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A new approach to the Korpelevich method for solving pseudomonotone equilibrium problems

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Abstract

The Korpelevich method is an algorithm which is used to find solutions to equilibrium problems. These problems are mathematical models which are used in economics, game theory, and engineering. Pseudomonotone equilibrium problems are a specific class of equilibrium problems that involve a weakened form of monotonicity. This work introduces a novel approach to applying the Korpelevich method to solving pseudomonotone equilibrium problems. We present a weak convergence theorem and linear convergence of the proposed method under some suitable conditions. Finally, a numerical example of a Nash-Cournot oligopolistic electricity market equilibrium model is given to complement the theoretical discussion and strengthen the evidence for the capabilities of our new approach.

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Data availability

The data sets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  4. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148(2), 318–335 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26(4–5), 827–845 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2012)

    MathSciNet  MATH  Google Scholar 

  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. Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets. EEE Trans. Power Syst. 19, 195–206 (2004)

    MATH  Google Scholar 

  9. Deng, L., Hu, R., Fang, Y.: Projection extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Numer. Algor. 86, 191–221 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Quoc, T.D.: Dual extragradient algorithms extended to equilibrium problems. Optimization 6, 57 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Dadashi, V., Iyiola, O.S., Shehu, Y.: The subgradient extragradient method for pseudomonotone equilibrium problems. Optimization 69, 901–923 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Dinh, B.V., Kim, D.S.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80(4), 1413–1436 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. 88(3), 399–415 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems. Optimization. 67, 2003–2029 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. RACSAM 111, 823–840 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80, 1413–1436 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Iusem, A.N., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. Ser. B 116, 259–273 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  22. Konnov, I.V.: Equilibrium models and variational inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  23. Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich’s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47(4), 631–639 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Lyashko, S.I., Semenov, V.V.: A new two-step proximal algorithm of solving the problem of equilibrium programming. In: Goldengorin, B. (ed.) Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications, vol. 115, pp. 315–325. Springer, Cham (2016)

    MATH  Google Scholar 

  25. Malitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Mastroeni, G.: On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica dell, Universita di Pisa 3, 1244–1258 (2000)

    MATH  Google Scholar 

  28. Mastroeni, G.: Gap function for equilibrium problems. J. Global Optim. 27, 411–426 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geometry 15, 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

  30. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)

    MathSciNet  MATH  Google Scholar 

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

  32. Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constraint equilibria. Nonlinear Analysis: Theory, Methods and Applications 18, 1159–1166 (1992)

    MATH  Google Scholar 

  33. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibria model. J. Optim. Theory Appl. 142(1)(2009), 185–204 (2009)

  34. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space. J. Optim. Theory Appl. 160, 809–831 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Peypouquet, J.: Convex optimization in normed spaces: theory, methods and examples. Springer, Berlin (2015)

    MATH  Google Scholar 

  37. Popov, L.D.: A modification of the Arrow-Hurwicz method for searching for saddle points. Mat. Zametki 28(5), 777–784 (1980)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Rehman, H., Kumam, P., Cho, Y., et al.: Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw. 36(1), 1–32 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Shehu, Y., Iyiola, O.S., Thong, D.V., et al.: An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems. Math. Meth. Oper. Res. 93, 213–242 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Global. Optim. 56(2), 373–397 (2013)

    MathSciNet  MATH  Google Scholar 

  42. Shehu, Y., Liu, L., Qin, X., et al.: Reflected iterative method for non-monotone equilibrium problems with applications to Nash-Cournot equilibrium models. Netw. Spat. Econ. 22, 153–180 (2022)

    MATH  Google Scholar 

  43. Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    MathSciNet  MATH  Google Scholar 

  44. 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, 429–451 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Yang, J., Liu, H.: The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim. Lett. 14, 1803–1816 (2020)

    MathSciNet  MATH  Google Scholar 

  46. Vinh, N.T., Muu, L.D.: Inertial extragradient algorithms for solving equilibrium problems. Acta Math. Vietnam. 44(3), 639–663 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Yao, Y., Shehu, Y., Li, X.H., Dong, Q.L.: A method with inertial extrapolation step for split monotone inclusion problems. Optimization 70, 741–761 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Iusem, A.N., Nasri, M.: Augmented Lagrangian methods for variational inequality problems. RAIRO-Oper. Res. 44, 5–25 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Iusem, A.N., Nasri, M.: Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 28, 1279–1308 (2007)

    MathSciNet  MATH  Google Scholar 

  50. Mashreghi, J., Nasri, M.: Forcing strong convergence of Korpelevich’s method in Banach spaces with its applications in game theory. Nonlinear Anal. Theory Methods Appl. 72, 2086–2099 (2010)

    MathSciNet  MATH  Google Scholar 

  51. Mashreghi, J., Nasri, M.: Strong convergence of an inexact proximal point algorithm for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 31, 1053–1071 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

All the authors are grateful to two anonymous referees for their useful comments and helpful suggestions.

Funding

Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion (Grant 2001893), and by the Technion General Research Fund (Grant 2016723). Xiao-Huan Li was partially supported by the National Natural Science Foundation of China (12371305) and ShandongProvince Natural Science Foundation (ZR2023MA020). This research has been done under the research projectQG.23.04 of Vietnam National University, Hanoi.

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Authors and Affiliations

  1. Faculty of Mathematical Economics, National Economics University, Hanoi City, Vietnam

    Duong Viet Thong & Dang Huy Ngan

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

    Xiao-Huan Li

  3. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

  4. Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Qiao-Li Dong

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  1. Duong Viet Thong
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Contributions

D. V. Thong, Simeon Reich, and Dang Huy Ngan wrote the main manuscript text, Xiao-Huan Li and Qiao-Li Dong prepared Tables 1 and 2 and Figs. 1 and 2. All authors reviewed the manuscript carefully. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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Correspondence to Duong Viet Thong.

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Dedicated to Professor Pham Ky Anh on the occasion of his 75th birthday.

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Thong, D.V., Li, XH., Reich, S. et al. A new approach to the Korpelevich method for solving pseudomonotone equilibrium problems. Numer Algor 98, 719–741 (2025). https://doi.org/10.1007/s11075-024-01813-z

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  • DOI: https://doi.org/10.1007/s11075-024-01813-z

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