γ-锥和严格γ-锥

引入有限维向量空间的γ－锥和严格γ－锥的概念．证明了γ－锥是一类闭锥,严格γ－锥是一类开锥．并且,研究了γ－锥和严格γ－锥与较多锥、严格较多锥及它们的闭包和内部之间的关系．     

r-锥和严格r-锥

引入有限维向量空间的r-锥和严格r-锥的概念,证明了r-锥是一类闭锥,严格r-锥是一类开锥。并且,研究了r-锥和严格r-锥与较多锥、严格较多锥及它们的闭包和内部之间的关系。     

ak-较多锥的有关性质

对于有限维Ｅｕｃｌｉｄ空间中带两个参数的ａｋ－较多锥,本文讨论了它与正锥以及有关较多锥类之间的若干关系．同时,还得到此类锥的一些基本性质．     

关于锥理论的一些注记

指出了再生锥和极小锥之间的关系,证明了一些具体锥的全正则性,探讨了半序空间中上确界的一些不同点,推广和发展了已有文献中的相关结论.     

张量空间中的多面体锥

本文利用文献 [1 ]中的结果给出了张量空间中多面体锥的性质 ,指出当 K1,K2 分别为 V1,V2中的多面体锥时 ,由它们所生成的 V1 V2 中的最小真正锥也是多面体锥 ,这一条件不仅是充分的也是必要的 .并利用这一结果对多面体锥的已有结论给出了新的证明     

张量空间中的真正锥

本指出了献「１」在张量空间中定义的两种锥是一致的,证明了它们是张量空间中的最小真正锥,并可用来表示有限维实空间中由锥不变算子所组成的锥,因而可用来研究锥不变算子。     

体锥与满足角性质锥之间的关系

本文证明：当锥Λ的极锥是尖锥或由不含θ点的闭球体张成时，Λ是体锥；拟多面体锥Λ和极锥Λ*弱*内部非空锥类是一致的并指出了P．Q．Khanh（1992年）的一个错误．     

张量空间中锥的可分解性

在献[1]的基础上研究张量空间中锥的性质,得到了张量空间中射影锥的极端向量的表示形式,给出了张量空间中射影锥可分解的充分必要条件,并由此可得出有限维实空间中真正锥可分解的已有结论。     

张量空间中的多面体锥

本文利用文献[1]中的结果给出了张量空间中多面体锥的性质,指出当K1,K2分别为V1,V2中的多面体锥时,由它们所生成的V1&;#215;V2中的最小值正锥也是正多面体锥,这一条件不仅是充分的也是必要的,并利用这一结果对多面体锥的已有结论结出了新的证明。     

拓扑向量空间中锥半连续性与锥拟凸映射

对于拓扑向量空间中的几个锥拟凸映射,讨论了它们之间的若干关系,同时,对锥半连续及锥上半连续映射的锥拟凸性给出了一些结果。     

Generalization of the Dubovitzky-Milutyn optimality conditions

Various types of conical approximation of sets are discussed. The notions of an external cone and an internal cone result in a strengthened separation theorem. Based on this theorem, some generalizations of the Dubovitzky-Milutyn optimality conditions are given.This research was sponsored by the Institute of Automatic Control, Department of Electronics, Technical University, Warsaw, Poland.     

An LMI description for the cone of Lorentz-positive maps II

Let L _n be the n-dimensional second-order cone. A linear map from ? ^m to ? ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n,?m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n???1)(m???1) that describes this cone.     

Characterization of the Cone of Attainable Directions

A useful characterization of the cone of attainable directions is provided.This research was partially supported by the Ministerio de Ciencia y Tecnología (Spain), Project BFM2003-02194.     

Contingent cone to a set defined by equality and inequality constraints at a Fréchet differentiable point

In this paper, we prove equality expression for the contingent cone and the strict normal cone to a set determined by equality and/or inequality constraints at a Fréchet differentiable point. A similar result has appeared before in the literature under the assumption that all the constraint functions are of classC or under the assumption that the functions are strictly differentiable at the point in question. Our result has applications to the calculation of various kinds of tangent cones and normal cones.This research was supported, in part by the National Science and Engineering Research Council of Canada under Grant No. OGP-41983.The authors would like to thank D. E. Ward for his many helpful comments.     

Further relationship between second-order cone and positive semidefinite matrix cone

It is well known that second-order cone (SOC) programming can be regarded as a special case of positive semidefinite programming using the arrow matrix. This paper further studies the relationship between SOCs and positive semidefinite matrix cones. In particular, we explore the relationship to expressions regarding distance, projection, tangent cone, normal cone and the KKT system. Understanding these relationships will help us see the connection and difference between the SOC and its PSD reformulation more clearly.     

A Class of Cone Bounded Quasiconvex Mappings in Topological Vector Spaces

Within the context of cone-ordered topological vector spaces, this paper introduces the concepts of cone bounded point and cone bounded set for vector set. With their aid, a class of new cone quasiconvex mappings in topological vector spaces is defined, and their fundamental properties are presented. The relationships between the cone bounded quasiconvex mapping defined in this paper and cone convex mapping, and other known cone quasiconvex mapping are also discussed.     

拓扑向量空间中的几个锥广义凸映射

本文引进拓扑向量空间中的几个锥伪凸映射的概念,并且讨论了它们之间及其它的锥广义凸映射之间的关系,最后,给出一个显示各种关系的图表。     

Some topological properties in optimization theory

We study some topological properties of the vector sum of two sets, and we introduce the concept of strong cone compactness, a slightly strengthened version of the notion of cone compactness, which is used frequently in vector optimization. Furthermore, we establish a generalization of the well-known fact that the continuous image of a compact set is compact.