关于真有效点集的连通性

在局部有界的Ｈａｕｓｄｏｒｆｆ局部凸空间中讨论了集合的真有效点集的连通性问题。证明了当序锥具有基底时,任何非空紧凸集的真有效点集是连通的。     

约束锥扰动多目标规划锥有效解集的闭性和半连续性

本文研究拓扑向量空间中目标映射和约束映射均为连续,约束映射的约束锥为半连续的条件下,受扰动可达目标集的锥有效点集和锥弱有效点集的闭性、半连续性和锥半连续性.在此基础上,得到了约束锥扰动多目标规划问题的锥有效解集和锥弱有效解集的闭性和半连续性.     

锥有效拟凹集的有效点集的连通性

本文引进了锥有效拟凹集的概念，讨论了R ^m-严格拟凹与R ^m－有效拟凹的关系，证明了一个紧集为锥有效拟凹且其有效点集为闭集时，这个有效点集是连通的。     

序扰动多目标规划的锥次微分稳定性

对于局部凸拓扑向量空间的多目标规划问题,本文研究并得到当确定空间序的控制锥受扰动,它们的锥有效点（解）集和锥弱有效点（解）集分别在锥次微分和锥弱次微分意义下的稳定性结果.     

逼近锥族和严有效点

本文建立了逼近锥族存在的必要充分条件，并利用这个概念讨论了严有效点的截口性质、稠密性质和连通性质     

严有效点与Henig真有效点

文章证明了严有效点等价于Henig真有效点．利用这个等价关系，得到了局部凸空间中Henig真有效点的存在性条件。纯量化特征和稠密性定理．并且改进了已知的有关结果．     

自反Banach空间中锥拟凸映射多目标最优化问题锥有效解集的弱连通性

本文在自反Banach空间中引进了锥弱连续映射和点集的弱连通概念．在讨论锥弱连续和锥拟凸映射以及锥最小上界的几个性质的基础上,证明了当象集为锥凸集时定义在自反Ba- nach空间中的有界闭凸集上的锥弱连续和锥拟凸映射多目标最优化问题的锥有效解集是弱连通的．     

点锥和锐锥的特性

本文证明了ｎ维欧氏空间中的点锥和锐锥的特征属性，由此，得到关于点锥和锐锥的描述局部性质的结果。     

弱锥2有效集连通性的一个等价定理

利用连通集上有关上半连续点集映射的连通性质，本文证明了拓扑向量空间中向量最优化问题的弱锥－有效解集与相应目标空间中的弱锥－有效点集连通性之间的一个等价定理．     

向量优化中有效点的存在性

分别在有pre-order的无线性结构的集合和拓扑空间中，给出了有效点的存在性。作为应用，讨论了向量优化问题中解的存在性。最后给出了紧、弱紧、锥紧、锥半紧、上序紧、下序紧、上序半紧、准上序半紧和准下序半紧等之间的关系。     

多目标规划的ακ-较多有效解的几何特性

In this paper, with the help of properties of P-linear below bounded points, tangent cones, cone k-interior points, projective cones, the geometric char-acterizations of ακ-major efficient solutions and ακ-major optimality solutions of multiobjective programming are obtained.     

Comparison of Existence Results for Efficient Points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

New existence results for efficient points in locally convex spaces ordered by supernormal cones

In this paper, the definition of supernormality for convex cones in locally convex spaces is discussed in detail on many interesting examples. Starting from the new direction for the study of the existence of efficient points (Pareto type optimums) in locally convex spaces offered by the concept of supernormal (nuclear) cone, we establish some existence results for the efficient points using boundedness and completeness of conical sections induced by non-empty subsets and we specify properties for the sets of efficient points beside important remarks     

一类新的二阶组合切导数及其应用

引进了一种新的切锥,讨论它与相依切锥的关系.借助这种新的切锥引进了一类新的二阶组合切导数,并讨论了它与其他二阶切导数的关系.利用这类新的二阶组合切导数,建立了集值优化分别取得Henig有效元和全局有效元的最优性必要条件.     

The Fermat rule for multifunctions on Banach spaces

Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat’s rules as necessary conditions for an optimal solution of the above problems. As applications, we also provide some necessary conditions (in terms of Clarke’s normal cones or the limiting normal cones) for Pareto efficient points.This research was supported by a postdoctoral fellowship scheme (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong. Research of the first author was also supported by the National Natural Science Foundation of P. R. China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, P. R. China (Grant No. 2003A002M).     

赋范线性空间集合的严有效点

本文引入一个新的有效点概念一严有效点,它是Borwein超有效点的推广.此外还讨论了严有效点的基本性质:存在性条件、纯量化特征、稠密性定理以及与Borwein超有效点的关系.     

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.     

Geoffrion type characterization of higher-order properly efficient points in vector optimization

We consider the constrained vector optimization problem minCf(x), x∈A, where X and Y are normed spaces, A⊂X₀⊂X are given sets, C⊂Y, C≠Y, is a closed convex cone, and is a given function. We recall the notion of a properly efficient point (p-minimizer) for the considered problem and in terms of the so-called oriented distance we define also the notion of a properly efficient point of order n (p-minimizers of order n). We show that the p-minimizers of higher order generalize the usual notion of a properly efficient point. The main result is the characterization of the p-minimizers of higher order in terms of “trade-offs.” In such a way we generalize the result of A.M. Geoffrion [A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (3) (1968) 618-630] in two directions, namely for properly efficient points of higher order in infinite dimensional spaces, and for arbitrary closed convex ordering cones.