Non-interior Point Homotopy Method For Solving Semi-infinite Variational Inequalities Problem

The semi-infinite variational inequality problem comes from practical applications,such as engineering,transportation,economics,communications,and other fields.It inherits certain properties of semi-infinite programming and finite variational inequality.At present,there are many methods for solving semi-infinite variational inequality problems,including exterior approximation method,discretization smoothing method,commutative method,entropy analysis center tangent plane method,algorithm combining external approximation method and regularization method,interior point homotopy method and so on.However,these methods are complex to calculate and converge with strong assumptions.On the one hand,some methods cannot solve nonlinear problems,but they can only solve linear problems;On the other hand,some methods limit the initial point to the feasible domain when selecting it,resulting in a decrease in solution efficiency.In order to overcome these difficulties,this paper uses the moving boundary group contract ethics method to solve the semi-infinite variational inequality problem.This paper mainly studies from the following three aspects:Firstly,this paper proposes a non-interior point homotopy method for the problem of semiinfinite variational inequalities in which the constrained function g(x,t)is a one-dimensional form.According to the KKT condition,this paper first constructs the corresponding homotopy equation,then proves the existence and global convergence of the homotopy path under appropriate assumptions,and finally compares the numerical results of the interior point method and the noninterior point method by example to verify the effectiveness and feasibility of the non-interior point homotopy method.Secondly,this paper proposes a non-interior point homotopy method for the problem of semiinfinite variational inequality in which variables are one-dimensional constraint functions g(x,t)and combinations of multiple inequality constraint functions g(x).According to the KKT condition,it is transformed into a corresponding equivalence problem.In this paper,the corresponding homotopy equation is constructed,and then the existence and global convergence of the homotopy path are proved under appropriate assumptions,and finally the numerical results of the examples given by the interior point method and the non-interior point method are compared to verify the efficiency and feasibility of the non-interior point homotopy method.Thirdly,this paper proposes a non-interior point homotopy method for the semi-infinite variational inequality problem in which variables are multidimensional constrained functions g(x,t).According to the KKT condition,this class of problem can be equivalent to solving convex multiobjective programming problems.First,the corresponding homotopy equation is constructed,and then due to theoretical proof and numerical experimental results,and finally the feasibility and effectiveness of the non-interior point homotopy method are proved.Therefore,the non-interior point homotopy method no longer constrains the selection of initial points within the feasible domain,thereby expanding the selection range of initial points and improving the operation efficiency of solving semi-infinite variational inequality problems.