The Numerical Solution To A Class Of Variational Inequality Problem

Variational inequality problem is one of the focal point problems paid close attention by scholars in the field of mathematics. It has been widely used in mathematics, physics, economics and engineering sciences.In this paper, we mainly study how to solve the variational inequality problem in convex polyhedron. The paper first transforms the variational inequality problem in convex polyhedron to a complementary problem. On this basis, we proposed two kinds of prediction-correction methods. The first one is the modified projection method. At each iteration, it needs two projection operations. Compared with previous methods, the new method is better use of the components that have been solved and gives the new Armijo criteria in the prediction step. It also gives the optimal steps in the correction. The second one is a lagarithmic-quadratic proximal(LQP) prediction-correction method which is based on the proximal point algorithm. At each iteration, we first find a predictor with the LQP method. Then we correct the predictor with the projection method. Compared with previous methods, the new algorithm does not continue to use the LQP method, but using the projection method in the correction step. It combines the advantages of the LQP algorithm and the projection. For two kinds of new algorithm the only requirement is that the funtion is continuous and monotone, which broadens the scope of application. Finally, the convergences are proved when they satisfy certain constraints. The numerical experiments show that the two algorithms are effective.The paper contains five parts. The first part is the introduction. In this section, the definition of variational inequality and the overview of various methods are introduced. In section two, the modified prediction-correction method for monotone variational inequality is given. Section three is the modified LQP prediction-correction method. In this section, the paper gives the specific steps of the algorithm and the proof of convergence. The fourth part is numerical test, which aims to verify the convergence of the algorithm. The fifth part is a summary. The paper not only offers a summary of the full text, but also gives prospect of future research.