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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2020. Vol. 32

An Exact Penalty Approach and Conjugate Duality for Generalized Nash Equilibrium Problems with Coupling and Shared Constraints

Author(s)

L. Altangerel, G. Battur

Abstract

Generalized Nash Equilibrium Problems (GNEP) have been attracted by many researchers in the field of game theory, operational research, engineering, economics as well as telecommunication in recent two decades. One of the most important classes of GNEP is a convex GNEP with jointly convex or shared constraints which has been studied extensively. It is considered to be one of the most challenging classes of problems in the field. Moreover, there is a gap in the studies on the GNEP with coupling and shared constraints. The aim of this paper is to investigate the relationship between an exact penalty approach and conjugate duality in convex optimization for the GNEP with coupling and shared constraints. In association with necessary optimality conditions, we obtained the parameterized variational inequality problems. This problem has provided an opportunity to solve many other GNEs. Some numerical results are also presented.

About the Authors

Lkhamsuren Altangerel, Prof., Faculty of Mathematics, Computer and Natural Sciences, German-Mongolian Institute for Resources and Technology, Nalaikh, Mongolia, tel.: 976-99031692, e-mail: altangerel@gmit.edu.mn

Gompil Battur, Assoc. Prof., Center of Mathematics for Applications and Department of Applied Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia, tel.: 976-91017812, e-mail: battur@seas.num.edu.mn

For citation

Altangerel L., Battur G. An Exact Penalty Approach and Conjugate Duality for Generalized Nash Equilibrium Problems with Coupling and Shared Constraints. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 32, pp. 3-16. https://doi.org/10.26516/1997-7670.2020.32.3

Keywords
generalized Nash equilibrium problems, exact penalty function, conjugate duality, coupling and shared constraints
UDC
518.517
MSC
91A10, 90C46
DOI
https://doi.org/10.26516/1997-7670.2020.32.3
References
  1. Altangerel L., Battur G. Perturbation approach to generalized Nash equilibrium problems with shared constraints. Optimization Letters, 2012, vol. 6, pp. 1379-1391. https://doi.org/10.1007/s11590-012-0510-8
  2. Bot R.I. Duality and optimality in multiobjective optimization. Faculty of Mathematics, Ph.D. Dissertation, Chemnitz University of Technology, 2003.
  3. Bot R.I., Wanka G. The conjugate of the pointwise maximum of two convex functions revisited. Journal of Global Optimization, 2008, vol. 41, no. 4, pp. 625-632. https://doi.org/10.1007/s10898-008-9282-1
  4. Facchinei F., Kanzow C. Generalized Nash equilibrium problems. Annals of Operations Research, 2010, vol. 175, no. 1, pp. 177-211. https://doi.org/10.1007/s10479-009-0653-x
  5. Facchinei F., Kanzow C. Penalty methods for the solution of generalized Nash equilibrium problems. SIAM Journal on Optimization, 2010, no. 5, pp. 2228-2253. https://doi.org/10.1137/090749499
  6. Facchinei F., Pang J.-S. Exact penalty functions for generalized Nash problems. Di Pillo, G., Roma, M.(eds.) Large scale nonlinear optimization, pp. 115-126. New York, Springer, 2006. https://doi.org/10.1007/0-387-30065-1_8
  7. Facchinei F., Lampariello L. Partial penalization for the solution of generalized Nash equilibrium problems. Journal of Global Optimization, 2011, vol. 50, pp. 39-57.
  8. Facchinei F., Pang J.-S. Finite-Dimensional Variational Inequalities and Complementarity Problems. New York, Springer Publ., 2003.
  9. Fukushima M. Restricted generalized Nash equilibria and controlled penalty algorithm. Technical Report 2008-007, Department of Applied Mathematics and Physics, Kyoto University, July 2008.
  10. Fukushima M., Pang J.-S. Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput.Manag.Sci., 2005, vol. 2, pp. 21-56.
  11. Kubota K., Fukushima M. Gap Function Approach to the Generalized Nash Equilibrium Problem. Journal of Optimization Theory and Applications, 2010, vol. 144, no. 3, pp. 511-531.
  12. Harker P.T. Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res.,1991, vol. 54, pp. 81-94. https://doi.org/10.1016/0377-2217(91)90325-P
  13. von Heusinger A., Kanzow C. Relaxation methods for generalized Nash equilibrium problems with inexact line search. Journal of Optimization Theory and Applications, 2009, vol. 143, no. 1, pp. 159-183. https://doi.org/10.1007/s10957-009-9553-0
  14. Nabetani K., Tseng P., Fukushima M. Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Computational Optimization and Applications, 2011, vol. 48, pp. 423-452. https://doi.org/10.1007/s10589-009-9256-3
  15. Wei J.-Y., Smeers Y. Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res.,1999, vol. 47, pp. 102-112. https://doi.org/10.1287/opre.47.1.102

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