Liang X B, Si J
Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA.
IEEE Trans Neural Netw. 2001;12(2):349-59. doi: 10.1109/72.914529.
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 obtained 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.
本文研究了一大类具有全局Lipschitz连续激活函数的神经网络平衡点的存在性、唯一性和全局指数稳定性（GES），其中包括广泛使用的Sigmoid激活函数和分段线性激活函数。所提供的GES充分条件较为宽松，还给出了一些在实践中易于检验的条件。在适当条件下，也得到了局部Lipschitz连续激活函数情况下神经网络的GES。本文给出的分析结果极大地扩展了文献中现有的相关稳定性结果，从而显著扩大了神经网络在解决优化问题中的应用范围。作为一个示例，我们将所获得的分析结果应用于设计一个递归神经网络（RNN），用于求解定义在任何非空闭盒集上的线性变分不等式问题（VIP），其中包括盒约束二次规划和线性互补问题作为特殊情况。可以推断，对于Lyapunov对角稳定矩阵类，线性VIP有唯一解，并且合成的RNN全局指数收敛到唯一解。还给出了一些说明性的仿真示例。
