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Gap functions and error bounds for quasi variational inequalities

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Abstract

The paper aims to obtain new local/global error bounds for quasi variational inequality problems in terms of the regularized gap function and the D-gap function. These bounds provide effective estimated distances between a specific point and the exact solution of quasi variational inequality problem.

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References

  1. Baiocchi C., Capelo A.: Variational and Quasi-variational Inequalities: Apllications to Free Boundary Problems. Wiley, Chichester (1984)

    Google Scholar 

  2. Bliemer M.C.J., Bovy P.H.L.: Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem. Transp. Res. Part: Methodol. 37, 501–519 (2003)

    Article  Google Scholar 

  3. Chan D., Pang J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  Google Scholar 

  4. Donato M.B., Milasi M., Vitanza C.: An existence result of a quasi-variational inequality associated to an equilibrium problem. J. Glob. Optim. 40, 87–97 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Facchinei F., Pang J.S.: Finite Dimensional Vriational Inequalities And Complimentarity Problems, vol. 1. Springer, Berlin (2003)

    Google Scholar 

  7. Facchinei F., Pang J.S.: Finite Dimensional Variational Inequalities And Complimentarity Problems, vol. 2. Springer, Berlin (2003)

    Google Scholar 

  8. Fukushima M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)

    Article  Google Scholar 

  9. Fukushima M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)

    Article  Google Scholar 

  10. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Springer, Berlin (2002)

    Google Scholar 

  11. Harker P.T.: Generalized Nash games and quasivariational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    Article  Google Scholar 

  12. Huang L.R., Ng K.F.: Equivalent optimization formulations and error bounds for variational inequality problems. J. Optim. Theory Appl. 125, 299–314 (2005)

    Article  Google Scholar 

  13. Jianghua, F., Xiaoguo, W.: Global bounds for cocoercive variational inequalities. Abstr. Appl. Anal. 2007, Article ID 37217 (2007). doi:10.1155/2007/37217

  14. Jian W.P., Soon Y.W.: The generalized Tykhonov well-posedness for system of vector quasi equilibrium problem. Optim. Lett. 4, 501–530 (2010)

    Article  Google Scholar 

  15. Kočvara M., Outrata J.V.: On a class of quasi-variational inequalities. Optim. Method. Softw. 5, 275–295 (1995)

    Article  Google Scholar 

  16. Kubota K., Fukushima F.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511–531 (2009)

    Article  Google Scholar 

  17. Li G., Ng K.F.: Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM J. Optim. 20, 667–690 (2009)

    Article  Google Scholar 

  18. Nesterov, Yu., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities, Core Discussion Paper 2006/107 (2006) http://www.core.ucl.ac.be/services/psfiles/dp06/dp2006_107.pdf

  19. Noor M.A.: On merit functions for quasivariational inequalities. J. Math. Inequal. 1, 259–268 (2007)

    Article  Google Scholar 

  20. Noor M.A., Noor K.I., Al-Said E.: Iterative methods for solving general quasi-variational inequalities. Optim. Lett. 4, 513–530 (2010)

    Article  Google Scholar 

  21. Noor M.A., Oettli W.: On genral nonlinear complimenterity problem and quasi equilibria. Le Mathematiche 99, 313–331 (1994)

    Google Scholar 

  22. Pang J.C., Fukushima M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    Article  Google Scholar 

  23. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems in Honor of George Isac, Series: Springer Optimization and Its Application, vol. 35. Springer, Berlin (2010)

    Google Scholar 

  24. Scrimali, L.: A quasi-variational inequality approach to the financial equilibrium problem, Core Discussion Paper 2006/108 (2006) http://www.core.ucl.ac.be/services/psfiles/dp06/dp2006_108.pdf

  25. Solodov M.V.: Merit functions and error bounds for generalized variational inequalities. J. Math. Anal. Appl. 287, 405–414 (2003)

    Article  Google Scholar 

  26. Taji, K.: On gap function for quasi-variational inequalities, Abstr. Appl. Anal. 2008, Article ID 531361 (2008). doi:10.1155/2008/531361; http://www.emis.de/journals/HOA/AAA/Volume2008/531361.pdf

  27. Yamashita N., Fukushima M.: Equivalent unconstrained minimization and global error bounds for variational inequality problems. SIAM J. Control Optim. 35, 273–284 (1997)

    Article  Google Scholar 

  28. Yamashita N., Taji K., Fukushima M.: Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 92, 439–456 (1997)

    Article  Google Scholar 

  29. Yao J.C.: The generalized quasi-variational inequality problem with applications. J. Math. Anal. Appl. 158, 139–160 (1991)

    Article  Google Scholar 

  30. Zhao Y.B., Hu J.: Global bounds for the distance to solutions of co-coercive variational inequalities. Oper. Res. Lett. 35, 409–415 (2007)

    Article  Google Scholar 

  31. Zhao Y.B., Li D.: Monotonicity of fixed point and normal mapping associated with variational inequality and its application. SIAM J. Optim. 11, 962–973 (2001)

    Article  Google Scholar 

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Correspondence to Rachana Gupta.

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The first author acknowledge the Council of Scientific Research (CSIR), India, for providing financial assistance for this research.

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Gupta, R., Mehra, A. Gap functions and error bounds for quasi variational inequalities. J Glob Optim 53, 737–748 (2012). https://doi.org/10.1007/s10898-011-9733-y

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  • DOI: https://doi.org/10.1007/s10898-011-9733-y

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