| Variational inequality and complementarity problems are the im- portant research topics in the optimization problems. They have a wider applica- tion in many fields, such as signal processing, system identification, filter design, robot control, economic science, transprotation science, operational research, and nonlinear analysis fields, meanwhile, many problems in mathematics, physics, and engineering can be formulated as it. As a special case ofâ…¥, complementarity prob- lem involves in the engineering physics, economy and the transportation balance and so on, they usually appear in the optimozation codition of the analysis of theoptimozation problem with restriction, so it get an extensive research.In many practical applications, real-time solutions of the variational inequality and complementarity problem are desired. However, traditional algorithms are not suitable for a real-time implementation on the computer since the required comput- ing time for a solution is greatly dependent on the dimension and the structure of the problem, and the complexity of the used algorithm. One promising approach to handle these problems with high dimension and dense structure is to employ artificial neural network based 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 exe- cuted on general-purpose digital computers, and it is of great interest in practice to develop some neural network models.In this thesis, we consider a class of linaer variational inequality problem and the horizontal linear complementarity problem. According to the inherent properties of their solutions, the neural networks for them are presented respectively, and the relationship between the solutions of the original problems and the equilibrium points of the neural networks are discussed. Meanwhile, the stability, especially asympotic stability and exponential stability of these networks are strictly proved; Some illustrative examples show the feasibility and effectivenessof these networks.The full text is divided into three parts.The first part sumrizes the significance and development of the concerned prob- lems, the basic characteristices of the neural network, and some fundmental theories. Finally the primary works in this thesis are showed.In the second part, we consider a kind of linear variational inequality problem, and present a projection neural network for it. With the aid of the projection theory, stability theory and LaSalle invariant set theorem, the neural network is shown to be stable in the sense of Lyapunov and three sufficient conditions are provided to ensure the finite-time convergence by defing a suitable function. Finally exponentiall stablity of the proposed network is also shown.In the third part, based on the inherent properties of horizontal linear comple- mentarity problems, this paper presents a neural network with a one-layer structure for solving a class of horizontal linear complementarity problem in real-time by in- troducing the new vectors. We show that the proposed neural network is stable in the sense of Lyapunov, and will converge to an exact equilibrium point in finite time. Furthermore, global exponential stability of the proposed neural network is also shown under mild conditions. The size of the proposed neural network is only half of the original problem, and it has a low complexity, finite-time convergence and can be applied to solve some nonmonotone complementarity problems. Thus the proposed neural network is more suitable for parallel implementation by using simple hardware units. |
