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A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

  • Le Dung Muu
  • Xuan Thanh Le
Article
  • 17 Downloads

Abstract

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Hölder continuity of the bifunctions involved.

Keywords

Monotone equilibria Fixed point Common solution Splitting algorithm 

Mathematics Subject Classification

47H05 47H10 90C33 

Notes

Acknowledgements

The authors would like to thank the associate editor and anonymous referee for their constructive comments and helpful remarks. This work is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.01-2017.315.

References

  1. 1.
    Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algorithm 76(1), 67–91 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62(2), 271–283 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anh, P.N., Muu, L.D.: A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems. Optim. Lett. 8(2), 727–738 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1–4), 123–145 (1994)MathSciNetMATHGoogle Scholar
  6. 6.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithm 8(2), 221–239 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cruz, J.Y.B., Millán, R.D.: A direct splitting method for nonsmooth variational inequalities. J. Optim. Theory Appl. 161(3), 728–737 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65(10), 1855–1866 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Duc, P.M., Muu, L.D., Quy, N.V.: Solution-existence and algorithms with their convergence rate for strongly pseudomonotone equilibrium problems. Pac. J. Optim. 12(4), 833–845 (2016)MathSciNetMATHGoogle Scholar
  10. 10.
    Eckstein, J., Svaiter, A.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Optim. 48(2), 787–811 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality, vol. III, pp. 103–113. Academic Press, New York (1972)Google Scholar
  12. 12.
    Hai, T.N., Vinh, N.T.: Two new splitting algorithms for equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. 111(4), 1051–1069 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hieu, D.V., Moudafi, A.: A barycentric projected-subgradient algorithm for equilibrium problems. J. Nonlinear Var. Anal. 1(1), 43–59 (2017)Google Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74(17), 6121–6129 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58(2), 251–261 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5(2), 269–278 (1998)MathSciNetMATHGoogle Scholar
  18. 18.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27(4), 411–426 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359(2), 508–513 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18(12), 1159–1166 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142(1), 185–204 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nikaidô, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5(5), 807–815 (1955)CrossRefMATHGoogle Scholar
  25. 25.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72(2), 383–390 (1979)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52(1), 139–159 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Reich, S.: A limit theorem for projections. Linear Multilinear Algebra 13(3), 281–290 (1983)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 5, 877–890 (1976)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Santos, P.S.M., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30(1), 91–107 (2011)MathSciNetMATHGoogle Scholar
  32. 32.
    Sun, S.: An alternative regularization method for equilibrium problems and fixed point of nonexpansive mappings. J. Appl. Math. 2012, Article ID 202860 (2012).  https://doi.org/10.1155/2012/202860
  33. 33.
    Svaiter, B.F.: On weak convergence of the Douglas–Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133(3), 359–370 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Tan, K.-K., Xu, H.-K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Tseng, P.: A modied forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Tuy, H.: Convex Analysis and Global Optimization, 2nd edn. Springer, Berlin (2016)CrossRefMATHGoogle Scholar
  39. 39.
    Vuong, P.T., Strodiot, J.-J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64(2), 429–451 (2015)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Applied SciencesThang Long UniversityHanoiVietnam
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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