, 56:3 | Cite as

A new splitting method for monotone inclusions of three operators

  • Yunda DongEmail author
  • Xiaohuan Yu


In this article, we consider monotone inclusions in real Hilbert spaces and suggest a new splitting method. The associated monotone inclusions consist of the sum of one bounded linear monotone operator and one inverse strongly monotone operator and one maximal monotone operator. The new method, at each iteration, first implements one forward–backward step as usual and next implements a descent step, and it can be viewed as a variant of a proximal-descent algorithm in a sense. Its most important feature is that, at each iteration, it needs evaluating the inverse strongly monotone part once only in the forward–backward step and, in contrast, the original proximal-descent algorithm needs evaluating this part twice both in the forward–backward step and in the descent step. Moreover, unlike a recent work, we no longer require the adjoint operation of this bounded linear monotone operator in the descent step. Under standard assumptions, we analyze weak and strong convergence properties of this new method. Rudimentary experiments indicate the superiority of our suggested method over several recently-proposed ones for our test problems.


Monotone inclusions Self-adjoint operator Inverse strongly monotone Splitting method Weak convergence 

Mathematics Subject Classification

58E35 65K15 



We are very grateful to Xixian Bai at Shandong University for his help in numerical experiments.


  1. 1.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, H.G., Rockafellar, R.T.: Convergence rates in forward-backward splitting. SIAM J. Optim. 7, 421–444 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lawrence, J., Spingarn, J.E.: On fixed points of non-expansive piecewise isometric mappings. Proc. Lond. Math. Soc. 55, 605–624 (1987)CrossRefGoogle Scholar
  5. 5.
    Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3), 293–318 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dong, Y.D., Fischer, A.: A family of operator splitting methods revisited. Nonlinear Anal. 72, 4307–4315 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dong, Y.D.: Douglas–Rachford splitting method for semi-definite programming. J. Appl. Math. Comput. 51, 569–591 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dong, Y.D.: Splitting Methods for Monotone Inclusions. Ph.D. dissertation, Nanjing University (2003)Google Scholar
  11. 11.
    Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91(1), 123–140 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Solodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34, 1814–1830 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dong, Y.D.: A variable metric proximal-descent algorithm for monotone operators. J. Appl. Math. Comput. 60, 563–571 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set Valued Anal. 7, 323–345 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huang, Y.Y., Dong, Y.D.: New properties of forward–backward splitting and a practical proximal-descent algorithm. Appl. Math. Comput. 237, 60–68 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Noor, M.A.: Mixed quasi-variational inequalities. Appl. Math. Comput. 146, 553–578 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dong, Y.D.: An LS-free splitting method for composite mappings. Appl. Math. Lett. 18(8), 843–848 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    He, B.S.: Solving a class of linear projection equations. Numer. Math. 68, 71–80 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Irschara, A., Zach, C., Klopschitz, M., Bischof, H.: Large-scale, dense city reconstruction from user-contributed photos. Comput. Vis. Image Underst. 116, 2–15 (2012)CrossRefGoogle Scholar
  21. 21.
    Alotaibi, A., Combettes, P.L., Shahzad, N.: Solving coupled composite monotone inclusions by successive Fejér approximations of their Kuhn-Tucker set. SIAM J. Optim. 24(4), 2076–2095 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Combettes, P.L., Eckstein, J.: Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions. Math. Program. 168(1–2), 645–672 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Latafat, P., Patrinos, P.: Asymmetric forward–backward-adjoint splitting for solving monotone inclusions involving three operators. Comput. Optim. Appl. 68(1), 57–93 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Briceno-Arias, L.M., Davis, D.: Forward-backward-half forward algorithm with non self-adjoint linear operators for solving monotone inclusions. arXiv:1703.03436 (2017)
  25. 25.
    Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667–681 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460–479 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, River Edge, NJ (2002)CrossRefGoogle Scholar
  28. 28.
    Brezis, H., Crandall, M.G., Pazy, A.: Perturbation of nonlinear maximal monotone sets in Banach space. Comm. Pure Appl. Math. 23, 123–144 (1970)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Agarwal, R.P., Verma, R.U.: Inexact A-proximal point algorithm and applications to nonlinear variational inclusion problems. J. Optim. Theory Appl. 144, 431–444 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

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