Journal of Optimization Theory and Applications

, Volume 170, Issue 3, pp 818–837 | Cite as

On the Existence of Projected Solutions of Quasi-Variational Inequalities and Generalized Nash Equilibrium Problems

  • Didier Aussel
  • Asrifa Sultana
  • Vellaichamy Vetrivel


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.


Quasi-variational inequality Generalized Nash equilibrium Non-self map 

Mathematics Subject Classification

49J40 90C26 90B10 



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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Didier Aussel
    • 1
  • Asrifa Sultana
    • 2
  • Vellaichamy Vetrivel
    • 2
  1. 1.Lab. PROMES UPR CNRS 8521Université de Perpignan Via DomitiaPerpignanFrance
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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