Differential Equation Approaches For Solving Equilibrium Problems

Equilibrium problem plays an important role in the field of operations research.It is an important mathematical model in the fields of economy and finance,network transportation,game theory,mathematical economy and so on.In recent years,it has gradually become the focus of scholar’s attention.The essence of using differential equation method to solve equilibrium problem is to transform solving optimization problem into solving differential equation model.The solution of optimization problem is equivalent to the equilibrium point of differential equation,and the effectiveness of the method is guaranteed through consistency and stability.Although differential equation methods for solving optimization problems have existed for a long time,the differential equation solving models for generalized vertical linear complementarity problems and box constrained variational inequality problems established in this thesis are not mentioned,and the validity of their theories and methods needs to be studied urgently.In this thesis,a differential equation method for solving generalized vertical linear complementarity problems and box constrained variational inequalities is proposed.The consistency and stability of the method are analyzed,and the effectiveness of the method is verified by numerical simulation.The main content of this article is as follows:Firstly,the research background of equilibrium problem and the development history and research status of solving variational inequality and complementarity problem by differential equation method are introduced.Secondly,a differential equation method for solving generalized vertical linear complementarity problems is proposed.We construct a new smoothing function of generalized vertical linear complementarity problem by using F-B function,and propose a first-order differential equation model to solve the problem by using the properties of smoothing function and the equivalent transformation of complementarity problem.The existence and consistency of solutions are analyzed by using the properties of complementary problems,and the exponential stability theory of equilibrium point of differential equations is established under the condition of nonsingular Jacobian matrix.Thirdly,a differential equation method for solving box constrained variational inequalities is proposed.The box constrained variational inequality inequality problem is transformed into a complementary problem through its KKT system,and the complementary problem is equivalent to an unconstrained minimization problem by using nonsmooth equations method and smoothing function method.Then a differential equation model is established to solve the problem.The equivalence and consistency of solutions are studied by using box constrained variational inequality,and the stability theory of stable points of differential equations is established.Finally,combined with the generalized double matrix game problem,a specific example of differential equation method for solving the generalized vertical linear complementarity problem and box constrained variational inequality problem is given,and numerical experiments are carried out to verify the validity of the method.