Global exponential stability of neural networks with globally Lipschitz continuous activations and its application to linear variational inequality problem

Xue-Bin Liang, J. Si
2001 IEEE Transactions on Neural Networks  
This paper investigates the existence, uniqueness, and global exponential stability (GES) of the equilibrium point for a large class of neural networks with globally Lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations. The provided sufficient condition for GES is mild and some conditions easily examined in practice are also presented. The GES of neural networks in the case of locally Lipschitz continuous activations is also
more » ... ed under an appropriate condition. The analysis results given in the paper extend substantially the existing relevant stability results in the literature, and therefore expand significantly the application range of neural networks in solving optimization problems. As a demonstration, we apply the obtained analysis results to the design of a recurrent neural network (RNN) for solving the linear variational inequality problem (VIP) defined on any nonempty and closed box set, which includes the box constrained quadratic programming and the linear complementarity problem as the special cases. It can be inferred that the linear VIP has a unique solution for the class of Lyapunov diagonally stable matrices, and that the synthesized RNN is globally exponentially convergent to the unique solution. Some illustrative simulation examples are also given. Index Terms-Global exponential stability, global Lipschitz continuous activations, linear variational inequality problems, recurrent neural networks. I. INTRODUCTION S TABILITY analysis of neural networks has been an important topic in the neural-network field since the Hopfield network model was proposed in [1] and [2] which demonstrated the great potential of the model in applications of associative memory and optimization. For the two applications of Hopfield network, the underlying qualitative property of the network model is the local or global stability of the network equilibrium points. In general, we require that the equilibrium points of the network correspond to the memory pattern vectors for associative memory application or the local optimal solutions for optimization application. The common requirement of the two applications is that these equilibrium points are stable in the sense Manuscript
doi:10.1109/72.914529 pmid:18244389 fatcat:gsbsyjftwvcajkyvppawlsv6dm