Linear Variational Inequality And The Level Of Complementarity Problems, The Neural Network

Variational inequality has significant applications in many fields including non-linear analysis, operational research, transportation science, economic science, robot control, signal processing, system identification, filter design fields and so on, and many problems in mathematics, physics and engineering can be transformed into it. Thus it is important in the field of optimization. As a special case of varia-tional inequality, complementarity problem has a wider application in the fields of engineering physics, economy and the transportation balance. Therefore it is impor-tant in theory and practice to intensively study the method for solving variational inequality and complementarity problems.Many traditional numerical iterative methods can be applied to solve the varia-tional inequality and the complementarity problems. However, these methods might not be effective to obtain the solution in real-time because the computing time al-ways depends on the dimension and structure of the problem, and the complexity of the algorithms used. The artificial neural network based on circuit implemen-tation has the abilities of massively parallel processing and distributed information storage. Moreover, its algorithms have faster convergence speed and good stability. Therefore, it is suitable for a real-time implementation and can be viewed as an effective and promising approach to handle the optimization problems. It is well known that in the hardware implementation of neural networks.time delays occur in the signal communication among the neurons. This may lead to the oscillation phenomenon or instability of neural networks. So it becomes more and more im-portant to construct delayed neural networks for solving optimization problems and studying their dynamical behaviorsIn this thesis, a class of linear variational inequality problem and the horizon-tal linear complementarity problem are further investigated. The neural network models for solving them are construted. respectively. By the theory of functional differential equation. Lyapunov functionals and the linear matrix inequality method. the existence and uniqueness of the solution of the new models have been proved. Some sufficient conditions are provided to cnsure the stability of the proposed neural networks, especially for the exponential stability. Some numerical examples show the feasibility and effectiveness of these networks The full text is divided into four chapters.In chapter one, we summarize the significance and development of the con-cerned problems, and introduce analysis methods and research situations on the stability of the neural networks, and some preliminaries. Finally the main work is also summarized at the end of this chapter.In chapter two, we consider the linear variational inequality problem, and present a new delayed projection neural network model for it. By the theory of functional differential equation, we prove the existence and uniqueness of the solu-tion of the new model, and provide some sufficient conditions to ensure the global exponential stability of the proposed model. Finally, numerical simulations show the good characteristics of the new model.In chapter three, a novel delayed projection neural network model for solving the linear variational inequality problem is presented. By the theory of functional differential equation, the existence and uniqueness of the solution of the new model has been proved. By constructing appropriate Lyapunov function and using the linear matrix inequality (LMI) method, some sufficient conditions are provided to ensure the exponential stability and global asymptotic stability of the proposed delayed projection neural network. Compared with the existing delayed projection neural networks for solving the linear variational inequality, these criterions of the new model guarantee that it has more wide application and its architecture is very simple. Finally, the feasibility and efficiency of the presented neural network are showed by some numerical examplesIn chapter four, we consider the horizontal linear complementarity problem. Based on its equivalent equation, a new and simple neural network model for solv-ing it is constructed. This new model has the half size as the original problem. The existence and uniqueness of the solution of the problem is proved. By construct-ing appropriate Lyapunov function and using the linear matrix inequalities(LMIs) method, some sufficient conditions are given to ensure the global exponential sta-bility of the neural network model. Besides, the limited numerical simulations show the characteristics of the constructed neural network model.