Neural Networks For Two Kinds Of Generalized Linear Complementarity Problems

The generalized linear complementarity problem comes from engineering physics, mechanics, economics and operational research. It has important application in economic equilibrium problems, noncooperative games, traffic assignment problems and optimization problems. It is related to the variational inequality problem, bilinear programming, nonlinear equation. So there is actual meaning and theories value to study how to solve it.In recent years, many algorithms for solving the generalized linear complementarity problem are put forward. But they basically belong to the traditional iterative methods, and may not be efficient since the computing time required for a solution is greatly dependent on the dimension, structure of the problem and the complexity of the algorithm used. Different from the traditional algorithms, the neural network has many advantages on calculations and real-time applications for its inherent massive parallelism and electric circuit implementation. Since Hopfield proposed a famous artificial neural network, named Hopfield neural network, and successfully applied it to optimization problems, the theory, methodology, and application of neural network have been widely investigated, and achieved many significant results.The thesis primarily deals with two kinds of the generalized linear complementarity problems. Based on their inherent properties, we present several neural networks to solve them respectively. Then the relationship between the equilibrium point of the network and the solution of the problem is analyzed. Finally, we prove the stability and convergence of the proposed network. The full thesis is divided into three chapters.In chapter 1, some research activities of the generalized linear complementarity problem are introduced, followed by the basic properties, theories and the progresses of the neural network. The main work is summarized at the end of this chapter.In chapter 2, the first kind of the general linear complementarity problem is discussed. For its two special cases: N = 0 and R=0, based on nongradient and gradient methods respectively, neural networks for the two cases are constructed. With the Lyapunov theorem and the LaSalle invariant set principle, the proposed networks are proved to be...