Optimization Letters

, Volume 12, Issue 7, pp 1625–1638 | Cite as

On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities

  • Nguyen Thi Thu Ha
  • J. J. StrodiotEmail author
  • Phan Tu Vuong
Original Paper


We investigate the global exponential stability of equilibrium solutions of a projected dynamical system for variational inequalities. Under strong pseudomonotonicity and Lipschitz continuity assumptions, we prove that the dynamical system has a unique equilibrium solution. Moreover, this solution is globally exponentially stable. Some examples are given to analyze the effectiveness of the theoretical results. The numerical results confirm that the trajectory of the dynamical system globally exponentially converges to the unique solution of the considered variational inequality. The results established in this paper improve and extend some recent works.


Global exponential stability Projected dynamical system Strong pseudomonotonicity Variational inequality 



The authors would like to thank the Editor and the anonymous referee for their useful comments. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.01-2017.315 and the Austrian Science Foundation (FWF), grant P26640-N25. Support provided by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST) is also gratefully acknowledged.


  1. 1.
    Cavazzuti, E., Pappalardo, M., Passacantando, M.: Nash equilibria, variational inequalities, and dynamical systems. J. Optim. Theory Appl. 114, 491–506 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Facchinei, F., Pang, S.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I, II. Springer, New York (2003)zbMATHGoogle Scholar
  3. 3.
    Friesz, T.L.: Dynamic Optimization and Differential Games. Springer, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Hopfield, J.J., Tank, D.W.: Neural computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985)zbMATHGoogle Scholar
  5. 5.
    Huang, B., Zhang, H., Gong, D., Wang, Z.: A new result for projection neural networks to solve linear variational inequalities and related optimization problems. Neural Comput. Appl. 23, 357–362 (2013)CrossRefGoogle Scholar
  6. 6.
    Hu, X., Wang, J.: Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17, 1487–1499 (2006)CrossRefGoogle Scholar
  7. 7.
    Hu, X., Wang, J.: Global stability of a recurrent neural network for solving pseudomonotone variational inequalities. In: Proceedings of IEEE International Symposium on Circuits and Systems, Island of Kos, Greece, May 21–24, pp. 755–758 (2006)Google Scholar
  8. 8.
    Jiang, S., Han, D., Yuan, X.: Efficient neural networks for solving variational inequalities. Neurocomputing 86, 97–106 (2012)CrossRefGoogle Scholar
  9. 9.
    Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341–350 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kim, D.S., Vuong, P.T., Khanh, P.D.: Qualitative properties of strongly pseudomonotone variational inequalities. Opt. Lett. 10, 1669–1679 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic, New York (1980)zbMATHGoogle Scholar
  13. 13.
    Konnov, I.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  14. 14.
    Kosko, B.: Neural Networks for Signal Processing. Prentice-Hall, Englewood Cliffs, NJ (1992)zbMATHGoogle Scholar
  15. 15.
    Liu, Q., Cao, J.: A recurrent neural network based on projection operator for extended general variational inequalities. IEEE Trans. Syst. Man Cybern. B Cybern. 40, 928–938 (2010)CrossRefGoogle Scholar
  16. 16.
    Liu, Q., Yang, Y.: Global exponential system of projection neural networks for system of generalized variational inequalities and related nonlinear minimax problems. Neurocomputing 73, 2069–2076 (2010)CrossRefGoogle Scholar
  17. 17.
    Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic, Dordrecht (1996)CrossRefGoogle Scholar
  19. 19.
    Pappalardo, M., Passacantando, M.: Stability for equilibrium problems: from variational inequalities to dynamical systems. J. Optim. Theory Appl. 113, 567–582 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Salmon, G., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14, 869–893 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tank, D.W., Hopfield, J.J.: Simple neural optimization networks: an A/D converter, and a linear programming circuit. IEEE Trans. Circuits Syst. 33, 533–541 (1986)CrossRefGoogle Scholar
  22. 22.
    Xia, Y., Leung, H., Wang, J.: A projection neural network and its application to constrained optimization problems. IEEE Trans. Circuits Syst. I Reg. Papers 49, 447–458 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Xia, Y., Wang, J.: A general methodology for designing globally convergent optimization neural networks. IEEE Trans. Neural Netw. 9, 1331–1343 (1998)CrossRefGoogle Scholar
  24. 24.
    Xia, Y., Wang, J.: Global exponential stability of recurrent neural networks for solving optimization and related problems. IEEE Trans. Neural Netw. 11, 1017–1022 (2000)CrossRefGoogle Scholar
  25. 25.
    Xia, Y., Wang, J.: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans. Neural Netw. 15, 318–328 (2004)CrossRefGoogle Scholar
  26. 26.
    Yan, Z., Wang, J., Li, G.: A collective neurodynamic optimization approach to bound-constrained nonconvex optimization. Neural Netw. 55, 20–29 (2014)CrossRefGoogle Scholar
  27. 27.
    Yoshikawa, T.: Foundations of Robotics: Analysis and Control. MIT Press, Cambridge, MA (1990)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ho Chi Minh City University of Technology and EducationThu Duc, Ho Chi Minh CityVietnam
  2. 2.University of NamurNamurBelgium
  3. 3.Institute of Statistics and Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria

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