The connectedness of the solutions set for generalized vector equilibrium problems

In this paper, we establish the connectedness of the sets of Henig efficient solutions, globally efficient solutions, weak efficient solutions, superefficient solutions and efficient solutions for a class of generalized vector equilibrium problems without the assumptions of monotonicity and compactness.     

Generalized vector variational-like inequalities and nonsmooth vector optimization problems

In this article, we establish some relationships between a solution of generalized vector variational-like inequalities and an efficient solution or a weakly efficient solution to the nonsmooth vector optimization problem under the assumptions of pseudoinvexity or invariant pseudomonotonicity. Our results extend and improve the corresponding results in the literature.     

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (E_n, ξ_n) be an inductive limit of a sequence (E_n, ξ_n)_{n∈ N} of locally convex spaces and let every step (E_n, ξ_n) be endowed with a partial order by a pointed convex (solid) cone S_n. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (S_n)_{n∈ N} of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.     

关于局部凸拓扑向量空间中广义双拟变分不等式

In this paper we investigate generalized bi-quasi-variational inequalities in locally convex topological vector spaces. Motivated and inspired by the recent research work in this field,we establish several existence theorems of solutions for generalized bi-quasi-variational inequalities,which are the extension and improvements of the earlier and recent results obtained previously by many authors including Sun and Ding [18],Chang and Zhang [23] and Zhang [24].     

Existence of weakly efficient solutions in vector optimization

In this paper, we present an existence result for weak efficient solution for the vector optimization problem. The result is stated for invex strongly compactly Lipschitz functions.     

Existence of Solutions to Generalized Vector Quasi-variational-like Inequalities with Set-valued Mappings

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.     

Existence of Solutions to Generalized Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings

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.     

函数空间上的模糊线性拓扑

本文给出了模糊拓扑向量空间（Ｘ，Ｗ）到（Ｙ，Ｊ）的函数族Ｆ上的模糊线性拓扑，证明了若值域空间（Ｙ，Ｊ）是（Ｑ）型的、局部凸的模糊拓扑向量空间，则（Ｆ，），也是型的、局部凸的模糊拓扑向量空间。     

Change of generalized indices of stationary solutions. to multiparametric optimization

We study the local change of the generalized index, which is a modification of the Morse index and the stationary index, for the multiparametric optimization. Under the Regular Value Condition, the change of the generalized index around a triplet (x,v,t) is locally bounded by the dimension of the parameter vector t, where x is a variable vector and v a vector of the Lagrange multiplier space. We also discuss the local change of the generalized index around a pair (x,t). Received: March 27, 1998 / Accepted: January 29, 2000?Published online April 20, 2000     

Generalized Bi quasi variational Inequalities in Locally Convex Topological Vector Spaces

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,Φｄｅｎｏｔｅｓｅｉｔｈｅｒｔｈｅｒｅａｌｆｉｅｌｄｏｒｔｈｅｃｏｍｐｌｅｘｆｉｅｌｄ．Ｆｏｒａｎｏｎｅｍｐ－ｔｙｓｅｔＹ,２ＹｗｉｌｌｓｔａｎｄｆｏｒｔｈｅｆａｍｉｌｙｏｆａｌｌｎｏｎｅｍｐｔｙｓｕｂｓｅｔｓｏｆＹ．ＬｅｔＥ,ＦｂｅｖｅｃｔｏｒｓｐａｃｅｓｏｖｅｒΦ,〈,〉：Ｆ×Ｅ→Φｂｅａｂｉｌｉｎｅａｒｆｕｎｃｔｉｏｎａｌ,ａｎｄＸｂｅａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＥ．Ｇｉｖｅｎａｍｕｌ－ｔｉ－ｖａｌｕｅｄｍａｐｐ…     

Existence of solutions for vector optimization

In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.     

局部凸空间中多值单调算子的隐补问题

应用Fan Ky定理讨论了局部凸空间中多值单调算子的隐补问题.     

Convexifactors,generalized convexity and vector optimization

In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.     

向量优化问题中的弱S-有效解

在局部凸拓扑线性空间中, 提出了集值向量优化问题的弱S-有效解和S-次似凸性概念. 在S-次似凸性假设下建立了择一性定理, 并利用择一性定理建立了弱S-有效解的标量化定理. 此外, 通过几个具体例子解释了主要结果.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Existence of solutions for a generalized vector quasivariational inequality

The paper deals with a generalization of a vector quasivariational inequality. An existence theorem for its solutions is established; it is based on a kind of minimax inequality, which is here established for continuous affine mappings and differs from previous results. Fan's section theorem for set-valued mappings is extended. An application for an equilibrium problem of a network with vector-valued cost functions is given.This research was supported by the National Nature Science Foundation of China.     

非紧集上的Hartman-Stampacchia变分不等式及其应用

在非紧集上证明了两类多值算子的 Hartman-Stampacchia 变分不等式, 作为应用, 讨论了多值算子的相补问题.     

Existences of solutions for generalized vector quasi-equilibrium problems

In the paper, existence theorems for two classes of generalized vector quasi-equilibrium problems are established by using a fixed point theorem and some examples are given to illustrate them. An application to vector quasi-optimization problems is shown. This research was partially supported by the National Natural Science Foundation of China (Grant numbers: 60574073 and 10471142).     

Sensitivity analysis in convex vector optimization

We consider a parametrized convex vector optimization problem with a parameter vectoru. LetY(u) be the objective space image of the parametrized feasible region. The perturbation mapW(u) is defined as the set of all minimal points of the setY(u) with respect to an ordering cone in the objective space. The purpose of this paper is to investigate the relationship between the contingent derivativeDW ofW and the contingent derivativeDY ofY. Sufficient conditions for MinDW=MinDY andDW=W minDY are obtained, respectively. Therefore, quantitative information on the behavior of the perturbation map is provided.The author would like to thank the anonymous referees for their helpful comments which improved the quality of this paper. The author would also like to thank Professor P. L. Yu for his encouragement.     

A type of Strassen's theorem for positive vector measures with values in dual spaces

In this paper, we extend a type of Strassen's theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.