Global stability of a recurrent neural network for solving pseudomonotone variational inequalities

Solving variational inequality problems by using neural networks are of great interest in recent years. To date, most work in this direction focus on solving monotone variational inequalities. In this paper, we show that an existing recurrent neural network proposed originally for solving monotone variational inequalities can be used to solve pseudomonotone variational inequalities with proper choice of a system parameter. The global convergence, global asymptotic stability and global exponential stability of the neural network are discussed under various conditions. The existing stability results are thus extended in view of the fact that pseudomonotonicity is a weaker condition than monotonicity

[1]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[2]  Shuzhi Sam Ge,et al.  A unified quadratic-programming-based dynamical system approach to joint torque optimization of physically constrained redundant manipulators , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[3]  P. Hartman.  Ordinary Differential Equations , 1965 .

[4]  Jun Wang,et al.  A general methodology for designing globally convergent optimization neural networks , 1998, IEEE Trans. Neural Networks.

[5]  Jun Wang,et al.  A projection neural network and its application to constrained optimization problems , 2002 .

[6]  Jun Wang,et al.  Global exponential stability of recurrent neural networks for solving optimization and related problems , 2000, IEEE Trans. Neural Networks Learn. Syst..

[7]  L. Vandenberghe,et al.  The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits , 1989 .

[8]  Xue-Bin Liang,et al.  Global exponential stability of neural networks with globally Lipschitz continuous activations and its application to linear variational inequality problem , 2001, IEEE Trans. Neural Networks.

[9]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[10]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[11]  Y. Xia,et al.  Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems , 2004 .

[12]  P. Tseng,et al.  Modified Projection-Type Methods for Monotone Variational Inequalities , 1996 .

[13]  Youshen Xia,et al.  A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..

[14]  S. Karamardian,et al.  Seven kinds of monotone maps , 1990 .

[15]  Naihua Xiu,et al.  Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities , 2001 .