A new system of nonlinear variational inequalities with relaxed cocoercive mappings in reflexive Banach spaces

A new system for relaxed cocoercive non-linear variational inequalities in uniformly smooth Banach spaces is introduced and studied using the convergence of projection methods.Our results generalize and improve the corresponding results of recent works.

Three-step relaxed hybrid steepest-descent methods for variational inequalities

The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.

SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS

Abstract Some modified Levitin Polyak projection methods are proposed in this paper for solving monotone linear variational inequalityx∈Ω,(x′-x) T(Hx+c)≤0,\ x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.

AN EXTRAGRADIENT METHOD FOR RELAXED COCOERCIVE VARIATIONAL INEQUALITY AND EQUILIBRIUM PROBLEMS

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of solutions of the variational inequality prob- lem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao[17], Takahashi[12] and many others.

Generalized system for strongly g-r-pseudomonotonic nonlinear variational inequalities in Hilbert spaces

The approximation solvability of a generalized system for strongly g-r- pseudomonotonic nonlinear variational inequalities in Hilbert spaces is studied based on the convergence of the projection method. The results presented in this paper improve, generalize and unify some recent results in the literature.

Comparison of two approximal proximal point algorithms for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for solving monotone variational inequalities (MVI). Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as predic- tion-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm I; in the same way, Algorithm II is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm II usually outperforms Algorithm I. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm II to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some nu- merical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.

General Convergence Analysis for Three-step Projection Methods and Applications to Variational Problems

为一个三步的设计方法的第一个一一般模型被介绍，并且第二，它在 Hilbert 空格设置被用于非线性的变化不平等问题的一个系统的近似解决之可能性。让 H 是一个真实 Hilbert 空格和 K nonempty 被关上 H 的凸的子集。为任意地选择的起始的点 x0 ， y0 ， z0 K ，计算序列 xn ， yxn ， zxn 以便{xn+1=( 1-n-rn ) xn+xPk [ yn-Tyn ] +rnun ，yn=( 1--n ) xn+nPk [ zn-Txn ] +nun ，zn=( 1-an-n ) xn+akPk [ xn-Txn ] +nwn .For ，，> 0 是常数，{ n }，{ n }，{一}，{ rn }，{ n }，{ n } C [ 0,1 ]，{ un }，{ vn }，{ wn }是在 K 的序列，并且 0n + rn 1,0 n + n 1,0 一+ n 1 ，(A) n 0 ，在此 T ：K H 是到 K 上的非线性的印射。最后三步的模型被用于一些变化不平等问题。

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

Generalized Variational Inequalities and Wiener-Hopf Equations in Hilbert Spaces

在这份报纸，我们考虑在 Hilbert 空格的框架包含多珍视的放松的单调操作员的概括变化不平等的解决之可能性。我们的结果主要改进 Verma 宣布的相应结果[R U Verma，包含 multivalued 的概括变化不平等放松了单调操作员， Appl 数学 Lett， 1997， 10：107-109 ] 并且许多其它。

A Projection and Contraction Method for P-Order Cone Constraint Stochastic Variational Inequality Problem

In this paper, we study the p-order cone constraint stochastic variational inequality problem. We first take the sample average approximation method to deal with the expectation and gain an approximation problem, further the rationality is given. When the underlying function is Lipschitz continuous, we acquire a projection and contraction algorithm to solve the approximation problem. In the end, the method is applied to some numerical experiments and the effectiveness of the algorithm is verified.

Modified Inertial Projection Method for Solving Pseudomonotone Variational Inequalities with Non-Lipschitz in Hilbert Spaces

This paper deals with a class of inertial gradient projection methods for solving a vari-ational inequality problem involving pseudomonotone and non-Lipschitz mappings in Hilbert spaces.The proposed algorithm incorporates inertial techniques and the projection and contraction method.The weak convergence is proved without the condition of the Lipschitz continuity of the mappings.Meanwhile,the linear convergence of the algorithm is established under strong pseudomonotonicity and Lipschitz continuity assumptions.The main results obtained in this paper extend and improve some related works in the literature.

