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

  • Francisco J. Aragón Artacho
  • Rubén CampoyEmail author
Article
  • 38 Downloads

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.

Keywords

Maximally monotone operator Resolvent Averaged alternating modified reflections algorithm Douglas–Rachford algorithm Splitting method 

Mathematics Subject Classification

47H05 47J25 65K05 47N10 

Notes

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”.

References

  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Svaiter, B.F.: On weak convergence of the Douglas–Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16(4), 727–748 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Combettes, P.L.: A Dykstra-like algorithm for two monotone operators. Pac. J. Optim. 4(3), 383–391 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Combettes, P.L.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380(2), 680–688 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 11.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Minty, G.A.: A theorem on monotone sets in Hilbert spaces. J. Math. Anal. Appl. 14, 434–439 (1967)MathSciNetzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bauschke, H.H., Moursi, W.M.: On the Douglas–Rachford algorithm. Math. Program. Ser. A 164(1—-2), 263–284 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of AlicanteAlicanteSpain

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