Neural Networks For Two Kinds Of Optimization Problems

Minimax problem is a kind of improtantly nondifferentiable opti- mization problems, and arises many fields including engineering design, electronic circuit, game theory, optimization theory, variational inequality, differential equation and so on. Specially, many problems such as nonlinear equation, nonlinear inequal- ities, nonlinear programming and multiobjective programming can be formulated as it. Also semi-infinite programming(SIP) is another kind of typical optimization problems, which has widely and directly applications in economic equilibrium, op- timal control, information technology, computer network, engineering design, robot trajetory planning, environment pollution control system and so on. So the investi- gation how to solve them is significant in both theory and application.In many practical applications, real-time solutions of the minimax problem and semi-infinite problem are often desired. However, traditional algorithms are not suit- able for a real-time implementation on the computer since the required computing 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 han- dle 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.The thesis primarily deals with minimax problem and semi-infinite problem. Based on their inherent properties, we present two neural networks to solve them respectively. Then the relationships between the equilibrium point of the networks and the solution of the problems is analyzed. Finally, we prove the stability and convergence of the proposed network. The full thesis is divided into three parts.In Chapter 1, scientific background and development of neural network, basic feature and research situation of the maximum entropy method are first introduced. Then we cite some preliminary knowledge and fundmental theories, such as stability theory of ordinary differential equation and LaSalle invariant theory. Finally the main work is stated.In Chapter 2, we first discuss the minimax problem's model, its some algo- rithms and classification. Then a new neural netwok is proposed, it is shown to be Lyapunov stable, and convergent to an exact solution of the problem in finite time by the Lyapunov theorem and the LaSalle invariant set principle. Compared with the existing neural networks, the proposed model has simple structure and can be implemented in hardware. Illustrative examples demonstrate the feasibility and efficiency of the network.In Chapter 3, we elaborete semi-infinite programming's classification, their models, and some existing algorithms. By using maximal entropy method, we converts many constraints semi-infinite problem into single constraint semi-infinite problem. Then we propose a new neural network for semi-infinite problem, and analyze the stability and convergence. Illustrative examples show the feasibility and efficiency of the network.