Skip to main content
Log in

Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems

Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The globally projected dynamical system has received considerable attention due to its low complexity for variational inequality and optimization computation. This paper obtains further results on the global convergence, asymptotic stability, and exponential stability of this system, respectively under monotonicity of the mapping, strict monotonicity of the mapping, and positive definiteness of the Jacobian matrix of the mapping. The new results obtained improve existing ones and cover the classical stability results of autonomous dynamical systems as special cases. An application to constrained optimization and complementarity problems is given to show the applied significance of the results obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Praprost, K. L., and Loparo, K. A., A Stability Theory for Constrained Dynamic Systems with Applications to Electric Power Systems, IEEE Transactions on Automatic Control, Vol. 41, pp. 1605-1617, 1996.

    Google Scholar 

  2. Xia, Y. S., and Wang, J., On the Stability of Globally Projected Dynamical Systems, Journal of Optimization Theory and Applications, Vol. 106, pp. 129-150, 2000.

    Google Scholar 

  3. Kinderlehrer, D., and Stampcchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.

    Google Scholar 

  4. Yamashita, H., A Differential Equation Approach to Nonlinear Programming, Mathematical Programming, Vol. 18, pp. 155-168, 1980.

    Google Scholar 

  5. Hirsch, M. W., and Smale, S., On Algorithms for Solving f(x)=0, Communications on Pure and Applied Mathematics, Vol. 55, pp. 1-12, 1979.

    Google Scholar 

  6. Kloeden, P. E., and Lorenz, J., Stable Attracting Sets in Dynamical Systems and Their One-Step Discretizations, SIAM Journal on Numerical Analysis, Vol. 23, pp. 987-995, 1986.

    Google Scholar 

  7. Dupuis, P., and Nagurney, A., Dynamical Systems and Variational Inequalities, Annals of Operations Research, Vol. 44, pp. 19-42, 1993.

    Google Scholar 

  8. Friesz, T. L., Bernstein, D. H., Mehta, N. J., Tobin, R. L., and Ganjlizadeh, S., Day-to-Day Dynamical Network Disequilibria and Idealized Traveler Information Systems, Operations Research, Vol. 42, pp. 1120-1136, 1994.

    Google Scholar 

  9. Friesz, T. L., Bernstein, D. H., and Stough, R., Dynamic Systems, Variational Inequalities, and Control Theoretic Models for Predicting Time-Varying Urban Network Flows, Transportation Science, Vol. 30, pp. 14-31, 1996.

    Google Scholar 

  10. Pappalardo, M., and Passacantando, M., Stability for Equilibrium Problems: From Variational Inequalities to Dynamical Systems, Journal of Optimization Theory and Applications, Vol. 113, pp. 567-582, 2002.

    Google Scholar 

  11. Cichocki, A., and Unbehauen, R., Neural Networks for Optimization and Signal Processing, John Wiley, Chichester, England, 1993.

    Google Scholar 

  12. Xia, Y. S., A New Neural Network for Solving Linear and Quadratic Programming Problems, IEEE Transactions on Neural Networks, Vol. 7, pp. 1544-1547, 1996.

    Google Scholar 

  13. Xia, Y. S., and Wang, J., A Recurrent Neural Network for Solving Linear Projection Equations, Neural Networks, Vol. 13, pp. 337-350, 2000.

    Google Scholar 

  14. Xia, Y. S., and Wang, J., Global Asymptotic and Exponential Stability of a Class of Dynamical Neural Systems with Asymmetric Connection Weights, IEEE Transactions on Automatic Control, Vol. 46, pp. 635-638, 2001.

    Google Scholar 

  15. Xia, Y. S., Leung, H., and Wang, J., A Projection Neural Network and Its Application to Constrained Optimization Problems, IEEE Transaction on Circuits and Systems, Part I, Vol. 49, pp. 447-458, 2002.

    Google Scholar 

  16. Fukushima, M., Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99-110, 1992.

    Google Scholar 

  17. Hirsch, M. W., and Smale, S., Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, London, England, 1974.

    Google Scholar 

  18. Miller, R. K., and Michel, A. N., Ordinary Differential Equations, Academic Press, New York, NY, 1982.

    Google Scholar 

  19. Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming:Theory and Algorithms, 2nd Edition, John Wiley, New York, NY, 1993.

    Google Scholar 

  20. Ferris, M. C., and Bang, J. S., Engineering and Economic Applications of Complementarity Problems, SIAM Review, Vol. 39, pp. 669-713, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, Y.S. Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems. Journal of Optimization Theory and Applications 122, 627–649 (2004). https://doi.org/10.1023/B:JOTA.0000042598.21226.af

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTA.0000042598.21226.af

Navigation