Neural Networks For Two Kinds Of Variational Inequality Problems

Variational inequality problem(VIP) is one of the most important optimization problems, it arises in a widely fields of signal processing, system identification, filter design, robot control, economic science, transprotation science, operational research, and nonlinear analysis fields. Many problems in mathematics, physics, and engineering can be formulated as it, thus they have to be solved in real time, so the investigation of solving it is significant in both theory and application. Since 1960s, people began to study the theory and methodology of it in detail, and there are many methods for solving it. But almost all are sequential iteration, and these conventional numerical methods may not obtain its solutions in real-time due to stringent requirment on computational time. One promising approach to handle these problems is to employ artificial neural network based on circuit implementation. Because of the dynamic nature and the potential of electronic implementation, neural networks can be implemented physically by designated hardware such as application-specific integrated circuits where the computational procedure is truly distributed and in parallel. Therefore, the neural network approach can solve optimization problems in running times at the order of magnitude much faster than conventional optimization algorithms executed on general-purpose digital computers, and it is of great interest in practice to develop some neural network models which could provide a real-time on-line solution for variational inequality problems.In this paper, optimization theory, projection theory, stability theory of ordinary differential equation and LaSalle invariant theory are used to analyze the stabilities of neural networks for VIP. And in theory, the stability, especially asym-potic stability and exponential stability of these networks are strictly proved; in pratice, illustrative examples show that these networks are not only feasible, but also effective.The first part is introduction. In this section, there are significance and development of VIP, and some fundmental theories, such as optimization theory, projection theory, stability theory of ordinary differential equation and LaSalle invariant theory.In the second part, we analyze the general projection neural network and its...