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On ergodic algorithms for equilibrium problems

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  • P. Anh
  • T. Hai
  • P. Tuan

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. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • P. Anh & T. Hai & P. Tuan, 2016. "On ergodic algorithms for equilibrium problems," Journal of Global Optimization, Springer, vol. 64(1), pages 179-195, January.
    • Handle: RePEc:spr:jglopt:v:64:y:2016:i:1:p:179-195
    DOI: 10.1007/s10898-015-0330-3
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    References listed on IDEAS

    as
    1. P. Anh & H. Le Thi, 2013. "An Armijo-type method for pseudomonotone equilibrium problems and its applications," Journal of Global Optimization, Springer, vol. 57(3), pages 803-820, November.
    2. M. A. Noor, 2004. "Auxiliary Principle Technique for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 371-386, August.
    3. P. N. Anh, 2012. "Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 303-320, July.
    4. I.V. Konnov, 2003. "Application of the Proximal Point Method to Nonmonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 317-333, November.
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    Cited by:

    1. Trinh Ngoc Hai, 2020. "Two modified extragradient algorithms for solving variational inequalities," Journal of Global Optimization, Springer, vol. 78(1), pages 91-106, September.
    2. Le Quang Thuy & Trinh Ngoc Hai, 2017. "A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 411-431, November.
    3. Pham Ngoc Anh & Qamrul Hasan Ansari & Ho Phi Tu, 2023. "DC auxiliary principle methods for solving lexicographic equilibrium problems," Journal of Global Optimization, Springer, vol. 85(1), pages 129-153, January.
    4. Pham Ky Anh & Trinh Ngoc Hai, 2019. "Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications," Journal of Global Optimization, Springer, vol. 73(3), pages 637-657, March.

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    More about this item

    Keywords

    Equilibrium problem; Monotone; Ergodic algorithm; Auxiliary problem principle; 65 K10; 90 C25;
    All these keywords.

    JEL classification:

    • K10 - Law and Economics - - Basic Areas of Law - - - General (Constitutional Law)
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities

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