Hu Xiaolin, Wang Jun
Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China.
IEEE Trans Neural Netw. 2006 Nov;17(6):1487-99. doi: 10.1109/TNN.2006.879774.
In recent years, a recurrent neural network called projection neural network was proposed for solving monotone variational inequalities and related convex optimization problems. In this paper, we show that the projection neural network can also be used to solve pseudomonotone variational inequalities and related pseudoconvex optimization problems. Under various pseudomonotonicity conditions and other conditions, the projection neural network is proved to be stable in the sense of Lyapunov and globally convergent, globally asymptotically stable, and globally exponentially stable. Since monotonicity is a special case of pseudomononicity, the projection neural network can be applied to solve a broader class of constrained optimization problems related to variational inequalities. Moreover, a new concept, called componentwise pseudomononicity, different from pseudomononicity in general, is introduced. Under this new concept, two stability results of the projection neural network for solving variational inequalities are also obtained. Finally, numerical examples show the effectiveness and performance of the projection neural network.
近年来，一种名为投影神经网络的递归神经网络被提出来用于解决单调变分不等式及相关的凸优化问题。在本文中，我们表明投影神经网络也可用于解决伪单调变分不等式及相关的伪凸优化问题。在各种伪单调性条件和其他条件下，投影神经网络在李雅普诺夫意义下被证明是稳定的，并且是全局收敛的、全局渐近稳定的以及全局指数稳定的。由于单调性是伪单调性的一种特殊情况，投影神经网络可应用于解决与变分不等式相关的更广泛类别的约束优化问题。此外，还引入了一个与一般伪单调性不同的新概念，即逐分量伪单调性。在这一新概念下，也得到了投影神经网络用于解决变分不等式的两个稳定性结果。最后，数值例子展示了投影神经网络的有效性和性能。
