To solve a special case of the extended vertical linear complementarity problem (EVLCP), we propose the artificial neural network. The model is based on reconstructing the generalized vertical linear complementarity problem as an unconstrained minimization problem. In theory, we analyze the existence of equilibrium point of the model and prove that if the equilibrium point of the neural network exists, it has asymptotic stability for any initial state. By using Lyapunov function, the neural network model has Lyapunov stability. Finally, some numerical simulation results are given.

neural network, extended vertical linear complementarity problem, stability.

Received: August 2, 2021; Accepted: August 29, 2021; Published: November 10, 2021

How to cite this article: Jie Zhang, Chen Qiu and Bin Hou, A neural network for a special case of extended vertical linear complementarity problem, Far East Journal of Applied Mathematics 110(2) (2021), 91-98. DOI: 10.17654/AM110020091

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References

[1] L. O. Chua and G. N. Lin, Nonlinear programming without computation, IEEE Trans. Circuits Syst. 31 (1984), 182-188.
[2] J. Zhang, W. B. Shan and N. Shi, A smoothing SAA method for a special case of stochastic generalized vertical linear complementary problem, Journal of Liaoning Normal University 4(30) (2017), 301-306.
[3] M. S. Gowda and R. Sznajder, The generalized order linear complementarity problem, SIAM J. Matrix Anal. Appl. 15 (1994), 779-795.
[4] S. Effati, A. Ghomashi and A. R. Nazemi, Application of projection neural network in solving convex programming problems, Appl. Math. Comput. 188(2) (2007), 1103-1114.
[5] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
[6] W. E. Lillo, M. H. Loh, S. Hui and S. H. Zak, On solving constrained optimization problems with neural networks: a penalty method approach, IEEE Trans. Neutal Networks 4 (1993), 931-940.
[7] T. D. Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Program. 75 (1996), 407-439.
[8] L. Z. Liao, H. D. Qi and L. Qi, Solving nonlinear complementarity problems with neural networks: a reformulation method approach, J. Comput. Appl. Math. 131(1-2) (2001), 343-359.