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


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.

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  1. 1.

    Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algorithm 76(1), 67–91 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62(2), 271–283 (2013)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1–4), 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithm 8(2), 221–239 (1994)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MATH  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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5(2), 269–278 (1998)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27(4), 411–426 (2003)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Nikaidô, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5(5), 807–815 (1955)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Reich, S.: A limit theorem for projections. Linear Multilinear Algebra 13(3), 281–290 (1983)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 5, 877–890 (1976)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Santos, P.S.M., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30(1), 91–107 (2011)

    MathSciNet  MATH  Google 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

    MathSciNet  Google Scholar 

  33. 33.

    Svaiter, B.F.: On weak convergence of the Douglas–Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Tseng, P.: A modied forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Tuy, H.: Convex Analysis and Global Optimization, 2nd edn. Springer, Berlin (2016)

    Book  Google 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)

    MathSciNet  Article  Google Scholar 

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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.

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Correspondence to Le Dung Muu.

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Muu, L.D., Le, X.T. A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems. J. Fixed Point Theory Appl. 20, 130 (2018). https://doi.org/10.1007/s11784-018-0612-8

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  • Monotone equilibria
  • Fixed point
  • Common solution
  • Splitting algorithm

Mathematics Subject Classification

  • 47H05
  • 47H10
  • 90C33