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

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Abstract

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.

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References

  1. Cavazzuti, E., Pappalardo, M., Passacantando, M.: Nash equilibria, variational inequalities, and dynamical systems. J. Optim. Theory Appl. 114, 491–506 (2002)

    Article  MathSciNet  Google Scholar 

  2. Facchinei, F., Pang, S.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I, II. Springer, New York (2003)

    MATH  Google Scholar 

  3. Friesz, T.L.: Dynamic Optimization and Differential Games. Springer, New York (2010)

    Book  Google Scholar 

  4. Hopfield, J.J., Tank, D.W.: Neural computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  8. Jiang, S., Han, D., Yuan, X.: Efficient neural networks for solving variational inequalities. Neurocomputing 86, 97–106 (2012)

    Article  Google Scholar 

  9. Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)

    Article  MathSciNet  Google Scholar 

  10. Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341–350 (2014)

    Article  MathSciNet  Google Scholar 

  11. Kim, D.S., Vuong, P.T., Khanh, P.D.: Qualitative properties of strongly pseudomonotone variational inequalities. Opt. Lett. 10, 1669–1679 (2016)

    Article  MathSciNet  Google Scholar 

  12. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic, New York (1980)

    MATH  Google Scholar 

  13. Konnov, I.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  14. Kosko, B.: Neural Networks for Signal Processing. Prentice-Hall, Englewood Cliffs, NJ (1992)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  17. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)

    Article  MathSciNet  Google Scholar 

  18. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic, Dordrecht (1996)

    Book  Google Scholar 

  19. Pappalardo, M., Passacantando, M.: Stability for equilibrium problems: from variational inequalities to dynamical systems. J. Optim. Theory Appl. 113, 567–582 (2002)

    Article  MathSciNet  Google Scholar 

  20. Salmon, G., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14, 869–893 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  23. Xia, Y., Wang, J.: A general methodology for designing globally convergent optimization neural networks. IEEE Trans. Neural Netw. 9, 1331–1343 (1998)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  26. Yan, Z., Wang, J., Li, G.: A collective neurodynamic optimization approach to bound-constrained nonconvex optimization. Neural Netw. 55, 20–29 (2014)

    Article  Google Scholar 

  27. Yoshikawa, T.: Foundations of Robotics: Analysis and Control. MIT Press, Cambridge, MA (1990)

    Google Scholar 

Download references

Acknowledgements

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.

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Authors and Affiliations

  1. Ho Chi Minh City University of Technology and Education, Thu Duc, Ho Chi Minh City, Vietnam

    Nguyen Thi Thu Ha

  2. University of Namur, Namur, Belgium

    J. J. Strodiot

  3. Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria

    Phan Tu Vuong

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  1. Nguyen Thi Thu Ha
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  2. J. J. Strodiot
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  3. Phan Tu Vuong
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Correspondence to J. J. Strodiot.

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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. Optim Lett 12, 1625–1638 (2018). https://doi.org/10.1007/s11590-018-1230-5

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  • DOI: https://doi.org/10.1007/s11590-018-1230-5

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