The Method For A Class Of Monotone Variational Inequality Problem

Variational inequality problem originates in mathematical physics and non-linear programmingproblems．It has been widely used in mathematics, physics, economics and engineering sciences．In this paper, we mainly study how to solve the monotone variational inequality problem inconvex polyhedron. The paper first transforms the variational inequality problem in convexpolyhedron to a complementary problem by dual linear programming. On this basis, two kinds ofmethods are proposed. The first one is Non-smooth Newton method, at first, the complementaryproblem is transformed into an Non-smooth equations which has the same results, then we can get theanswer to the original problem by using the generalized Newton type method. The second one ispointwise approximation method. We construct the algorithm mainly by using the fixed pointequation of the projection properties. At each step, the iteration point is calculated by implicitexpression. Though the calculation is complex, it has the super linear convergence property.The paper contains five parts. The first part is the introduction. In this section, the definition ofvariational inequality and the overview of various methods are introduced. In section two, weintroduce how to transform the variational inequality problem into a complementary problem. Insection three, the Non-smooth Newton method to the original problem is introduced. The fourth partis pointwise approximation method. The fifth part is a summary. The paper not only offers a summaryof the full text, but also gives prospect of the future research.