Abstract
A quasi-variational inequality is a variational inequality, in which the constraint set is depending on the variable. However, as shown by a motivating example in electricity market, the constraint map may not be a self-map, and then, there is usually no solution. Thus, we define the concept of projected solution and, based on a fixed point theorem, we establish some results on existence of projected solution for quasi-variational inequality problem in a finite-dimensional space where the constraint map is not necessarily self-map. As an application of our results, we obtain an existence theorem for quasi-optimization problems. Finally, we introduce the concept of projected Nash equilibrium and study the existence of such equilibrium for noncooperative games as another application.
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References
Kien, B.T., Wong, N.C., Yao, J.C.: On the solution existence of generalized quasivariational inequalities with discontinuous multifunctions. J. Optim. Theory Appl. 135, 515–530 (2007)
Bhattacharyya, P., Vetrivel, V.: An existence theorem on generalized quasi-variational inequality problem. J. Math. Anal. Appl. 188, 610–615 (1994)
Tan, N.X.: Quasi-variational inequality in topological linear locally convex Hausdorff spaces. Math. Nachr. 122, 231–245 (1985)
Lassonde, M.: Fixed points for Kakutani factorizable multifunctions. J. Math. Anal. Appl. 152, 46–60 (1990)
Aussel, D., Cotrina, J.: Quasimonotone quasivariational inequalities: existence results and applications. J. Optim. Theory Appl. 158, 637–652 (2013)
Aussel, D., Cervinka, M., Marechal, M.: Deregulated electricity markets with thermal losses and production bounds: models and optimality conditions. RAIRO-Oper. Res. 50, 19–38 (2015)
Aussel, D., Bendotti, P., Pištěk, M.: Nash Equilibrium in a Pay-as-bid Electricity Market: Part 1—Existence and Characterisation. Prepint (2015)
Aussel, D., Bendotti, P., Pištěk, M.: Nash Equilibrium in a Pay-as-bid Electricity Market Part 2—Best Response of a Producer. Prepint (2015)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Hadjisavvas, N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 10, 459–469 (2003)
Aussel, D., Hadjisavvas, N.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)
Aussel, D., Cotrina, J.: Stability of quasimonotone variational inequality under sign-continuity. J. Optim. Theory Appl. 158, 653–667 (2013)
Aussel, D., Hadjisavvas, N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121, 445–450 (2004)
Ait Mansour, M., Aussel, D.: Quasimonotone variational inequalities and quasiconvex programming: qualitative stability. J. Convex Anal. 15, 459–472 (2008)
Dorsch, D., Jongen, H.T., Shikhman, V.: On structure and computation of generalized Nash equilibria. SIAM J. Optim. 23, 452–474 (2013)
Dreves, A., Kanzow, C., Stein, O.: Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems. J. Global Optim. 53, 587–614 (2012)
Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)
Aussel, D., Ye, J.J.: Quasiconvex programming with locally starshaped constraint region and applications to quasiconvex MPEC. Optimization 55, 433–457 (2006)
Facchinei, F., Pang, J.C.: Finite-Dimensional Variational Inequalities and Complementarity Problems-Volume I. Springer, New York (2003)
Acknowledgments
The first author would like to thank the research center VIASM, Hanoi, Vietnam, for its hospitality. Indeed, a part of this work has been accomplished while this author was invited professor in this center. The second author acknowledges the University Grants Commission (UGC), India, and the Erasmus Mundus Euphrates fellowship 2015 by Universidade de Santiago de Compostela for providing the financial support for this research work. The final print of the paper was prepared while the second author was at Université de Perpignan Via Domitia, France. The author also thanks Université de Perpignan Via Domitia, France. Moreover, both authors are really grateful to the referees for their valuable comments and suggestions.
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Communicated by Akhtar A. Khan.
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Aussel, D., Sultana, A. & Vetrivel, V. On the Existence of Projected Solutions of Quasi-Variational Inequalities and Generalized Nash Equilibrium Problems. J Optim Theory Appl 170, 818–837 (2016). https://doi.org/10.1007/s10957-016-0951-9
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DOI: https://doi.org/10.1007/s10957-016-0951-9