一般多值混合隐拟变分不等式的解的存在性与算法

引入了实Hilbert空间中一类新的一般多值混合隐拟变分不等式.它概括了丁协平教授引入与研究过的熟知的广义混合隐拟变分不等式类成特例.运用辅助变分原理技巧来解这类一般多值混合隐拟变分不等式.首先,定义了具真凸下半连续的二元泛函的新的辅助变分不等式,并选取了一适当的泛函,使得其唯一的最小值点等价于此辅助变分不等式的解.其次,利用此辅助变分不等式,构造了用于计算一般多值混合隐拟变分不等式逼近解的新的迭代算法.在此,等价性保证了算法能够生成一列逼近解.最后,证明了一般多值混合隐拟变分不等式解的存在性与逼近解的收敛性.而且,给算法提供了新的收敛判据.因此,结果对M.A.Noor提出的公开问题给出了一个肯定答案,并推广和改进了关于各种变分不等式与补问题的早期与最近的结果,包括最近文献中涉及单值与集值映象的有关混合变分不等式、混合拟变不等式与拟补问题的相应结果.     

Differential quasi-variational inequalities in finite dimensional spaces

In this paper, we introduce and study a new class of differential quasi-variational inequalities in finite dimensional Euclidean spaces. First, we prove existence theorems for Carathéodory weak solutions of the differential quasi-variational inequalities under various conditions. Furthermore, we establish a convergence result on Euler time-dependent procedure for solving the initial-value differential set-valued variational inequalities.     

Regularization of quasi-variational inequalities

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others.     

The <Emphasis Type="Italic">p</Emphasis>-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with (<Emphasis Type="Italic">H</Emphasis>,η)-monotone operators

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (H,η)-monotone operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (H,η)-monotone operators, we prove the existence of solutions and the convergence of some new p-step iterative algorithms for this system of generalized mixed quasi-variational inclusions and its special cases. The results in this paper unifies, extends and improves some known results in the literature.     

A class of quasi-variational inequalities for adaptive image denoising and decomposition

We introduce a class of adaptive non-smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set-valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extension gives rise to a class of quasi-variational inequalities. We provide sufficient conditions for the existence of fixed points as solutions, and an algorithm based on solving a sequence of variational problems. Denoising experiments with spatial and spatio-temporal image data and an adaptive total variation regularizer illustrate our approach.     

A New Existence Theorem of Maximal Elements in Non-Compact H-Spaces with Applications to Minimax Inequalities and Variational Inequalities

We give a new existence theorem of maximal elements in non-compact H-spaces. As applications, two extended versions of Fan-Yen minimax inequality are obtained and the existence problems of solutions of variational inequalities and quasi-variational inequalities are studied.     

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.     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.     

广义隐向量拟变分不等式解的存在性

本文研究一类广义隐式向量拟变分不等式问题,利用Fan-Kakutani不动点定理证明其解存在,推广了相关的文献中的结论.     

Set-valued mixed quasi-variational inequalities and implicit resolvent equations

In this paper, we introduce and study a new class of variational inequalities, which is called the set-valued mixed quasi-variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued mixed quasi-variational inequalities, fixed-point problems and the set-valued implicit resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.     

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.     

Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems

The aim of this paper is to obtain the continuity of solutions to time-dependent nonlinear variational and quasi-variational inequalities which express many dynamic equilibrium problems. To prove our results, we make use of Minty’s Lemma and of the notion of the Mosco’s convergence.     

Variational convergence of bivariate functions: lopsided convergence

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, of solutions to inclusions, the primal and dual solutions of convex optimization problems and of zero-sum games. These applications will be dealt with in a couple of accompanying papers.     

Existence and iterative approximation of solutions of some systems of variational inequalities and inclusions

In 1968, Brézis [Ann. Inst. Fourier (Grenoble), 18 (1) (1968) 115-175] initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximal-point mappings due to Moreau [Bull. Soc. Math. France, 93 (1965) 273-299]. Variational inclusions include variational, quasi-variational, variational-like inequalities as special cases. In 1985, Pang [Math. Prog. 31 (1985) 206-219] showed that a variety of equilibrium models can be uniformly modelled as a variational inequality defined on the product sets equivalent to a system of variational inequalities and discuss the convergence of method of decomposition for system of variational inequalities. Motivated by the recent research work in this directions, we consider some systems of variational (-like) inequalities and inclusions; develop the iterative algorithms for finding the approximate solutions and discuss their convergence criteria. Further, we study the sensitivity analysis of solution of the system of variational inclusions. The techniques and results presented here improve the corresponding techniques and results for the variational inequalities and inclusions in the literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs with affine operator by using a continuously differentiable dual gap function. This extends an idea from Dietrich (J. Math. Anal. Appl. 235:380–393 [24]). Some numerical results illustrate the practical behavior of this dual gap function approach.     

Iterative methods for solving general quasi-variational inequalities

In this paper, we introduce a new class of variational inequalities, which is called the general quasi-variational inequality. We establish the equivalence among the general quasi variational inequality and implicit fixed point problems and the Wiener–Hopf equations. We use this equivalent formulation to discuss the existence of a solution of the general quasi-variational inequality. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general quasi-variational inequality. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.     

Self-adaptive methods for mixed quasi-variational inequalities

It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed-point problems and the resolvent equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving mixed quasi-variational inequalities. Our results can be viewed as significant extensions of the previously known results for mixed quasi-variational inequalities. An example is given to illustrate the efficiency of the proposed method.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

约束图像拓扑下的向量值拟变分不等式解集的通有稳定性

主要运用研究通有性质的方法研究向量值拟变分不等式解的稳定性.首先引入约束映射在图像拓扑意义下的Hausdorff度量,这是一种有别于通常一致度量的新度量,然后在此弱图像拓扑下,给出并证明了关于向量值拟变分不等式解的通有稳定性的几个结论.结论表明,在Baire分类的意义下,大多数的向量值拟变分不等式问题的解关于新定义的度量都是本质的.     

EXISTENCE AND DUALITY OF IMPLICIT VECTOR VARIATIONAL PROBLEMS*

In this paper, we consider implicit vector variational problems which contain vector equilibrium problems and vector variational inequalities as special cases. The existence of solutions of implicit vector variational problems and vector equilibrium problems have been established. As a special case, we derive some existence results for a solution of vector variational inequalities. We also study the duality of implicit vector variational problems and discuss the relationship between solutions of dual and primal problems. Our results on duality contains known results in the literature as special cases.