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About the Links Between Equilibrium Problems and Variational Inequalities

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Mathematical Programming and Game Theory

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Abstract

In this chapter, we seek to study the interrelation between an equilibrium problem and the variational inequality problem. Under most natural assumption, the equilibrium problem is equivalent to an associated variational inequality. Hence, the existence results for equilibrium problems can be obtained from the existence results for variational inequality problems and vice versa. We study a problem of existence of Nash equilibrium in an oligopolistic market and show that it is equivalent to a variational inequality under the most natural economic assumption. We also study the relation between quasi-equilibrium problem and quasi-variational inequality.

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References

  1. Aussel, D.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)

    Article  MathSciNet  Google Scholar 

  2. Aussel, D., Ye, J.: Quasiconvex minimization on locally finite union of convex sets. J. Optim. Theory Appl. 139, 1–16 (2008)

    Article  MathSciNet  Google Scholar 

  3. Le, D., Muu, D., Nguyen, V.H., Quy, N.V.: On Nash Cournot oligopolistic market equilibrium models with concave cost functions. J. Glob. Optim. 41, 351–364 (2008)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44, 175–192 (2009)

    Article  MathSciNet  Google Scholar 

  8. Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)

    Article  MathSciNet  Google Scholar 

  9. Dinh, N., Strodiot, J.J., Nguyen, V.H.: Duality and optimality conditions for generalized equilibrium problems involving DC functions. J. Glob. Optim. 48, 183–208 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Charitha, C.: A note on D-gap functions for equilibrium problems. Optimization 62, 211–226 (2013)

    Article  MathSciNet  Google Scholar 

  12. Konnov, I.V.: On penalty methods for non monotone equilibrium problems. J. Glob. Optim. 59, 131–138 (2014)

    Article  MathSciNet  Google Scholar 

  13. Anh, P.N., Hai, T.N., Tuan, P.M.: On ergodic algorithms for equilibrium problems. J. Glob. Optim. 64, 179–195 (2016)

    Article  MathSciNet  Google Scholar 

  14. Eaves, B.C.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971)

    Article  MathSciNet  Google Scholar 

  15. Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 153–188 (1966)

    Article  Google Scholar 

  16. Aussel, D.: Quasimonotone quasivariational inequalities: existence results and applications. J. Optim. Theory Appl. 158, 637–652 (2013)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. I. Springer, New York (2003)

    MATH  Google Scholar 

  20. Dhara, A., Dutta, J.: Optimality Conditions in Convex Optimization: A finite-Dimensional View. With a Foreword by Stephan Dempe. CRC Press, Boca Raton (2012)

    MATH  Google Scholar 

  21. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  22. Borwein, J.M., Dutta, J.: Maximal monotone inclusions and Fitzpatrick functions. J. Optim. Theory Appl. 171, 757–784 (2016)

    Article  MathSciNet  Google Scholar 

  23. Borwein, J.M., Lewis, A.S.: Convex analysis and nonlinear optimization. Theory and examples. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3, Second edition edn. Springer, New York (2006)

    Google Scholar 

  24. Chan, D., Pang, J.S.: The generalized quasivariational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  MathSciNet  Google Scholar 

  25. Bianchi, M., Pini, R.: A note on equilibrium problems with properly quasimonotone bifunctions. J. Glob. Optim. 20, 67–76 (2001)

    Article  MathSciNet  Google Scholar 

  26. Nasri, M., Sosa, W.: Equilibrium problems and generalized Nash games. Optimization 60, 1161–1170 (2011)

    Article  MathSciNet  Google Scholar 

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Aussel, D., Dutta, J., Pandit, T. (2018). About the Links Between Equilibrium Problems and Variational Inequalities. In: Neogy, S., Bapat, R., Dubey, D. (eds) Mathematical Programming and Game Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-3059-9_6

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