Inertial Extragradient Algorithms for Solving Equilibrium Problems

Abstract

We introduce two new algorithms based upon the extragradient and inertial methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. Strong converge and weak convergence of the proposed algorithms are established under some mild assumptions. Numerical results show that the proposed algorithms are more efficient than some existing methods for equilibrium problems.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. 1.

    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator damping. Set-valued. Anal. 9, 3–11 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces. SIAM J. Optim. 9, 773–782 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. European. J. Oper. Res. 227, 1–11 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–146 (1994)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bot, R.I., Csetnek, E.R., Nimana, N.: Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data. Optim. Lett. 12, 17–33 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Dinh, B.V., Muu, L.D.: A projection algorithm for solving pseudomonotone equilibrium problems and it’s application to a class of bilevel equilibria. Optimization 64, 559–575 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Dong, Q.-L., Lu, Y.-Y., Yang, J.-F.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217–2226 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    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. 12, 87–102 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Dong, Q.-L., Cho, Y.J., Zhong, L.L., Rassias, Th.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Duc, P.M., Muu, L.D., Quy, N.V.: Solution-existence and algorithms with their convergence rate for strongly pseudomonotone equilibrium problems. Pacific J. Optim. 12, 833–845 (2016)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65, 1855–1866 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities, vol. III, pp 103–113. Academic Press, New York (1972)

  15. 15.

    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    Google Scholar 

  16. 16.

    Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algorithms. https://doi.org/10.1007/s11075-018-0532-0 (2018)

  17. 17.

    Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. https://doi.org/10.1007/s00186-018-0640-6 (2018)

  18. 18.

    Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems, Optimization. https://doi.org/10.1080/02331934.2018.1505886(2018)

  19. 19.

    Iofee, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)

    Google Scholar 

  20. 20.

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

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Kassay, G., Miholca, M., Vinh, N.T.: Vector quasi-equilibrium problems for the sum of two multivalued mappings. J. Optim. Theory Appl. 169, 424–442 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kassay, G., Hai, T.N., Vinh, N.T.: Coupling Popov’s algorithm with subgradient extragradient method for solving equilibrium problems. J. Nonlinear Convex Anal. 19, 959–986 (2018)

    MathSciNet  Google Scholar 

  24. 24.

    Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Li, M., Yao, Y.: Strong convergence of an iterative algorithm for λ-strictly pseudocontractive mappings in Hilbert spaces. Analele Ştiinţifice ale Universitatii Ovidius constanţa. Seria Mathematica 18, 219–228 (2010)

    Google Scholar 

  26. 26.

    Maingé, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Maingé, P.E.: Strong convergence of projected subgradientmethods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16, 899–911 (2009)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris 255, 2897–2899 (1962)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4, Article 18 (2003)

  32. 32.

    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Pham, K.A., Trinh, N.H.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algor. 76, 67–91 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms to equilibrium Problems. J. Glob. Optim. 52, 139–159 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

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

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  38. 38.

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

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Shehu, Y., Iyiola, O.S.: Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J. Fixed Point Theory Appl. 19, 2483–2510 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Strodiot, J.J., Vuong, P.T., Van, N.T.T.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Global Optim. 64, 159–178 (2016)

  41. 41.

    Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag Optim. https://doi.org/10.3934/jimo.2018023 (2017)

  42. 42.

    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    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  MATH  Google Scholar 

  45. 45.

    Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Vuong, P.T., Strodiot, J.J., Hien, N.V.: Extragradient methods and linearsearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and referee for careful reading, and constructive suggestions that allowed to improve significantly the presentation of this paper.

Funding

The first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 101.01-2017.08.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Le Dung Muu.

Additional information

Dedicated to Professor Le Tuan Hoa on the occasion of his 60th birthday

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vinh, N.T., Muu, L.D. Inertial Extragradient Algorithms for Solving Equilibrium Problems. Acta Math Vietnam 44, 639–663 (2019). https://doi.org/10.1007/s40306-019-00338-1

Download citation

Keywords

  • Inertial method
  • Extragradient algorithm
  • Variational inequality
  • Equilibrium problem
  • Pseudomonotone
  • Lipschitz-type inequality

Mathematics Subject Classification (2010)

  • 47H10
  • 47J25
  • 47N10
  • 90C25