Advertisement

Inertial algorithms for a system of equilibrium problems and fixed point problems

  • Prashanta Majee
  • Chandal NahakEmail author
Article
  • 111 Downloads

Abstract

In this paper, we introduce two new algorithms (one parallel and another sequential) with inertial effect for approximating a common solution of a system of equilibrium problems and fixed point problems. Under suitable conditions, we establish weak convergence results for the proposed algorithms. Finally, we give a numerical example to demonstrate the convergence and performance of the proposed algorithms.

Keywords

Nonexpansive mapping Averaged mapping Equilibrium problem Fixed point problem Inertial method 

Mathematics Subject Classification

47H10 90C33 47J20 

Notes

Acknowledgements

The authors thank anonymous referees and the editor for their constructive comments which helped to improve the paper.

References

  1. 1.
    Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection–proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773–782 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1), 3–11 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62(2), 271–283 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Glob. Optim. 52(3), 627–639 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Anh, P.N., Muu, L.D.: A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems. Optim. Lett. 8(2), 727–738 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Attouch, H., Cabot, A.: Convergence rates of inertial forward–backward algorithms. HAL-01453170 (2017)Google Scholar
  7. 7.
    Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 158, 123–175 (2018)Google Scholar
  8. 8.
    Baillon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4, 1–9 (1978)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Beck, A., Teboulle, M.: A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90(1), 31–43 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. India: Indian Math. Soc. 63(1), 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Boţ, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 3(4), 459–470 (1977)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20(1), 103–120 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148(2), 318–335 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117–136 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 1–18 (2017)MathSciNetGoogle Scholar
  19. 19.
    Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65(12), 2217–2226 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, T.M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 1–16 (2018)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Giannessi, F.: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, vol. 38. Springer Science & Business Media, Berlin (2013)zbMATHGoogle Scholar
  22. 22.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory, Vol. 28 Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)Google Scholar
  23. 23.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  24. 24.
    Hieu, D.V.: Projected subgradient algorithms on systems of equilibrium problems. Optim. Lett. 1–16 (2017).  https://doi.org/10.1007/s11590-017-1127-8
  25. 25.
    Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73(1), 197–217 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Liu, Y.: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. Theory Methods Appl. 71(10), 4852–4861 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lorenz, D.A., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51(2), 311–325 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Maingé, P.E.: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219(1), 223–236 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155(2), 447–454 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73(4), 591–597 (1967)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Plubtieng, S., Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336(1), 455–469 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)Google Scholar
  33. 33.
    Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)MathSciNetzbMATHGoogle 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(1), 506–515 (2007)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. Theory Methods Appl. 69(3), 1025–1033 (2008)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58(5), 486–491 (1992)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62(11), 4007–4014 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Zhou, H.: Convergence theorems of fixed points for \(k\)-strict pseudocontractions in Hilbert spaces. Nonlinear Anal. Theory Methods Appl. 69(2), 456–462 (2008)Google Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations