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On ergodic algorithms for equilibrium problems

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Abstract

In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159–164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371–386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303–320 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anh, P.N.: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math. Vietnam. 34, 183–200 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Anh, P.N.: An LQP regularization method for equilibrium problems on polyhedral. Vietnam. J. Math. 36, 209–228 (2008)

    MATH  MathSciNet  Google Scholar 

  5. Anh, P.N., Hien, N.D.: Fixed point solution methods for solving equilibrium problems. B. Korean Math. Soc. 51, 479–499 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comp. Math. Appl. 61, 2588–2595 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient method for solving bilevel variational inequalities. J. Glob. Optim. 52, 627–639 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Anh, P.N., Le Thi, H.A.: An Armijo-type method for pseudomonotone equilibrium problems and Its applications. J. Glob. Optim. 57, 803–820 (2013)

    Article  MATH  Google Scholar 

  9. Bigi, G., Castellani, M., Pappalardo, M.: A new solution method for equilibrium problems. Optim. Method Softw. 24, 895–911 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)

    MathSciNet  Google Scholar 

  11. Cho, Y.J., Qin, X., Kang, J.I.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cho, Y.J., Qin, X.: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 69, 4443–4451 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cohen, G.: Auxiliary principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1–18 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59, 1259–1274 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Kim, J.K., Anh, P.N., Hyun, H.G.: A proximal point-type algorithm for pseudomonotone equilibrium problems. Bull. Korean Math. Society 49, 749–759 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)

    Google Scholar 

  21. Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  22. Mastroeni, G.: Gap function for equilibrium problems. J. Global. Optim. 27, 411–426 (2004)

    Article  MathSciNet  Google Scholar 

  23. Moudafi, A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geom. 15, 91–100 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Ronald, E., Bruck, R.E.: 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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ronald, E., Bruck, R.E.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 32, 107–116 (1979)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We are grateful to the anonymous referees for their really helpful and constructive comments that helped us very much to improve the original version of the paper substantially.

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Authors and Affiliations

  1. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    P. N. Anh

  2. School of Applied Mathematics and Informatics, Ha Noi University of Science and Technology, Hanoi, Vietnam

    T. N. Hai

  3. Academy of Military Science and Technology, Hanoi, Vietnam

    P. M. Tuan

Authors
  1. P. N. Anh
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  2. T. N. Hai
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  3. P. M. Tuan
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Corresponding author

Correspondence to P. N. Anh.

Additional information

This work was completed during the stay of the first author at the Vietnam Institute for Advanced Study in Mathematics, and supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED).

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Anh, P.N., Hai, T.N. & Tuan, P.M. On ergodic algorithms for equilibrium problems. J Glob Optim 64, 179–195 (2016). https://doi.org/10.1007/s10898-015-0330-3

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  • DOI: https://doi.org/10.1007/s10898-015-0330-3

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