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.

Abstract. The alternating directions method (ADM) is an effective method for solving a class of variational inequalities (VI) when the proximal and penalty parameters in sub-VI problems are properly selected. In this paper, we propose a new ADM method which needs to solve two strongly monotone sub-VI problems in each iteration approximately and allows the parameters to vary from iteration to iteration. The convergence of the proposed ADM method is proved under quite mild assumptions and flexible parameter conditions.

The problem concerned in this paper is the set valued equation 0 ∈T (z) where T is a maximal monotone operator. For given xk and βk>0, some existing approximate proximal point algorithms take xk+l=xk such that xk+ek∈xk+βkT(xk) and ‖ek‖≤nk‖xk-xk‖, where {nk} is a non-negative summable sequence. Instead of xk+1=xk, the new iterate of the proposing method is given by xk+1=pΩ[xk-ek], where Ω is the domain of T and PΩ(·) denotes the projection on Ω The convergence is proved under a significantly relaxed restriction supk>0 nk<1.

设Ω∈Rn是一个闭凸集，F是从Ω到Rn的一个映射变分不等式是求一个向量u*∈Ω，使得对所有的u∈Ω都有（u-u*）TF（u*）≥0.本文给出求解算子F为单调的变分不等式的一类预测一校正方法的统一框架。对给定的uk∈Ω，预测点uk可以用不同的方法产生，但都可以用公式（预测）uk=Pn[uk-βkq（uk, uk, βk）] 来表示，其中βk>0，q（uk, u, βk）∈Rπ是依赖于uk, uk和βk的向量并满足一些简单统一的条件新的迭代点uk+l由统一的校正公式（校正）uk+1=PΩ[uk-akβkF（uk）]。 产生，其中ak是最优步长参数，它使得在确定预测点的前提下，这一步迭代所取得的进步尽可能大。已有的一些方法可以看作是这个框架的特殊形式此外，它也为构造求解单调变分不等式新的预测一校正类方法提供了启示与帮助。

Solving a variational inequality problem VI (Ω, F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and/or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI (Ω, F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.

In this paper, we study the relationship between the forward-backward splitting method and the extra-gradient method for monotone variational inequalities. Both of the methods can be viewed as predictioncorrection methods. The only difference is that they use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods usually outperform the forward-backward splitting methods. We suggest some modifications for the two methods and numerical results are given to verify the superiority of the modified methods.