ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Further Results on the Stability of Neural Network for Solving Variational Inequalities

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 341--353 | DOI:10.5890/DNC.2016.12.001

Mi Zhou$^{1}$, Xiaolan Liu$^{2}$,$^{3}$

1School of Science and Technology, Sanya College, Sanya, Hainan 572022, China

2College of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China

3Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing, Zigong, Sichuan 643000, China

Abstract

This paper analyzes and proves the global Lyapunov stability of the neural network proposed by Yashtini and Malek when the mapping is continuously differentiable and the Jacobian matrix of the mapping is positive semi-definite. Furthermore, the neural network is shown to be exponentially stable under stronger conditions. In particular, the stability results can be applied to the stability analysis of variational inequalities with linear constraints and bounded constraints. Some examples show that the proposed neural network can be used to solve the various nonlinear optimization problems. The new results improve the existing ones in the literature.

References

1.  [1] Luenberger, D.G. (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, MA.
2.  [2] Xia, Y.S. and Wang, J. (2000), On the stability of globally projected dynamical systems, Journal of Optimization Theory and Applications, 106(1), 129-150.
3.  [3] Xia, Y.S. (2004), Further results on global convergence and stability of globally projected dynamical systems, Journal of Optimization Theory and Applications, 122(3), 627-649.
4.  [4] Kennedy,M.P. and L.O.Chua (1988), Neural networks for nonlinear programming, IEEE Transactions on Circuits and Systems, 35, 554-562.
5.  [5] Xia, Y.S. andWang, J. (1998), A generalmethodology for designing globally convergent optimization neural networks, IEEE Trans. Neural networks, 9(6), 1331-1343.
6.  [6] Malek, A. and Oskoei, H.G. (2005), Primal-dual solution for the linear programming problems using neural networks, Appl.Math.Comput, 169, 451-471.
7.  [7] Xia, Y.S. (1996), A new neural network for solving linear and quadratic programming problems, IEEE Trans. Nerual networks , 7(6), 1544-1547.
8.  [8] Tao, Q., Cao, J.D., Xue,M.S., and Qiao, H. (2001), A high performance neural network for solving nonlinear programming problems with hybrid constraints, Phys. Lett.A, 288(2), 88-94.
9.  [9] Wang, J., Hu, Q., and Jiang, D. (1993), A Lagrangian neural network for kinematics control of redundant robot manipulators, IEEE Trans. Nerual networks, 10(5), 1123-1132.
10.  [10] Yashtini, M. and Malek, A. (2007), Solving complementarity and variational inequalities problems using neural network, Appl.Math.Comput, 190, 216-230.
11.  [11] Xia, Y.S. (2004), An Extended Projection neural network for constrained optimization, Neural Computation, 16, 863- 883.
12.  [12] Kinderlehrer, D. and Stampcchia, G. (1980), An introduction to variational inequalities and their applications, Academic Press, New York.
13.  [13] Fukushima, M. (1992), Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53, 99-110.
14.  [14] Pang, J.S. (1987), A Posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res, 12, 474-484.
15.  [15] Slotine, J.J. and Li, W. (1991), Applied nonlinear control, Englewood Cliffs, NJ: Prentice Hall.
16.  [16] Ortega, J.M. and Rheinboldt, W. C. (1970), Iterative solution of nonlinear in several variables, Academic Press, New York.
17.  [17] Xia, Y.S. and Wang, J. (2000), A recurrent neural network for solving linear projection equations, Neural Networks, 13, 337-350.