, 55:49 | Cite as

Tseng type methods for solving inclusion problems and its applications

  • Aviv Gibali
  • Duong Viet ThongEmail author


In this paper, we introduce two modifications of the forward–backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergence of the proposed algorithms. We present two numerical examples, the first in infinite dimensional spaces, which illustrates mainly the strong convergence property of the algorithm. For the second example, we illustrate the performances of our scheme, compared with the classical forward–backward splitting method for the problem of recovering a sparse noisy signal. Our result extend some related works in the literature and the primary experiments might also suggest their potential applicability.


Forward–backward splitting method Viscosity approximation method Mann-type method Zero point 

Mathematics Subject Classification

65Y05 65K15 68W10 47H06 47H09 47H10 



The authors would like to thank the referees for their comments on the manuscript which helped in improving earlier version of this paper.


  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.
    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
  3. 3.
    Brézis, H., Chapitre, I.I.: Operateurs maximaux monotones. North-Holland Math. Stud. 5, 19–51 (1973)CrossRefGoogle Scholar
  4. 4.
    Bruck, R.: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, H.G., Rockafellar, R.T.: Convergence rates in forward–backward splitting. SIAM J. Optim. 7, 421–444 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, S., Donoho, D.L., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics 2057. Springer, Berlin (2012)Google Scholar
  8. 8.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Combettes, P.L., Wajs, V.: Signal recovery by proximal forward–backward splitting. SIAM Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Dong, Q., Jiang, D., Cholamjiak, P., Shehu, Y.: A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions. J. Fixed Point Theory Appl. 19, 3097–3118 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dong, Y.D., Fischer, A.: A family of operator splitting methods revisited. Nonlinear Anal. 72, 4307–4315 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T.: Efficient projections onto the \(l_1\)-ball for learning in high dimensions. In: Proceedings of the 25 th International Conference on Machine Learning, Helsinki, Finland (2008)Google Scholar
  14. 14.
    Duchi, J., Singer, Y.: Efficient online and batch learning using forward–backward splitting. J. Mach. Learn. Res. 10, 2899–2934 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  16. 16.
    Goldstein, A.A.: Convex programming in Hilbert spaces. Bull. Am. Math. Soc. 70, 709–710 (1964)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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
  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.
    Maingé, F.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Moudafi, A.: Viscosity approximating methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Raguet, H., Fadili, J., Peyré, G.: A generalized forward–backward splitting. SIAM J. Imaging Sci. 6, 1199–1226 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Takahashi, W.: Nonlinear Functional Analysis-Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  26. 26.
    Tibshirami, R.: Regression shrinkage and selection via lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1996)MathSciNetGoogle Scholar
  27. 27.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim. 38, 431–446 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Department of MathematicsORT Braude CollegeKarmielIsrael
  2. 2.The Center for Mathematics and Scientific ComputationUniversity of HaifaHaifaIsrael
  3. 3.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations