非线性混合似变分不等式的扰动近似点-投影算法

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.     

广义非凸变分不等式解的存在性与投影算法

本文运用Banach压缩映象原理和投影技巧研究一类新的广义非凸变分不等式问题解的存在唯一性,并在非凸集上建立一个逼近广义非凸变分不等式解的三步投影算法,在一定条件下证明了该投影算法所产生的迭代序列的收敛性.     

向量变分不等式与广义变分不等式之间的某些关系

文章首先建立了向量变分不等式与广义变分不等式之间的等价关系,然后利用这个结论,建立了向量变分不等式的Levitin-Polyak适定性与广义变分不等式的Levitin-Polyak适定性之间的等价关系.     

广义混合似变分不等式组的两步迭代算法

对H ilbert空间中一类广义混合似变分不等式组进行了研究;利用次微分算子的预解式技术,建立了广义混合似变分不等式组与不动点问题之间的等价关系;给出了一个求解这种广义混合似变分不等式组的显式两步迭代算法;并证明了该算法在适当的条件下收敛.     

Hilbert空间中关于松弛协强制映射的一类新的广义变分不等式组

在本文中,我们首先引入了一类新的关于松弛协强制映射的广义变分不等式组,通过用度量投影的方法,我们证明了这类广义变分不等式组解的存在性和唯一性.我们也建立了一类新的算法来逼近这一不等式组的解,本文的结果使得近期相关结果成为本文的特殊情况.     

一类变分不等式和非扩张映像不动点问题的一个迭代算法

该文提出了关于含松弛强制映像变分不等式和不动点问题解的一个投影迭代算法,所得结果改进和推广了目前一些作者的研究结果.     

辅助原理技术在广义混合似变分不等式中的应用

通过使用辅助原理技术证明了Hilbert空间中-类广义混合似变分不等式解的存在性,并给出一种算法计算此类变分不等式的近似解.     

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.     

一般伪单调变分不等式的改进投影算法

本文基于算子的分裂技巧给出了解一般伪单调变分不等式几种新的投影算法，包括三步和走步迭代算法．在算子T是g-伪单调和g-Lipschitz连续的条件下，即可证明新提出算法的收敛性．     

一类广义变分不等式组的隐迭代算法

依赖于投影映射的性质,许多学者在Hilbert空间研究了具不同映射的变分不等式组解的逼近问题,但在Banach空间的研究比较少.其主要原因是因为在Banach空间投影映射缺乏很好的性质.本文利用向阳非扩张保核映射(the sunny nonexpansiveretraction mapping)Q_K的性质,导出了一种隐迭代方法.用这一方法,本文的结果把[M.A.Noor,K.I.Noor,Projection algorithms for solving a system of generalvariational inequalities,Nonlinear Analysis,70(2009)2700-2706]的主要成果从Hilbert空间推广到了Banach空间.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.     

模糊映射的广义混合型强变分不等式

研究了新的一类模糊映射的广义混合型强变分不等式问题。证明了这类问题解的存在定理和收敛定理,给出解的带误差的Ishikawa型迭代算法。     

Generalized Mixed Variational Inequalities and Resolvent Equations

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized mixed variational inequality. Using essentially the resolvent operator concept, we establish the equivalence between the generalized mixed variational inequalities and the system of resolvent equations. This equivalence is used to suggest a number of new iterative algorithms for solving the variational inequalities. Several special cases are discussed which can be obtained from the main results of this paper.     

模糊映射的完全广义混合型强变分不等式

研究关于模糊映射的一类新的变分不等式－模糊映射的完全广义混合型强变分不等式,得到此类变分不等式解的存在定理分解的一个逼近算法,推广了文[4]和文[8]的主要结果。     

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.     

关于广义变分不等式与广义拟变分不等式

本文在非常一般的框架下,建立了极大极小不等式,广义变分不等式和广义拟变分不等式,证明了解的存在定理,且它们是在非紧集上得到的,从而推广和改进了[3～13]中的相应结果.     

非扩张映射和广义变分不等式的粘滞逼近法

应用已提出的非扩张映射的粘滞逼近方法,给定初值x_0∈C,考虑一般迭代过程{x_n},g(x_(n+1))=α_nf(x_n)+(1-α_n)SP_C(g(x_n)-λ_nAx_n),n≥0,其中{α_n}■(0,1),S:C→C是非扩张映射,C是实Hilbert空间H的非空闭凸子集.在{α_n}满足合适的条件下可证明,{x_n}强收敛到非扩张映射的不动点集和广义变分不等式解的公共元,且满足某变分不等式.     

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

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.