Research On Solving Constrained Variational Inequality Problems Based On Several Kinds Of NCP Functions

Variational inequalities and complementarity problems are widely used in many fields such as engineering,economics,and mechanics.Problems such as structural optimization and frictional contact can be solved in the framework of complementary problems.Based on this,since the 1980s,the existence and numerical algorithms of solutions to the variational inequality problem and the complementary problem have become the research objects of many scholars.In this thesis,the numerical method for solving the problem of constrained variational inequalities based on several NCP functions is studied.The main contents of the paper can be summarized as follows:The first chapter introduces the background and research status of the variational inequality problem,including the origin,application and current research hotspots of variational inequalities.Secondly,the relevant definitions and conclusions are given,including the definition of constrained variational inequalities.Related conclusions;Finally,several types of nonlinear complementarity problem?NCP?functions and their related properties are introduced in detail,as well as their applications in this paper.In the second chapter,the augmented Fischer-Burmeister?FB?function is used to transform the KKT condition of the constrained variational inequality problem into a nonlinear equations problem.The non-singularity of the Jacobian matrix of the system operator is proved under certain constraint specifications.The semi-smooth Newton algorithm is constructed to solve the nonlinear equations.Chapter 3 is based on the FB function.Chen^[1]proposed several new NCP functions.This chapter studies the definition and related properties of four new NCP functions,and studies the variational variables based on these four types of NCP functions.Algorithm for inequality problems.The general idea is to transform the KKT condition of the constrained variational inequality problem into a system of equations problem by using the new NCP function,then calculate the Jacobian matrix of the system of equations,and prove the operator of the system of equations under certain conditions.The non-singularity of the Jacobian matrix.The corresponding Newton algorithm is constructed to solve the nonlinear equations.Chapter 4 gives numerical examples,obtains numerical results based on different types of NCP functions,and compares numerical results to obtain corresponding conclusions.