Discrete-time projection neural network methods for computing the solution of variational inequalities
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Authors
Liping Zhang
- School of Science, Sichuan University of Science and Engineering, Zigong 643000, China.
Shu-Lin Wu
- School of Science, Sichuan University of Science and Engineering, Zigong 643000, China.
Abstract
Neural networks are useful tools to solve mathematical and engineering problems. By using the implicit-explicit-method and the method proposed recently by Mohamad to discretize the continuous-time neural networks, we formulate two classes of
discrete-time analogues to solve a system of variational inequalities. By adopting suitable Lyapunov functions and Razumikhintype
techniques, exponential stability of the discrete neural networks are established in terms of linear matrix inequalities (LMIs).
Several numerical experiments are performed to compare the convergence rates of the proposed discrete neural networks and it
is shown that:
(a) all of the discrete neural networks converge faster as the step size becomes larger,
(b) the discrete neural networks derived by the semi-implicit Euler method performs best.
Share and Cite
ISRP Style
Liping Zhang, Shu-Lin Wu, Discrete-time projection neural network methods for computing the solution of variational inequalities, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1896--1907
AMA Style
Zhang Liping, Wu Shu-Lin, Discrete-time projection neural network methods for computing the solution of variational inequalities. J. Nonlinear Sci. Appl. (2017); 10(4):1896--1907
Chicago/Turabian Style
Zhang, Liping, Wu, Shu-Lin. "Discrete-time projection neural network methods for computing the solution of variational inequalities." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1896--1907
Keywords
- Neural networks
- linear matrix inequalities (LMIs)
- variational inequalities
- discretization.
MSC
References
-
[1]
A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers] , New York-London (1979)
-
[2]
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994)
-
[3]
W.-H. Chen, X.-M. Lu, D.-Y. Liang, Global exponential stability for discrete-time neural networks with variable delays, Phys. Lett. A, 358 (2006), 186–198.
-
[4]
F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Vol. I and II, Springer Series in Operations Research, Springer-Verlag, New York (2003)
-
[5]
K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput., 154 (2004), 783–813.
-
[6]
P. T. Harker, J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 48 (1990), 161–220.
-
[7]
X.-L. Hu, J. Wang, Solving generally constrained generalized linear variational inequalities using the general projection neural networks, IEEE Trans. Neural Netw., 18 (2007), 1697–1708.
-
[8]
Z.-K. Huang, X.-H.Wang, F. Gao, The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A, 350 (2006), 182–191.
-
[9]
T. Koto, Stability of IMEX Runge-Kutta methods for delay differential equations, J. Comput. Appl. Math., 211 (2008), 201–212.
-
[10]
H.-Y. Lan, Y.-S. Cui, A neural network method for solving a system of linear variational inequalities, Chaos Solitons Fractals, 41 (2009), 1245–1252.
-
[11]
J.-L. Liang, J.-D. Cao, Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays, Chaos Solitons Fractals, 22 (2004), 773–785.
-
[12]
Q.-S. Liu, J. Wang, A projection neural network for constrained quadratic minimax optimization, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2891–2900.
-
[13]
T.-T. Liu, X. Zhang, X.-Y. Gao, Stability analysis for continuous-time and discrete-time genetic regulatory networks with delays, Appl. Math. Comput., 274 (2016), 628–643.
-
[14]
S. Mohamad, Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks, Phys. D, 159 (2001), 233–251.
-
[15]
S. Mohamad, Global exponential stability in discrete-time analogues of delayed cellular neural networks, J. Difference Equ. Appl., 9 (2003), 559–575.
-
[16]
S. Mohamad, Exponential stability preservation in discrete-time analogues of artificial neural networks with distributed delays, J. Comput. Appl. Math., 215 (2008), 270–287.
-
[17]
S. Mohamad, K. Gopalsamy, Dynamics of a class of discrete-time neural networks and their continuous-time counterparts, Math. Comput. Simulation, 53 (2000), 1–39.
-
[18]
S. Mohamad, K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17–38.
-
[19]
S. Mohamad, A. G. Naim, Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks, J. Comput. Appl. Math., 138 (2002), 1–20.
-
[20]
C.-L. Sha, H.-Y. Zhao, T.-W. Huang, W. Hu, A projection neural network with time delays for solving linear variational inequality problems and its applications, Circuits Syst. Signal Process., 35 (2016), 2789–2809.
-
[21]
C.-Y. Sun, C.-B. Feng, Exponential periodicity of continuous-time and discrete-time neural networks with delays, Neural Process. Lett., 19 (2004), 131–146.
-
[22]
C.-Y. Sun, C.-B. Feng, Discrete-time analogues of integrodifferential equations modeling neural networks, Phys. Lett. A, 334 (2005), 180–191.
-
[23]
L.-Y. Wu, L. Ju, L. Guo, Dynamics of continuous-time neural networks and their discrete-time analogues with distributed delays, Adv. Neural Netw., 4491 (2007), 1054–1060.
-
[24]
Y.-S. Xia, Further results on global convergence and stability of globally projected dynamical systems, J. Optim. Theory Appl., 122 (2004), 627–649.
-
[25]
Y.-S. Xia, J. Wang, A general projection neural network for solving monotone variational inequalities and related optimization problems, IEEE Trans. Neural Netw., 15 (2004), 318–328.
-
[26]
R.-J. Yang, B. Wu, Y. Liu, A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays, Appl. Math. Comput., 265 (2015), 696–707.
-
[27]
Z.-G. Zeng, X.-X. Liao, The solution to linear variational inequality by neural network, (chinese) J. HuaZhong Univ. Sci. Technol. Nat. Sci., 20 (2002), 18–20.
-
[28]
H. Zhang, L.-S. Chen, Asymptotic behavior of discrete solutions to delayed neural networks with impulses, Neurocomputing, 71 (2008), 1032–1038.