## A survey on solution approaches for the equilibrium problem defined by the nikaido-isoda-fan inequality

## Main Article Content

### Abstract

We provide a brief survey on basic solution approaches for solving the equilibirum problem defined by the Nikaido-Isoda-Fan inequality. Namely, first we state the problem and consider its most important special cases including the optimization, inverse optimization, Kakutani fixed point, variational inequality, Nash equilibrium problems. Next, we present some basic solution approaches for the problem. Finally, as an application, we consider the famous Cournot-Nash oligopolistic equilibrium model and discuss algorithms for solving it.

## Article Details

### Keywords

Nikaido-Isoda-Fan equilibrium problem, variational inequality, Cournot-Nash equilibrium model

### References

[2] Anh, P. N., Muu, L. D., Nguyen, V. H., and Strodiot, J. J., Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities, Journal of Optimization Theory and Applications, 124 (2003), 285–306.

[3] Anh, P. N., Hai, T. N., and Tuan, P. M., On ergodic algorithms for equilibrium problems, Journal of Global Optimization, 64 (2016), 179–195.

[4] Anh, T. V. and Muu, L. D., Quasi-nonexpansive mappings involving pseudomonotone bifunctions on convex sets, Journal of Convex Analysis, 25(4)

(2018), 1105–1119.

[5] Aussel, D., Adjusted sublevel set, normal operator, and quasi-convex programming, SIAM Journal on Optimization, 16(2) (2005), 358–367.

[6] Aussel, D., Dutta, J., and Pandit, T., About the links between equilibrium problems and variational inequalities, Pages 115–130 in Neogy, S., Bapat, R., and Dubey, D. (editors), Mathematical Programming and Game Theory, Indian Statistical Institute Series, Springer, 2018.

[7] Bauschke, H. and Combettes, P., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2011.

[8] Bigi, G., Castellani, M., Pappalardo, M., and Passacantando, M., Nonlinear Programming Techniques for Equilibria, Springer, 2019.

[9] Bigi, G. and Passacantando, M., Descent and penalization techniques for equilibrium problems with nonlinear constraint, Journal of Optimization Theory and Applications, 164 (2015), 804–818.

[10] Blum, E. and Oettli. W., From optimization and variational inequalities to equilibrium problems, Mathematics Student, 63(1) (1994), 123–145.

[11] Bresis, H., Nirenberg, L., and Stampachia, G., A remark on Ky Fan’s minimax principle, Bollletino della Unione Matematica Italiana, 6 (1972), 293–300.

[12] Cohen, G., Auxiliary problem principle extended to variational inequalities, Journal of Optimization Theory and Applications, 59 (1988), 325–333.

[13] Cournot, A. A., Recherches sur les principles mathematiques de la theorie des richeness. (English translation by Bacon, N. T.: Researches into the Mathematical Principles of the Theory of Wealth), New York: Macmillan, 1927.

[14] Dinh, B. V., Hung, P. G., and Muu, L. D., Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numerical Functional Analysis and Optimization, 35(5) (2014), 539–563.

[15] Dinh, B. V. and Kim, D. S., Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space, Journal of Computational and Applied Mathematics, 302 (2016), 106–117.

[16] Duc, P. M., Muu, L. D., and Quy, N. V., Solution existence and algorithms with their convergence rate for strongly pseudomonotone equilibrium problems, Pacific Journal of Optimization, 12(4) (2016), 833–845.

[17] Duc, P. M. and Le, X. T., A splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions and application to CournotNash model, RAIRO Operations Research, 55 (2021), S1395–S1410.

[18] Facchinei, F. and Pang, J. S., Finite Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin, 2002.

[19] Fan, K., A minimax inequality and applications, pages 103–113 in Shisha, O.(editor), Inequalities, Academic Press, New York, 1972.

[20] Fukushima, M., A class of gap functions for quasi-variational inequality problems, Journal of Industrial and Management Optimization, 3(2), (2007), 165–

174.

[21] Hieu, D. V., Strodiot, J. J., and Muu, L. D., Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems, Journal

of Computational and Applied Mathematics, 376 (2020), 112844.

[22] Hung, P. G. and Muu, L. D., The Tikhonov regularization extended to equilibrium problem involving pseudomonotne bifunctions, Nonlinear Analysis: Theory, Methods & Applications, 74(17) (2011), 6121–6129.

[23] Hung, P. G. and Muu, L. D., On inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problem, Vietnam Journal of

Mathematics, 40 (2012), 255–274.

[24] Kassay, G. and Radulescu, V. D., ˇ Equilibrium Problems and Applications, Academic Press, 2018.

[25] Konnov, I., Combined Relaxation Methods for Variational Inequalities, Springer, New York, 2001.

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

[27] Mastroeni, G., Gap functions for equilibrium problems, Journal of GlobalOptimization, 27 (2004), 411–426.

[28] Moudafi, A., Proximal methods for a class of bilevel monotone equilibrium problems, Journal of Global Optimization, 47 (2010), 287–292.

[29] Muu, L. D., Stability property of a class of variational inequalities, Optimization, 15(3) (1984), 347–351.

[30] Muu, L. D. and Le, X. T., On fixed point approach to equilibrium problem, Journal of Fixed Point Theory and Applications, 23 (2021), 50.

[31] Muu, L. D., Nguyen, V. H., and Quy, N. V., On Nash-Cournot oligopolistic market models with concave cost functions, Journal of Global Optimization, 41 (2008), 351–364.

[32] Muu, L. D. and Oettli, W., Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis, 18(12) (1992), 1159–1166.

[33] Muu, L. D. and Quoc, T. D., Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, Journal of Optimization Theory and Applications, 142 (2009), 185–204.

[34] Muu, L. D. and Quoc, T. D., One step from DC optimization to DC mixed variational inequalities, Optimization, 59(1) (2010), 63–76.

[35] Muu, L. D. and Quy, N. V., Global optimization from concave minimization to concave mixed variational inequality, Acta Mathematica Vietnamica, 45 (2020), 449–462.

[36] Nguyen, T. T. V., Strodiot, J. J., and Nguyen, V. H., A bundle method forsolving equilibrium problems, Mathematical Programming, 116 (2009), 529–552.

[37] Nikaido, H. and Isoda, K., Note on noncooperative convex games, Pacific Journal of Mathematics, 5(5) (1955), 807–815.

[38] Pham, K. A. and Hai, T. N., Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems, Numerical Algorithms, 76 (2017), 67–91.

[39] Quoc, T. D., Anh, P. N., and Muu, L. D., Dual extragradient algorithms extended to equilibrium problems, Journal of Global Optimization, 52 (2012),139–159.

[40] Quoc, T. D. and Muu, L. D., Iterative methods for solving monotone equilibrium problems via dual gap functions, Computational Optimization and Applications, 51 (2012), 709–718.

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

[42] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14(5) (1976), 877–899.

[43] Scarf, H. E., The approximation of fixed points of a continuous mapping, SIAM Journal on Applied Mathematics, 15(5) (1967), 1328–1343.

[44] Vinh, N. T. and Muu, L. D., Inertial extragradient algorithms for solving equilibrium problems, Acta Mathematica Vietnamica, 44 (2019), 639–663.

[45] Yen, L. H. and Muu, L. D., A subgradient method for equilibrium problems involving quasiconvex bifunctions, Operations Research Letters, 48(5) (2020),579–583.