Skip to main content
SpringerLink
Log in

Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed and convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aragón Artacho, F.J., Campoy, R.: A new projection method for finding the closest point in the intersection of convex sets. Comput. Optim. Appl. 69(1), 99–132 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aragón Artacho, F.J., Campoy, R.: Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces. Numer. Algor. 1–25 (2018). https://doi.org/10.1007/s11075-018-0608-x

  3. Bauschke, H.H.; Burachik, R.S., Kaya, C.Y.: Constraint splitting and projection methods for optimal control of double integrator. ArXiv e-prints: arXiv:1804.03767 (2018)

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Svaiter, B.F.: On weak convergence of the Douglas–Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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  MATH  Google Scholar 

  7. Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16(4), 727–748 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Bauschke, H.H., Combettes, P.L.: A Dykstra-like algorithm for two monotone operators. Pac. J. Optim. 4(3), 383–391 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Combettes, P.L.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380(2), 680–688 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Adly, S., Bourdin, L., Caubet, F.: On a decomposition formula for the proximal operator of the sum of two convex functions. J. Convex Anal. 26(3) (2019). http://www.heldermann.de/JCA/JCA26/JCA263/jca26037.htm

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

    Book  MATH  Google Scholar 

  12. Minty, G.A.: A theorem on monotone sets in Hilbert spaces. J. Math. Anal. Appl. 14, 434–439 (1967)

    MATH  Google Scholar 

  13. Bauschke, H.H., Hare, W.L., Moursi, W.M.: Generalized solutions for the sum of two maximally monotone operators. SIAM J. Control Optim. 52, 1034–1047 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bauschke, H.H., Moursi, W.M.: On the Douglas–Rachford algorithm. Math. Program. Ser. A 164(1—-2), 263–284 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bauschke, H.H., Lukens, B., Moursi, W.M.: Affine nonexpansive operators, Attouch–Théra duality and the Douglas–Rachford algorithm. Set-Valued Var. Anal. 25(3), 481–505 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We greatly appreciate the constructive comments of two anonymous reviewers which helped us to improve the paper. This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”.

Author information

Authors and Affiliations

  1. University of Alicante, Alicante, Spain

    Francisco J. Aragón Artacho & Rubén Campoy

Authors
  1. Francisco J. Aragón Artacho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rubén Campoy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rubén Campoy.

Additional information

Communicated by Nicolas Hadjisavvas.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aragón Artacho, F.J., Campoy, R. Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm. J Optim Theory Appl 181, 709–726 (2019). https://doi.org/10.1007/s10957-019-01481-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01481-3

Keywords

Mathematics Subject Classification

Search

Navigation