ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKLY COMPACT CONVEX SET

This papeer studies the known density theorem of Arrow-Barankin-Blackwell. The following main result is obtained: If X is a Hausdorff locally convex Topological space and C belong to X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.     

On the existence of efficient points in locally convex spaces

We study the existence of efficient points in a locally convex space ordered by a convex cone. New conditions are imposed on the ordering cone such that for a set which is closed and bounded in the usual sense or with respect to the cone, the set of efficient points is nonempty and the domination property holds.     

关于Arrow-Barankin-Blackwell定理的一点注记

关于Ａｒｒｏｗ－Ｂａｒａｎｋｉｎ－Ｂｌａｃｋｗｅｌｌ定理的一点注记傅万涛（南昌大学，３３００４７）设Ｘ是一个局部凸拓扑向量空间，其对偶空间为为闭凸锥，ＡＸ．我们称是Ａ的有效点，如果不存在，使得，即一个点Ａ称为Ａ的正真有效点，如果存在一个严格正的线性连...     

Contractibility and Connectedness of Efficient Point Sets

Using the technique of space theory and set-valued analysis, we establish contractibility results for efficient point sets in a locally convex space and a path connectedness result for a positive proper efficient point set in a reflexive space. We also prove a connectedness result for a positive proper efficient point set in a locally convex space; as an application, we give a connectedness result for an efficient solution set in a locally convex space.     

Continuity at the extreme points

We prove that a bounded convex lower semicontinuous function defined on a convex compact set K is continuous at a dense subset of extreme points. If there is a bounded strictly convex lower semicontinuous function on K, then the set of extreme points contains a dense completely metrizable subset.     

An Analog of the Krein--Mil'man Theorem for Strongly Convex Hulls in Hilbert Space

We prove the following theorem: in Hilbert space a closed bounded set is contained in the strongly convex R-hull of its R-strong extreme points. R-strong extreme points are a subset of the set of extreme points (it may happen that these two sets do not coincide); the strongly convex R-hull of a set contains the closure of the convex hull of the set.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Banach空间中关于有界集的同时远达问题的适定性

本文研究Banach空间中关于有界集的同时远达问题的适定性,在集合的Hausdorff距离下,证明了：对自反局部一致凸Banach空间中的闭有界集K,使所有关于K的同时远达问题是适定的紧凸子集A全体在紧凸子集全体中是Gδ型集．     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.     

Efficiency in constrained continuous location

We present a geometrical characterization of the efficient, weakly efficient and strictly efficient points for multi-objective location problems in presence of convex constraints and when distances are measured by an arbitrary norm. These results, established for a compact set of demand points, generalize similar characterizations previously obtained for uncontrained problems. They are used to show that, in planar problems, the set of constrained weakly efficient points always coincides with the closest projection of the set of unconstrained weakly efficient points onto the feasible set. This projection property which are known previously only for strictly convex norms, allows to easily construct all the weakly efficient points and provides a useful localization property for efficient and strictly efficient points.     

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.     

Henig proper efficient points and generalized Henig proper efficient points

Applying the theory of locally convex spaces to vector optimization, we investigate the relationship between Henig proper efficient points and generalized Henig proper efficient points. In particular, we obtain a sufficient and necessary condition for generalized Henig proper efficient points to be Henig proper efficient points. From this, we derive several convenient criteria for judging Henig proper efficient points.     

极大单调算子和非扩展映射的迭代收敛定理及其应用

In this paper, two iterative schemes for approximating common element of the set of zero points of maximal monotone operators and the set of fixed points of a kind of generalized nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Two strong convergence theorems are obtained and their applications on finding the minimizer of a kind of convex functional are discussed, which extend some previous work.     

关于Banach空间中渐近非扩张型半群的不动点的存在性

设C是具有弱一致正规结构的Banach空间X的非空弱紧凸子集, T={T(t):t∈S}是渐近非扩张型半群, 且每个T(t)在C上连续. 该文证明了如下结论:(i) 若X是一致凸的, 则F(T) 非空;(ii) 若T={T(t):t∈S}满足lim inf_{t→∞,t in S}|‖T(t)‖|<+∞, 且在C上弱渐近正则, 则F(T)非空, 其中|‖T(t)‖|是T(t)的精确的Lipsch itz常数,F(T)是T(t),t∈S的所有公共不动点之集.     

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     

On a Helly-type question for central symmetry

Periodica Mathematica Hungarica - We study a certain Helly-type question by Konrad Swanepoel. Assume that X is a set of points such that every k-subset of X is in centrally symmetric convex...     

On the X-Y convex hull of a set of X-Y polygons

We study the class of rectilinear polygons, calledX – Y polygons, with horizontal and vertical edges, which are frequently used as building blocks for very large-scale integrated (VLSI) circuit layout and wiring. In the paper we introduce the notion of convexity within the class ofX – Y polygons and present efficient algorithms for computing theX – Y convex hulls of anX – Y polygon and of a set ofX – Y polygons under various conditions. Unlike convex hulls in the Euclidean plane, theX – Y convex hull of a set ofX – Y polygons may not exist. The condition under which theX – Y convex hull exists is given and an algorithm for testing if the given set ofX – Y polygons satisfies the condition is also presented.     

Banach upper density recurrent points of <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-flows

Let X denote a compact metric space with distance d and F:X×R→X or F_t:X→X denote a C⁰-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C⁰-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.