Viscosity Modification with Inertial Forward-Backward Splitting Methods for Solving Inclusion Problems

  • D. Yambangwai
  • S. Suantai
  • H. Dutta
  • W. CholamjiakEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 123)


In this paper, we introduce the two new different modified forward-backward algorithms combining the viscosity approximation method with the inertial technical term for solving the inclusion problem. The strongly convergent theorems are established under some suitable conditions in Hilbert spaces. The application of algorithms in this study work is use to find the minimum-norm least-squares solution of an unconstrained linear system and test some numerical experiments. Moreover, the efficiency and the implementation of our methods have been shown through the examples.


Inertial method Inclusion problem Maximal monotone operator Forward-backward algorithm Minimum-norm least-squares solution. 

Mathematics Subject Classification (2010)

15A06 47H04 47H05 47H10 47J22 



S. Suantai would like to thank Chiang Mai University. D. Yambangwai and W. Cholamjiak would like to thank University of Phayao(Grant No. UoE62001).


  1. 1.
    Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Banach, S.: Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fund. Math. 3, 133–181 (1922)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)CrossRefGoogle Scholar
  4. 4.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Browder, F.E.: Convergence of approximants to fixed points of non-expansive maps in Banach spaces. Arch. Rational Mech. Anal. 24, 82–90 (1967)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Prob. 18, 441–453 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Byrne, C.: A unified treatment of some iterative algorithm in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projection in a product space. Numer. Algorithms 71, 915–932 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cholamjiak, P.: A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numer. Algorithms 8, 221–239 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Douglas, J., Rachford, H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Amer. Math. Soc. 82, 421–439 (1956)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dang, Y., Sun, J., Xu, H.: Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim.
  13. 13.
    Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, Th.M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett.
  14. 14.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)MathSciNetCrossRefGoogle Scholar
  15. 15.
    He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013 (2013). Art ID. 942315Google Scholar
  16. 16.
    Lakshmikantham, V., Sen, S.K.: Computational Error and Complexity in Science and Engineering. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
  17. 17.
    López, G., Martín-Marquez, V., Wang, F., Xu, H.K.: Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012 (2012). Art ID 109236Google 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.
    Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Maingé, P.E.: Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set-Valued Anal. 15, 67–79 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nesterov, Y.: A method for solving the convex programming problem with convergence rate \(O(1/k^2)\). Dokl. Akad. Nauk SSSR 269, 543–547 (1983)MathSciNetGoogle Scholar
  24. 24.
    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
  25. 25.
    Peaceman, D.H., Rachford, H.H.: The numerical solution of parabolic and elliptic differentials. J. Soc. Indust. Appl. Math. 3, 28–41 (1955)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)zbMATHGoogle Scholar
  27. 27.
    Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)MathSciNetGoogle Scholar
  28. 28.
    Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpasnsive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sen, S.K., Shaykhian, G.A.: Preprocessing inconsistent linear system for a meaningful least squares solution. J. Neural Parallel Sci. Comput. 19, 211–228 (2011)zbMATHGoogle Scholar
  32. 32.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38, 431–446 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  34. 34.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. (Basel) 58, 486–491 (1992)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yao, Y., Yao, J.C., Zhou, H.Y.: Approximation methods for common fixed points of infinite countable family of nonexpansive mappings. Comput. Math. Appl. 53, 1380–1389 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • D. Yambangwai
    • 1
  • S. Suantai
    • 2
  • H. Dutta
    • 3
  • W. Cholamjiak
    • 1
    Email author
  1. 1.School of ScienceUniversity of PhayaoPhayaoThailand
  2. 2.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  3. 3.Department of MathematicsGauhati UniversityGuwahatiIndia

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