Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

Projection-Splitting Algorithms for General Monotone Variational Inequalities

We suggest and analyze some new splitting type projection methods for solving general variational inequalities by using the updating technique of the solution. The convergence analysis of these new methods is considered and the proof of convergence is very simple. These new methods are versatile.     

On a Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Verma introduced a system of nonlinear variational inequalities and proposed projection methods to solve it. This system reduces to a variational inequality problem under certain conditions. So, at least in form, it can be regarded as a extension of a variational inequality problem. In this note, we show that solving this system coincides exactly with solving a variational inequality problem. Therefore, we conclude that it suffices to study the corresponding variational inequalities.This work was supported by the National Natural Science Foundation of China, Grant 10571134.Communicated by M. J. Balas     

Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Let K be a nonempty closed convex subset of a real Hilbert space H. The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element (x*, y*)K×K such that

Extragradient Methods for Pseudomonotone Variational Inequalities

We consider and analyze some new extragradient-type methods for solving variational inequalities. The modified methods converge for a pseudomonotone operator, which is a much weaker condition than monotonicity. These new iterative methods include the projection, extragradient, and proximal methods as special cases. Our proof of convergence is very simple as compared with other methods.     

变分不等式的一类二次投影算法

通过构造的一类严格分离当前点与解集的超平面得到了一类解伪单调变分不等式的修正二次投影算法,该算法对He Yiran的算法进行了修正.从而建立了解伪单调变分不等式二次投影算法的一种框架结构.证明了该算法生成的无穷序列具有的全局收敛性,在具备某种局部误差界和Lipchitz连续条件下给出了收敛率分析.并给出了该算法的数值演算结果.     

Hilbert空间中的广义非线性变分不等式组的投影算法

利用投影技术讨论了Hilbert空间中一类含松弛伪上强制映射的广义非线性变分不等式组的逼近解及其收敛性,所得到结果推广和统一了系列最新结果.     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

一类非线性拟变分不等式

本文利用伪单调算子理论研究如下变分不等式问题：求ｘ∈Ｍ，使得〈Ａｘ，ｙ－ｘ〉＋〈Ｇｘ，ｙ－ｘ〉≥〈ｆ，ｙ－ｘ〉，?ｙ∈Ｍ．并将所得结果应用于拟线性椭圆型边值问题的求解．     

三步投影算法的收敛性及其在变分不等式组中的应用

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K,compute sequences xn, yxn, zxn such that { xn+1=(1-αn-rn)xn+αxPk[yn-ρTyn]+rnun,yn=(1-β-δn)xn+βnPk[zn-ηTxn]+δnun,zn=(1-an-λn)xn+akPk[xn-γTxn]+λnwn.For η, ρ,γ>0 are constants,{αn}, {βn}, {an}, {rn}, {δn}, {λn} C [0,1], {un}, {vn}, {wn} are sequences in K, and 0≤n + rn ≤ 1,0 ≤βn + δn ≤ 1,0 ≤ an + λn ≤ 1,(A)n ≥ 0, where T : K → H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.     

Some New Iterative Methods for General Mixed Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.AMS Subject Classification (1991): 49J40, 90C33.     

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.     

Convergence Properties of Feasible Descent Methods for Solving Variational Inequalities in Banach Spaces

This work is concerned with the analysis of convergence properties of feasible descent methods for solving monotone variational inequalities in Banach spaces.     

两步投影对变分不等方程组解的迭代逼近

本文研究Hilbert空间中两步投影方法及其对变分不等方程组解具误差的迭代序列的收敛性.本文结果发展和改进了Verma等人的最新结果.     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

一般变分不等式的一个新的下降投影算法

本文运用辅助问题原理,给出一般变分不等式一个新的投影下降算法,并证明在伪单调条件下,算法是收敛的.     

Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities

In this paper, we study the relationship of some projection-type 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.     

A Class of Combined Iterative Methods for Solving Variational Inequalities

A general approach to constructing iterative methods that solve variational inequalities is proposed. It is based on combining, modifying, and extending ideas contained in various Newton-like methods. Various algorithms can be obtained with this approach. Their convergence is proved under weak assumptions. In particular, the main mapping need not be monotone. Some rates of convergence are also given.