A PROXIMAL POINT ALGORITHM FOR A SYSTEM OF GENERALIZED MIXED VARIATIONAL INEQUALITIES

这份报纸介绍并且在一个 Hilbert 空格考虑概括混合变化不平等的一个新系统，它作为特殊情况包括变化不平等和概括变化不平等的许多新、知道的系统。由使用 -subdifferential 的二个概念和合适的功能的近似 mappings，作者试着证明概括混合变化不平等的系统是有一个固定的点问题的等价。由使用等价，为发现概括混合变化不平等的系统的近似解决方案的一个新、创新的近似的点算法将被建议并且分析。作者也在弱得多的条件下面学习新反复的方法的集中分析。结果能被看作变化不平等的以前已知的结果的精炼和改进。

A GOLDSTEIN'S TYPE PROJECTION METHOD FOR A CLASSOF VARIANT VARIATIONAL INEQUALITIES

Some optimization problems in mathematical programming can be translated to a variant variational inequality of the following form: Find a vectoru*, such that This paper presents a simple iterative method for solving this class of variational inequalities. The method can be viewed as an extension of theGoldstein's projection method. Some results of preliminary numerical experiments are given to indicate its applications.

THEORETICAL AND NUMERICAL COMPARISON ON DOUBLE-PROJECTION METHODS FOR VARIATIONAL INEQUALITIES

Recently, double projection methods for solving variational inequalities have received much attention clue to their fewer projection times at each iteration. In this paper, we unify these double projection methods within two unified frameworks, which contain the existing double projection methods as special cases. On the basis of this unification, theoretical and numerical comparison between these double projection methods is presented.

Perturbed Proximal-Projection Methods for Nonlinear Mixed Variational-Like Inequalities

In this paper we introduce a new perturbed proximal-projection algorithm for finding the common element of the set of fixed points of non-expansive mappings and the set of solutions of nonlinear mixed variational-like inequalities. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. Our approach and results generalize many known results in this field.

一类拟单调变分不等式的惯性投影算法

2020年,Liu和Yang在Hilbert空间中提出了一种求解拟单调变分不等式的投影算法.该文介绍了一种新的惯性系数来加速Liu和Yang文中的算法,并在相同的假设条件下得到了算法的全局弱收敛性.数值实验表明适当选取参数后的惯性方法比Liu和Yang文中的算法有更少的迭代步数和计算机耗时.

Solution and applications of a class of general linear variational inequalities

Many problems in mathematical programming can be described as a general linear variational inequality of the following form: find a vector u*, such thatSome iterative methods for solving a class of general linear variational inequalities have been presented. It is pointed out that the methods can be used to solve some practical extended programming problems.

单调变分不等式问题的惯性松弛投影算法

在Hilbert空间中研究单调变分不等式问题的惯性松弛投影算法.在该算法的每一次迭代中,只需要向特殊结构的半空间进行2次投影.另外,采取一定的线搜索条件,在单调和Lipschitz连续且Lipschitz系数大小未知的假设下,证明该算法所产生的序列强收敛到变分不等式的解.

增广Lagrange方法求解二阶锥约束变分不等式问题

应用增广Lagrange方法求解了一类二阶锥约束变分不等式问题。首先,将二阶锥约束变分不等式问题转化为等价的优化问题,从而得到其不同的等价形式;其次,应用投影算子的性质,将二阶锥约束变分不等式问题转化为方程组问题,并针对方程组问题提出了增广Lagrange方法;再次,讨论了算法的全局收敛性,同时对算法的一个特殊情况进行了深入分析,并引入一类非精确牛顿法求解算法中蕴含的子问题;最后,给出3个算例的数值实验结果,验证了算法的可行性。

ON THE LINEAR CONVERGENCE OF PC-METHOD FOR ACLASS OF LINEAR VARIATIONAL INEQUALITIES

This paper studies the linear convergence properties of a class of the projection and contraction methods for the affine variational inequalities, and proposes a necessary and sufficient condition under which PC-Method has a globally linear convergence rate.