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On the Stability of Globally Projected Dynamical Systems

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  • Y. S. Xia

    (Chinese University of Hong Kong)

  • J. Wang

    (Chinese University of Hong Kong)

Abstract

Two types of projected dynamical systems, whose equilibrium states solve the corresponding variational inequality problems, were proposed recently by Dupuis and Nagurney (Ref. 1) and by Friesz et al. (Ref. 2). The stability of the dynamical system developed by Dupuis and Nagurney has been studied completely (Ref. 3). This paper analyzes and proves the global asymptotic stability of the dynamical system proposed by Friesz et al. under monotone and symmetric mapping conditions. Furthermore, the dynamical system is shown to be globally exponentially stable under stronger conditions. Finally, we show that the dynamical system proposed by Friesz et al. can be applied easily to neural networks for solving a class of optimization problems.

Suggested Citation

  • Y. S. Xia & J. Wang, 2000. "On the Stability of Globally Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 106(1), pages 129-150, July.
    • Handle: RePEc:spr:joptap:v:106:y:2000:i:1:d:10.1023_a:1004611224835
    DOI: 10.1023/A:1004611224835
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    References listed on IDEAS

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    1. Terry L. Friesz & David Bernstein & Nihal J. Mehta & Roger L. Tobin & Saiid Ganjalizadeh, 1994. "Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems," Operations Research, INFORMS, vol. 42(6), pages 1120-1136, December.
    2. Terry L. Friesz & David Bernstein & Roger Stough, 1996. "Dynamic Systems, Variational Inequalities and Control Theoretic Models for Predicting Time-Varying Urban Network Flows," Transportation Science, INFORMS, vol. 30(1), pages 14-31, February.
    3. Sjur D. Flåm & Adi Ben-Israel, 1990. "A Continuous Approach to Oligopolistic Market Equilibrium," Operations Research, INFORMS, vol. 38(6), pages 1045-1051, December.
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    Citations

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    Cited by:

    1. Wu, Zeng-bao & Zou, Yun-zhi & Huang, Nan-jing, 2016. "A class of global fractional-order projective dynamical systems involving set-valued perturbations," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 23-33.
    2. Ding, Ke & Huang, Nan-Jing, 2008. "A new class of interval projection neural networks for solving interval quadratic program," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 718-725.
    3. Jin-dong Li & Nan-jing Huang, 2018. "Asymptotical Stability for a Class of Complex-Valued Projective Neural Network," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 261-270, April.
    4. Diogo Pinheiro & Alberto A. Pinto & S. Z. Xanthopoulos & A. N. Yannacopoulos, 2011. "A projected gradient dynamical system modeling the dynamics of bargaining," CEMAPRE Working Papers 1101, Centre for Applied Mathematics and Economics (CEMAPRE), School of Economics and Management (ISEG), Technical University of Lisbon.
    5. Szilárd Nagy, 2013. "Stackelberg equilibria via variational inequalities and projections," Journal of Global Optimization, Springer, vol. 57(3), pages 821-828, November.
    6. Liu, Xiaolan & Zhou, Mi, 2016. "A one-layer recurrent neural network for non-smooth convex optimization subject to linear inequality constraints," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 39-46.
    7. Y. S. Xia, 2004. "Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 627-649, September.
    8. M. Pappalardo & M. Passacantando, 2002. "Stability for Equilibrium Problems: From Variational Inequalities to Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 113(3), pages 567-582, June.
    9. Farhat Suhel & S. K. Srivastava & Suhel Ahmad Khan, 2014. "A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-8, April.

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