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1.
In this paper, a key assumption is introduced by virtue of a parametric gap function. Then, by using the key assumption, sufficient conditions of the continuity and Hausdorff continuity of a solution set map for a parametric weak vector variational inequality are obtained in Banach spaces with the objective space being finite-dimensional. 相似文献
2.
In this paper, a key assumption similar to that of Li and Chen is introduced by virtue of a gap function for a class of parametric
set-valued weak vector variational inequalities in Banach spaces. By using this key assumption, sufficient and necessary conditions
of the continuity and Hausdorff continuity of the solution set mapping for such parametric set-valued weak vector variational
inequalities are given in Banach spaces when the image space is infinite dimensional. The results presented in this paper
generalize and improve some main results of Li and Chen. 相似文献
3.
4.
The paper mainly concerns the study of a parametric weak vector variational inequality. Robinson’s metric regularity and Lipschitzian
stability of the solution mapping for the parametric weak vector variational inequality are firstly established. Then sensitivity
analysis of the solution mapping for the parametric weak vector variational inequality is discussed. As applications, Lipschitzian
continuity and differentiability of the solution mapping are also investigated for a parametric variational inequality. 相似文献
5.
Chuqun Li 《Optimization》2016,65(8):1569-1584
In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping. 相似文献
6.
X.Q. Yang 《Journal of Optimization Theory and Applications》2003,116(2):437-452
Gap functions play a crucial role in transforming a variational inequality problem into an optimization problem. Then, methods solving an optimization problem can be exploited for finding a solution of a variational inequality problem. It is known that the so-called prevariational inequality is closely related to some generalized convex functions, such as linear fractional functions. In this paper, gap functions for several kinds of prevariational inequalities are investigated. More specifically, prevariational inequalities, extended prevariational inequalities, and extended weak vector prevariational inequalities are considered. Furthermore, a class of gap functions for inequality constrained prevariational inequalities is investigated via a nonlinear Lagrangian. 相似文献
7.
In this paper, we introduce and study a class of generalized vector quasivariational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space. 相似文献
8.
《Optimization》2012,61(9):1339-1352
In this article, by using the image space analysis, a gap function for weak vector variational inequalities is obtained. Its lower semicontinuity is also discussed. Then, these results are applied to obtain the error bounds for weak vector variational inequalities. These bounds provide effective estimated distances between a feasible point and the solution set of the weak vector variational inequalities. 相似文献
9.
Existence of Solutions to Generalized Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings 下载免费PDF全文
Dapeng Gao & Shiqiang Feng 《数学研究通讯:英文版》2014,30(1):90-96
In this paper, we introduce and study a class of generalized vector quasi-variational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and
generalized vector variational-like inequality problems as special cases. We use the
maximal element theorem with an escaping sequence to prove the existence results of
a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space. 相似文献
10.
Yunan Wu Yuchen Peng Long Peng Ling Xu 《Journal of Optimization Theory and Applications》2012,153(2):485-496
The super efficiency of a vector variational inequality is considered in this paper. We show that for both the single and
multiple criteria cases, a network equilibrium model can be recast as super efficient solutions to a kind of variational inequality.
For the network equilibrium model with a vector-valued cost function, we derive the necessary and sufficient condition in
terms of the super efficiency of a vector variational inequality by using the Gerstewitz’s function without any convex assumptions. 相似文献
11.
Jinlu Li 《Numerical Functional Analysis & Optimization》2019,40(2):178-193
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. 相似文献
12.
Mu Ming Wong 《Journal of Global Optimization》2010,46(3):435-446
This paper is concerned with the study of solution stability of a parametric vector variational inequality, where mappings
may not be strongly monotone. Under some requirements that the operator of a unperturbed problem is monotone or it satisfies
degree conditions then we show that the solution map of a parametric vector variational inequality is lower semicontinuous. 相似文献
13.
研究拓扑向量空间到连续线性映射空间映射的弱向量变分不等式和与之相关 的纯量型变分不等式解集的关系, 引入弱和强一致连续概念,利用纯量型变分不等式 解集所表征的集值映射的特性给出弱向量变分不等式解集连通的一个充分条件。 相似文献
14.
The set-valued variational inequality problem is very useful in economics theory and nonsmooth optimization. In this paper, we introduce some gap functions for set-valued variational inequality problems under suitable assumptions. By using these gap functions we derive global error bounds for the solution of the set-valued variational inequality problems. Our results not only generalize the previously known results for classical variational inequalities from single-valued case to set-valued, but also present a way to construct gap functions and derive global error bounds for set-valued variational inequality problems. 相似文献
15.
《Operations Research Letters》2021,49(2):283-289
In this paper, we first recall a class of parametric variational–hemivariational inequalities (PVHIs) introduced in Jiang et al. (2020). Then, based on the properties of the Clarke generalized gradient, we establish the Hölder continuity of the solution mapping for PVHIs in terms of regularized gap functions under some assumptions imposed on the data of PVHIs. Finally, an example is given to illustrate our main results. 相似文献
16.
Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities
Yong Zhao Jin Zhang Xinmin Yang Gui-Hua Lin 《Journal of Optimization Theory and Applications》2017,175(2):545-566
This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well. 相似文献
17.
Lkhamsuren Altangerel Gert Wanka 《Journal of Mathematical Analysis and Applications》2007,329(2):1010-1035
The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem. The conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality. The properties in the definition of the gap functions are verified by using the weak and strong duality theorems. 相似文献
18.
In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters. 相似文献
19.
For a convex program in a normed vector space with the objective function admitting the Gateaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gateaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.The research was supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable comments and constructive suggestions. 相似文献
20.
Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems
In this paper, we consider parametric primal and dual equilibrium problems in locally convex Hausdorff topological vector spaces. Sufficient conditions for the approximate solution maps to be Hausdorff continuous are established. We provide many examples to illustrate the essentialness of the imposed assumptions. As applications of these results, the Hausdorff continuity of the approximate solution maps for optimization problems, variational inequalities and fixed-point problems are derived. 相似文献