Three-operator splitting algorithm for a class of variational inclusion problems

  • Dang Van HieuEmail author
  • Le Van Vy
  • Pham Kim Quy
Original Paper


This paper concerns with a new three-operator splitting algorithm for solving a class of variational inclusions. The main advantage of the proposed algorithm is that it can be easily implemented without the prior knowledge of Lipschitz constant, strongly monotone constant and cocoercive constant of component operators. A reason explained for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. The strong convergence of the algorithm is established. Several fundamental numerical experiments are given to illustrate the behavior of the new algorithm and compare it with other algorithms.


Forward–backward method Tseng’s method Operator splitting method 

Mathematics Subject Classification

65J15 47H05 47J25 47J20 91B50 



The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The works of the first author are supported in part by the National Foundation for Science and Technology Development (NAFOS-TED) of Vietnam under Grant Number 101.01-2017.315.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Attouch, H., Peypouquet, J., Redont, P.: Backward–forward algorithms for structured monotone inclusions in Hilbert spaces. J. Math. Anal. Appl. 457, 1095–1117 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Combettes, P.L., Wajs, V.: Signal recovery by proximal forward–backward splitting. SIAM Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1457 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Davis, D., Yin, W.T.: A three-operator splitting scheme and its optimization applications. Set Valued Var. Anal. 25, 829–858 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dong, Y.D., Fischer, A.: A family of operator splitting methods revisited. Nonlinear Anal. 72, 4307–4315 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duchi, J., Singer, Y.: Efficient online and batch learning using forward–backward splitting. J. Mach. Learn. Res. 10, 2899–2934 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gibali, A., Hieu, D.V.: A new inertial double-projection method for solving variational inequalities. J. Fixed Point Theory Appl. (2019).
  8. 8.
    Hieu, D.V., Quy, P.K.: An inertial modified algorithm for solving variational inequalities. RAIRO Operations Research (2018).
  9. 9.
    Hieu, D.V., Thong, D.V.: New extragradient—like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385–399 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hieu, D.V., Thong, D.V.: A new projection method for a class of variational inequalities. Appl. Anal. (2018). MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hieu, D.V.: Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems. Numer. Algor. 77, 983–1001 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hieu, D.V.: New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl. Anal. 97, 811–824 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malaysian Math. Soc. 42, 201–221 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hieu, D.V., Gibali, A.: Strong convergence of inertial algorithms for solving equilibrium problems. Optim. Lett. (2019).
  15. 15.
    Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Golden ratio algorithms with new stepsize rules for variational inequalities. Math. Meth. Appl. Sci. (2019).
  16. 16.
    Huang, Y.Y., Dong, Y.D.: New properties of forward-backward splitting and a practical proximaldescent algorithm. Appl. Math. Comput. 237, 60–68 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Khanh, P.D.: A new extragradient method for strongly pseudomonotone variational inequalities. Numer. Funct. Anal. Optim. 37, 1131–1143 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Malitsky, Y., Tam, M. K.: A forward–backward splitting method for monotone inclusions without cocoercivity. (2018). arXiv:1808.04162
  20. 20.
    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
  21. 21.
    Raguet, H., Fadili, J., Peyré, G.: A generalized forward-backward splitting. SIAM J. Imaging Sci. 6, 1199–1226 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  23. 23.
    Ryu, E.K., Boyd, S.: A primer on monotone operator methods. Appl. Comput. Math. 15, 3–43 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38, 431–446 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zong, C., Tang, Y., Cho, Y.J.: Convergence analysis of an inexact three-operator splitting algorithm. Symmetry (2018). CrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of MathematicsCollege of Air ForceNha Trang CityVietnam

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