向量似变分不等式解的存在性及解集的稳定性

本文首先得到一类广义向量似变分不等式问题的解的存在性定理,然后利用usco映射的性质,讨论广义向量似变分不等式的解集的通有稳定性,得到大多数(在Baire分类意义下)广义向量似变分不等式问题的解集是稳定的;另外还引入广义向量似变分不等式解集的本质连通区的概念,并证明了满足一定连续性、凸性条件的广义向量似变分不等式的解集至少存在一个本质连通区.     

加权拟变分不等式

本文介绍了Stampacchia型和Minty型加权拟变分不等式以及加权拟变分不等式组,并讨论了它们之间的关系.利用这种关系研究了加权拟变分不等式和加权拟变分不等式组的解的存在性.     

关于混合拟似变分包含问题的解的迭代算法

本文对混合拟似变分包含问题提出新的辅助变分不等式,首先证明辅助变分不等式存在唯一解.然后,通过这一辅助形式建立混和拟似变分包含问题解的迭代算法.最后讨论在新的算法下迭代解的收敛性.     

广义向量似变分不等式解集的通有稳定性

本文首先得到广义向量似变分不等式问题的解的存在性定理,然后利用UＳＣＯ映射的性质（见ＴａｎＫＫ（ｅｔａｌ）（１９９５））,讨论广义向量似变分不等式的解集的通有稳定性,得到大多数（拓扑意义下）广义向量似变分不等式问题的解集是稳定的．     

广义向量似变分不等式解集的通有稳定性及本质连通区的存在性

本文研究广义向量似变分不等式解集的稳定性．证明了在满足一定的连续性和凸 性条件的广义向量似变分不等式问题构成的空间Ｍ中，大多数（在Ｂａｉｒｅ分类意下）广 义向量似变分不等式问题的解集是稳定的，并证明了Ｍ中的每个广义向量似变分不等式 的解集至少存在一个本质连通区。     

向量最优化弱有效解的特征和存在性

研究多目标凸向量优化问题在Gateaux可微条件下弱有效解的特性,并讨论一类非凸向量最优化问题弱有效解及与一变分不等式的等价性,给出了解的存在性。     

自反Banach空间内混合非线性似变分不等式解的算法*

本文在自反Banach空间内研究了一类混合非线性似变分不等式应用作者得到的一个极小极大不等式,对这类混合非线性似变分不等式的解,证明了几个存在唯一性定理其次由应用辅助问题技巧,作者建议了一个计算此类混合非线性似变分不等式的近似解的创新算法最后讨论收敛性准则.     

一类广义不变凸函数和向量变分不等式

首先在函数不变凸性的基础上,引进了一类广义不变凸函数,称之为(α,ρ,η)-不变凸函数,并给出实例说明了这类广义凸函数的存在性.其次,在(α,ρ,η)-不变凸性假设下,研究了Stampacchia型和Minty型向量变分不等式与多目标规划解之间的关系.最后,利用KKM定理讨论了向量变分不等式解的存在性问题.     

广义向量变分不等式和广义对偶模型

研究实Banach空间中带有不等式约束的非光滑向量优化问题(VP).首先,借助下方向导数引进了广义Minty型向量变分不等式,并通过变分不等式来探讨问题(VP)的最优性条件.接着,利用函数的上次微分构造了不可微向量优化问题(VP)的广义对偶模型,并且在适当的弱凸性条件下建立了弱对偶定理.     

一类混合似变分不等式问题解的存在唯一性

本文中,我们首先给出了一类混合似变分不等式问题．接着,在Banach空间中研究了它的解的存在性和唯一性．最后,讨论了混合似变分不等式问题的扰动问题,并证明了扰动问题的解的存在唯一性定理．     

On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel–Penot subdifferentials

In this paper, we establish some results which exhibit an application for Michel–Penot subdifferential in nonsmooth vector optimization problems and vector variational-like inequalities. We formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the Michel–Penot subdifferentials and use these variational-like inequalities as a tool to solve the vector optimization problem involving nonsmooth V-invex function. We also consider the corresponding weak versions of the vector variational-like inequalities and establish various results for the weak efficient solutions.     

Vector variational-like inequalities and vector optimization problems in Asplund spaces

In this paper, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of Mordukhovich limiting subdifferentials are introduced and studied in Asplund spaces. Some relations between the vector variational-like inequalities and vector optimization problems are established by using the properties of Mordukhovich limiting subdifferentials. An existence theorem of solutions for the weak Minty vector variational-like inequality is also given.     

Existence results for Stampacchia and Minty type vector variational inequalities

In this article, we consider the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for set-valued maps and prove the existence of their solutions in the setting of Banach spaces as well as topological vector spaces. We point out that our vector variational inequalities extend and generalize several vector variational inequalities that appeared in the literature. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.     

Differential and sensitivity properties of gap functions for Minty vector variational inequalities

The purpose of this paper is to investigate differential properties of a class of set-valued maps and gap functions involving Minty vector variational inequalities. Relationships between their contingent derivatives are discussed. An explicit expression of the contingent derivative for the class of set-valued maps is established. Optimality conditions of solutions for Minty vector variational inequalities are obtained.     

Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces

We consider two generalized Minty vector variational-like inequalities and investigate the relations between their solutions and vector optimization problems for non-differentiable α-invex functions.     

Generalized vector variational-like inequalities and vector optimization

In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems.     

Minty lemma for inverted vector variational inequalities

In this paper, we introduce Stampacchia-type inverted vector variational inequalities and Minty-type inverted vector variational inequalities and discuss Minty lemma for the inequalities showing the existence of solutions to them in Banach spaces. Next, we consider the equivalence of our Minty lemma with Brouwer’s fixed point theorem as an application.     

On vector variational-like inequalities and vector optimization problems with (G,α)-invexity

The aim of this paper is to study the relationship among Minty vector variationallike inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving(G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem,Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of(G, α)-invex functions. Examples are provided to elucidate our results.     

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.     

On minty variational principle for nonsmooth vector optimization problems with approximate convexity

In this paper, we consider a vector optimization problem involving locally Lipschitz approximately convex functions and give several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Stampacchia and Minty type and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem.