求解单调变分不等式的两类迭代算法

给出了求解单调变分不等式的两类迭代算法．通过解强单调变分不等式子问题,产生两个迭代点列,都弱收敛到变分不等式的解．最后,给出了这两类新算法的收敛性分析．     

2一致凸Banach空间中均衡问题、变分不等式问题和不动点问题的一个弱收敛定理

In this paper,we introduce a new iterative scheme for finding the common element of the set of solutions of an equilibrium problem,the set of solutions of variational inequalities for an α-inversely strongly monotone operator and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and 2-uniformly convex Banach space.Some weak convergence theorems are obtained,to extend the previous work.     

Hilbert空间中Lipschitz单调映像变分不等式解的另一个逼近定理

首先给出了Hilbert空间中Lipschitz单调映像变分不等式解的迭代格式,并证明了其收敛性.作为应用,证明了Hilbert空间中Lipschitz伪压缩映像的强收敛定理,扩展了已知的相关结果.     

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem

Split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present feasible algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for these algorithms. As application, we study the split feasibility problem in real Hilbert spaces. Final, numerical results are given for our main results.     

一类分裂变分不等式及其收敛算法

引入一种新的分裂变分不等式问题,构造了两种算法来求解,得到了相应迭代序列的弱收敛性和强收敛性.     

Hilbert空间中关于平衡与变分不等式问题及无限族非扩张映射的弱收敛定理

本文根据外梯度方法引进一新的迭代序列来寻找三个集合的公共元素.这三个集合分别是无限个非扩张映射的公共不动点集、平衡问题的解集与所含映射为单调、Lipschitz连续的变分不等式问题的解集.所得结果提高和推广了许多作者的相应结果.     

Strong Convergence of a Projection-Type Method for Mixed Variational Inequalities in Hilbert Spaces

Abstract

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results.     

广义平衡问题与非扩张映象不动点问题公解的一个新迭代方法

In this paper,we introduce a new iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems and the set of fixed points for nonexpansive mappings in Hilbert space.Under suitable conditions,some strong convergence theorems are proved.Our results extend and improve some recent results.     

集值B－H－S变分不等式及其应用

本文在Ｄａｎａｃｈ空间中，研究了集值Ｂ－Ｈ－Ｓ变分不等式问题和相补问题．将有限维空间、单值映象的若干结果改进并推广到无穷维空间．利用本文获得的结果．我们解决了无穷维空间中的鞍点问题和两类规划问题．     

Hilbert空间上新的变分不等式问题和不动点问题的粘性迭代算法

本文在Hilbert空间上引入了一个新的粘性迭代算法,找到了关于两个逆强单调算子的变分不等式问题的解集与非扩张映射的不动点集的公共元.通过修改的超梯度算法,得到了强收敛定理,也给出了一个数值例子.所得结果改进了许多最新结果.     

Banach空间中混合变分不等式解的存在性

By using the property of generalized f-projection operator and FKKM theorem, the existence theorem of solutions for the mixed variational inequality is proved under weaker assumption in reflexive and smooth Banach space. The results improve and extend the corresponding results shown recentlv.     

Banach空间中集值混和变分不等式问题解的迭代算法

介绍了集值映象的伪单调定义，并在Banach空间中构造了集值混和变分不等式问题近似解的迭代算法．应用伪单调映象定义，证明了该迭代算法收敛于集值混和变分不等式问题的近似解．特别值得注意的是：在文章中对集值映象没有Lipschitz连续性假设．     

Inverse Variational Inequality Approach and Applications

It is well known that a dynamic oligopolistic market equilibrium problem can be studied as an evolutionary variational inequality and this problem is approached as a problem of profit optimization for the firms. On the contrary, in this article, with the help of an inverse variational formulation, the behavior of control policies for an oligopolistic market equilibrium problem, whose aim is to regulate the exportation through the adjustment of taxes on the firms, is studied. This is considered as a policymaker optimization problem. More precisely, a definition of equilibrium for the firms by using the Lagrange multipliers is provided together to the optimal regulatory tax definition. Moreover an existence result is given and, at last, a numerical example is analyzed.     

Quadratic Convergence of a Long-Step Interior-Point Method for Nonlinear Monotone Variational Inequality Problems

This paper offers an analysis on a standard long-step primal-dual interior-point method for nonlinear monotone variational inequality problems. The method has polynomial-time complexity and its q-order of convergence is two. The results are proved under mild assumptions. In particular, new conditions on the invariance of the rank and range space of certain matrices are employed, rather than restrictive assumptions like nondegeneracy.     

Hilbert空间中混合平衡问题与非扩张半群的强收敛定理

在Hilbert空间中引进并研究了一种新的迭代算法,借以寻求混合平衡问题解集与非扩张半群不动点集的一公共元.所得到的结果,推广并改进了最近一些人所发布的新结果.     

集值映象的向量变分不等式和相补问题

研究了一类集值映象的广义向量变分不等式和相补问题 ,证明了解的一些存在性定理 .推广和改进了文 [1 ,4 6 ]的相关研究成果 .     

Approximating a Zero of Sum of Two Monotone Operators Which Solves a Fixed Point Problem in Reflexive Banach Spaces

Abstract

The purpose of this paper is to introduce an iterative method for approximating a point in the set of zeros of the sum of two monotone mappings, which is also a solution of a fixed point problem for a Bregman strongly nonexpansive mapping in a real reflexive Banach space. With our iterative technique, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a variational inclusion problem for sum of two monotone mappings and the set of solutions of a fixed point problem for Bregman strongly nonexpansive mapping. We give applications of our result to convex minimization problem, convex feasibility problem, variational inequality problem, and equilibrium problem. Our result complements and extends some recent results in literature.     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.