Neural Network Methods For Solving Two Kinds Of Complementarity Problems

Finite dimensional complementarity problem has become a relatively mature and fruitful subject in mathematical programming,and is widely used in mechanics,engineering,economics,game theory,network transportation and communication networks.In this thesis,we mainly study the generalized vertical complementarity problem and the second-order cone complementarity problem which have important applications in the generalized bimatrix game problem and the manipulator grasping problem.There are many well-known methods to solve these two kinds of problems,such as non-smooth Newton method,smooth Newton method,interior point method,etc.,but there are not many research achievements on neural network method to solve these two kinds of complementarity problems,and some theoretical and computational problems need to be solved.For example,the relationship between the existence of solution and the stability of the differential equation describing the neural network,the relationship between the equilibrium point obtained by the neural network and the solution of the original problem,the comparative analysis of neural network models based on different merit functions and the selection of parameters in the calculation,etc.In this thesis,using the theory of variational analysis and differential equation stability theory to study the generalized vertical complementarity problem and solving the second-order cone complementarity problem of neural network method,the consistency and stability analysis are proposed,and by using neural network method has important research value in solving the actual problems,to verify the effectiveness of the neural network method.This thesis is divided into four parts.Firstly,in the first chapter,the research background of the generalized vertical complementarity problem,the research background of second-order cone complementarity problem and the research progress of neural network are introduced.The purpose,significance,content and symbol of this thesis are given.Secondly,in the second chapter,the neural network method for solving generalized vertical complementarity problem is proposed by using the log-exponential function.The consistency relationship between the original problem and the unconstrained minimization problem,and between the original problem and equilibrium point of the neural network is expounded in detail.Based on stability theory of differential equation,the condition that equilibrium point of neural network is stable in lyapunov sense is given.Thirdly,in the third chapter,the neural network method for solving the second-order cone complementarity problem is proposed by using F-B function.Based on the second-order cone theory of the Jordan product,the consistency theory between the original problem and the equilibrium point of the neural network is obtained,and based on the properties of the second-order cone complementarity problem,the equilibrium point of the neural network is given to be stable in the sense of Lyapunov.Finally,in the fourth chapter,the results obtained in chapter 2 and Chapter 3 are applied to the generalized bimatrix game problem and the robust Nash equilibrium problem.The results obtained in Chapter 2 and Chapter 3 are verified by numerical experiments,and the effectiveness of the neural network method is illustrated.