Abstract
In this paper, we propose a new algorithm for solving bilevel variational inequalities. We consider a dynamical system and prove that the trajectory of this dynamical system converges to a desired solution.
Similar content being viewed by others
References
Anh, T.V., Muu, L.D.: A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1229–1243 (2016)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Banert, S., Bot, R.I.: A forward-backward-forward differential equation and its asymptotic properties. J. Conv. Anal. 25(2), 371–388 (2018)
Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46, 247–263 (2010)
Cavazzuti, E., Pappalardo, M., Passacantando, M.: Nash equilibria, variational inequalities, and dynamical systems. J. Optim. Theory Appl. 114, 491–506 (2002)
Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems. Springer, New York (2003)
Ha, N.T.T., Strodiot, J.J., Vuong, P.T.: On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities. Opt. Lett. 12, 1625–1638 (2018)
Haraux, A.: Systèmes Dynamiques Dissipatifs et applications. Masson, Recherches en Mathématiques Appliquées (1991)
Hieu, D.V., Moudafi, A.: Regularization projection method for solving bilevel variational inequality problem. Optim. Lett. 15, 205–229 (2021)
Hieu, D.V., Strodiot, J.J., Muu, L.D.: An explicit extragradient algorithm for solving variational inequalities. J. Optim. Theory Appl. 185, 476–503 (2020)
Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)
Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1–18 (2009)
Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341–350 (2014)
Kim, J.K., Anh, P.N., Hai, T.N.: The Brucks ergodic iteration method for the Ky Fan inequality over the fixed point set. Int. J. Comput. Math. 94, 2466–2480 (2017)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)
Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2000)
Maingé, P.E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010)
Nagurney, A., Zhang, D.: Projected dynamical systems and variational inequalities with applications. Kluwer Academic, Dordrecht (1996)
Pappalardo, M., Passacantando, M.: Stability for equilibrium problems: from variational inequalities to dynamical systems. J. Optim. Theory Appl. 113, 567–582 (2002)
Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, New Jersey (1970)
Thuy, L.Q., Hai, T.N.: A projected subgradient algorithm for bilevel equilibrium problems and applications. J. Optim. Theory Appl. 175, 411–431 (2017)
Vuong, P.T., Strodiot, J.J.: A dynamical system for strongly pseudo-monotone equilibrium problems. J. Optim. Theory Appl. 185, 767–784 (2020)
Vuong, P.T.: The global exponential stability of a dynamical system for solving variational inequalities. Netw. Spatial Econ. (2019). https://doi.org/10.1007/s11067-019-09457-6
Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)
Acknowledgements
The authors would like to thanks the referee for valuable remarks and comments which improved the quality of the paper. This work was supported by Vietnam Ministry of Education and Training under grant number B2020-BKA-21-CTTH. This work is also supported by Vietnam Institute for Advanced Study in Mathematics (VIASM).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Anh, P.K., Hai, T.N. Dynamical system for solving bilevel variational inequalities. J Glob Optim 80, 945–963 (2021). https://doi.org/10.1007/s10898-021-01029-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-021-01029-8