim. Theory Appl. 119(2), 235–259 (2003)

    Article  MathSciNet  Google Scholar 

  10. Boţ, R.I., Csetnek, E.R.: A dynamical system associated with the fixed points set of a nonexpansive operator. J. Dyn. Differ. Equ. 29(1), 155–168 (2017)

    Article  MathSciNet  Google Scholar 

  11. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  Google Scholar 

  12. Combettes, P.L., Pesquet, J.C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012)

    Article  MathSciNet  Google Scholar 

  13. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans.Am. Math. Soc. 82(2), 421–439 (1956)

    Article  MathSciNet  Google Scholar 

  14. Gavurin, M.K.: Nonlinear functional equations and continuous analogues of iterative methods (in Russian). Isvestiya Vuzov Matem 5, 18–31 (1958)

    MathSciNet  Google Scholar 

  15. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2013)

    MATH  Google Scholar 

  16. Johnstone, P.R., Eckstein, J.: Projective splitting with forward steps: asynchronous and block-iterative operator splitting. arXiv:1803.07043 (2018)

  17. Johnstone, P.R., Eckstein, J.: Single-forward-step projective splitting: Exploiting cocoercivity. arXiv:1902.09025 (2019)

  18. Latafat, P., Patrinos, P.: Primal-dual proximal algorithms for structured convex optimization: a unifying framework. In: Giselsson, P., Rantzer, A. (eds.) Large-Scale and Distributed Optimization, pp. 97–120. Springer, Cham (2018)

    Chapter  Google Scholar 

  19. Latafat, P., Stella, L., Patrinos, P.: New primal-dual proximal algorithm for distributed optimization. In: IEEE 55th Conference on Decision and Control (CDC), 2016 , pp. 1959–1964

  20. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)

    Article  MathSciNet  Google Scholar 

  21. Malitsky, Y.: Reflected projected gradient method for solving monotone variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)

    Article  MathSciNet  Google Scholar 

  22. Malitsky, Y., Tam, M.K.: A forward-backward splitting method for monotone inclusions without cocoercivity. arXiv:1808.04162 (2018)

  23. Peypouquet, J., Sorin, S.: Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time. J. Convex Anal. 17(3&4), 1113–1163 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comp. Math. Math. Phys. 4(5), 1–17 (1964)

    Article  Google Scholar 

  25. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Ernö Robert Csetnek

  2. Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16–18, 37083, Göttingen, Germany

    Yura Malitsky & Matthew K. Tam

Authors
  1. Ernö Robert Csetnek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yura Malitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew K. Tam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yura Malitsky.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Csetnek, E.R., Malitsky, Y. & Tam, M.K. Shadow Douglas–Rachford Splitting for Monotone Inclusions. Appl Math Optim 80, 665–678 (2019). https://doi.org/10.1007/s00245-019-09597-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09597-8

Keywords

  • Monotone operator
  • Operator splitting
  • Douglas–Rachford algorithm
  • Dynamical systems

Mathematics Subject Classification

  • 49M29
  • 90C25
  • 47H05
  • 47J20
  • 65K15