Some Methods Based on the D-Gap Function for Solving Monotone Variational Inequalities

The D-gap function has been useful in developing unconstrained descent methods for solving strongly monotone variational inequality problems. We show that the D-gap function has certain properties that are useful also for monotone variational inequality problems with bounded feasible set. Accordingly, we develop two unconstrained methods based on them that are similar in spirit to a feasible method of Zhu and Marcotte based on the regularized-gap function. We further discuss a third method based on applying the D-gap function to a regularized problem. Preliminary numerical experience is also reported.     

Solving variational inequalities by a modified projection method with an effective step-size

In this paper, we propose a new projection method for the solution of variational inequality problems. The method is simple, which uses only function evaluations and projections onto the feasible set. We adopt a new step-size rule and a new search direction in the new method. Under the mild conditions, we prove the proposed method is globally convergent. Preliminary numerical results are reported.     

Local Linear Convergence of an Outer Approximation Projection Method for Variational Inequalities

This paper considers an outer approximation projection method for variational inequalities, in which the projections are not performed on the original set that appears in the variational inequality, but on a polyhedral convex set defined by the linearized constraints. It shows that the method converges linearly, when the starting point is sufficiently close to the solution and the step lengths are sufficiently small.     

Existence of solutions for variational inequalities on Riemannian manifolds

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on Riemannian manifolds are studied.     

Iterative approximation of a common solution of a split equilibrium problem,a variational inequality problem and a fixed point problem

In this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area.     

Fixed point methods for pseudomonotone variational inequalities involving strict pseudocontractions

We introduce a new iteration method for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of strict pseudocontractions in a real Hilbert space. The weak convergence of the iterative sequences generated by the method is obtained thanks to improve and extend some recent results under the assumptions that the cost mapping associated with the variational inequality problem only is pseudomonotone and not necessarily inverse strongly monotone. Finally, we present some numerical examples to illustrate the behaviour of the proposed algorithm.     

A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints

We propose a method for solving bilevel split variational inequalities involving strongly monotone operators in the leader problems and nonexpansive mappings in the follower ones. The proposed method is a combination between the projection method for variational inequality and the Krasnoselskii–Mann scheme for fixed points of nonexpansive mappings. Strong convergence of the iterative process is proved. Special cases are considered.     

Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces

In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality.     

The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions

We extend the Tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems. It is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem. For pseudomonotone equilibrium problems, the Tikhonov regularized subproblems have a unique solution only in the limit, but any Tikhonov trajectory tends to the solution of the original problem, which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction.     

Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.     

Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.     

求解广义互补问题

1 引言 设为一闭凸锥,f是R~n到自身的一映射.广义互补问题,记作GCP(K,f),即找一向量x满足 GCP(K,f) x∈K,f(x)∈且x~Tf(x)=0,(1) 其中,是K的对偶锥(即对任一K中向量x,满足x~Ty≤0的所有y的集合).该问题首先 由Habetler和Price提出.当K=R_+~n(R~n空间的正卦限),此问题就是一般的互补问题.许多作者已经提出了很多求解线性或非线性互补问题的方法.例如:Dafermos,Fukushima,Harker和Price以及其它如参考文献所列.近年来,何针对单调线性变分不等式提出了一些投影收缩算法. Fang在函数是Lipschitz连续及强单调的条件下,在[3]给出一简单的迭代投影法,在[4]中给出一线性化方法去求解广义互补问题(1).在[3]中,他的迭代模式是     

混合向量变分不等式标量化及间隙函数误差界

利用Konnov对变分不等式问题的标量化方法,对一般的强变分不等式（SVI）和弱变分不等式（WVI）进行了进一步的推广.主要介绍了基于集值映射的强广义混合向量变分不等式（SGMVVI）和弱广义混合向量变分不等式（WGMVVI）,考虑了与它们相关的间隙函数,在合适的条件下讨论了强广义混合集值变分不等式（SGMVI）的间隙函数和SGMVVI的间隙函数之间的关系,以及WGMVVI和SGMVI的间隙函数之间的关系,最后讨论了它们的间隙函数的全局误差界.     

A New Extragradient Method for Strongly Pseudomonotone Variational Inequalities

This article proposes a new extragradient solution method for strongly pseudomonotone variational inequalities. A detailed analysis of the iterative sequences’ convergence and of the range of applicability of the method is provided. Moreover, an interesting class of strongly pseudomonotone infinite dimensional variational inequality problems is considered.     

Banach空间上变分不等式的一个超梯度方法

把王宜举等人[Modified extragradient—type method for variational inequali—ties and verification of the existence of solutions,J.Optim.Theory Appl.,2003,119:167-183]在欧氏空间上求解变分不等式的一个超梯度型方法推广到Banach空间.变分不等式中的算子不要求是一致连续的,其主要优点在于不管变分不等式是否有解,算法都是可执行的.此外,变分不等式的可解性可以通过算法产生的序列的性态来刻画.在适当的条件下,算法产生的序列强收敛于变分不等式的一个解,这是Bregman距离意义下离初始点最近的解.本文的主要结果推广和改善了近来文献中的相应结果.     

Banach空间上变分不等式问题的新投影算法北大核心CSCD

本文在Banach空间上提出一种关于伪单调变分不等式问题的新算法.在对参数强加适当的条件下,我们证明由算法生成的序列强收敛到变分不等式的一个元素,所得结果推广和提高了很多最新结果.     

A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality

In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.     

A new self-adaptive alternating direction method for variational inequality problems with linear equality and inequality constraints

In this paper, we present a new self-adaptive alternating direction method for solving a class of variational inequality problems with both linear equality and inequality constraints without the need to add any extra slack variables. The method is simple because it needs only to perform some projections and function evaluations. In addition, to further enhance its efficiency, we adopt a self-adaptive strategy to adjust parameter μ at each iteration. Convergence of the proposed method is proved under certain conditions. Numerical experience illustrates the efficiency of the new method.     

求解非单调变分不等式的一种二次投影算法北大核心CSCD

投影算法是求解变分不等式问题的主要方法之一.目前,有关投影算法的研究通常需要假设映射是单调且Lipschitz连续的,然而在实际问题中,往往不满足这些假设条件.该文利用线搜索方法,提出了一种新的求解非单调变分不等式问题的二次投影算法.在一致连续假设下,证明了算法产生的迭代序列强收敛到变分不等式问题的解.数值实验结果表明了该文所提算法的有效性和优越性.     

弱向量变分不等式解集的连通性

研究拓扑向量空间到连续线性映射空间映射的弱向量变分不等式和与之相关 的纯量型变分不等式解集的关系, 引入弱和强一致连续概念,利用纯量型变分不等式 解集所表征的集值映射的特性给出弱向量变分不等式解集连通的一个充分条件。