Many problems in mathcmatical programming can be described as a general linear variational inequality of the following form: find a vector u*, such that Nu*+t∈Ω, (v-(Nu*+t))T(Mu*+q)≥0, Av∈Ω. Some iterative methods for solving a class of general linear variationar inequalities have been presented. It is pointed out that the methods can be used to solve some practical extended programming problems.

In this paper we introduce a new class of iterative methods for solving the monotone variational inequalities u*∈Ω, (u+u*)T F(u*)≥0, Au∈Ω Each iteration of the methods presented consists essentially only of the computation of F (u) a projection toΩ, v:= PΩ[u-F(u)], and the mapping F(v). The distance of the iterates to the solution set monotonically converges to zero. Both the methods and the convergence proof are quite simple.

Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton-like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Q(u*)∈Ω, (v-Q(u*))T F(u*)≥0, Av∈Ω Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method.

Linear variational inequality is a uniform approach for some important problems in optimization and equilibrium problems. In this paper, we give a neural-network model for solving asymmetric linear variational inequalities. The model is based on a simple projection and contraction method. Computer simulation is performed for linear programming (LP) and linear complementarity problems (LCP). The test results for LP problem demonstrate that our model converges significantly faster than the three existing neural-network models examined in a recent comparative study paper.

In this paper, we study the relationship of some projectiontype methods for monotone nonlinear variational inequalities and investigate some improvements. If we refer to the Goldstein-Levitin-Polyak projection method as the explicit method, then the proximal point method is the corresponding implicit method. Consequently, the Korpelevich extragradient method can be viewed as a prediction-correction method, which uses the explicit method in the prediction step and the implicit method in the correction step. Based on the analysis in this paper, we propose a modified prediction-correction method by using better prediction and correction stepsizes. Preliminary numerical experiments indicate that the improvements are significant.