集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念, 讨论了它与各种锥凸性的关系. 然后对恰当锥拟凸性得到了某种水平集意义下的刻画. 同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则. 最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

含参集值向量均衡问题各种有效解映射的下半连续性

在实Hausdorff拓扑向量空间中,引进含参集值向量均衡问题,给出各种有效解的概念.在锥-次类凸的条件下,得到各种有效解的标量化结果.结合集值映射的弱f-性,在适当假设条件下,得到含参集值向量均衡问题各种有效解映射的下半连续性.     

集值优化问题在有效意义下的最优性条件

本文在赋范空间中,讨论集值优化问题的有效元导数型最优性条件.当目标映射和约束映射的下方向导数存在时,在近似锥次类凸假设下利用有效点的性质和凸集分离定理得到了集值优化问题有效元导数型Kuhn-Thcker必要条件,在可微Г-拟凸性的假设下得到了Kuhn-Tucker最优性充分条件;此外利用集值映射沿弱方向锥的导数的特性给出了有效解最优性的另一种刻画.     

锥-拟凸集值映射的标量刻画[英文]

利用非零连续线性泛函和Gerstewitz非线性标量函数,本文主要获得如下结论:集值映射的锥-拟凸性可以由实值函数的拟凸性完全刻画.本文所得结果改进了已有文献中的相应结果.     

锥拟凸集值映射与集值映射的锥半连续性

本文借助锥界集定义了一种新的锥拟凸集值映射,并且利用广义Minkowski泛函讨论了锥半连续及锥外上半连续集值映射的锥拟凸性．     

内部锥-凸集值映射Henig有效性的Morea-Rockafellar定理

本文利用集值映射弱次梯度的Morea-Rockafellar定理,在内部(锥)-凸性假设下,得到了集值映射关于Henig有效性的Morea-Rockafellar定理.其结论为:在内部(锥)-凸条件下,两个集值映射和的Henig有效次梯度可以表示成它们Henig有效次梯度的和.     

集值广义强向量拟均衡问题解的存在性

利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义以及Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了集值广义强向量拟均衡问题解的存在性定理.把相关文献中所得的关于单值映射解的存在性结果推广到了集值映射的情形.     

生成锥内部凸-锥-类凸集值优化问题的标量化和鞍点

讨论集值向量优化的标量化和鞍点问题.在生成锥内部凸-锥-类凸假设下,建立了集值向量优化问题在（弱）有效和Benson真有效意义下的标量化定理和鞍点定理.     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

集值映射向量优化问题的ε—超有效解

本文引进了集值映射向量优化问题的ε－超有效解概念,并在集值映射为近似广义锥次似凸的假设下,建立了关于ε－超有效解的标量化定理和Lagrange乘子定理。     

Variational characterization of the contingent epiderivative

In this paper the existence of the contingent epiderivative of a set-valued map is studied from a variational perspective. We give a variational characterization of the ideal minimal of a weakly compact set. As a consequence we characterize the existence of the contingent epiderivative in terms of an associated family of variational systems. When a set-valued map takes values in Rn we show that these systems can be formulated in terms of the contingent epiderivatives of scalar set-valued maps. By applying these results we extend some existing theorems.     

Epidifferentiability of the map of infima in Hilbert spaces

This article deals with derivatives for set-valued maps that take values in ordered vector spaces, in particular it concerns about the relationship between the epiderivatives of a set-valued map and its associated map of infima. When the image space is a real separable Hilbert space ordered by an orthonormal basis, by using a variational technique based on a decoupling of the ordering cone into half-spaces, we show that both epiderivatives coincide under certain hypothesis of compactness and stability. Furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set-valued maps.     

Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.     

TOPOLOGICAL STRUCTURE OF EFFICIENT SET OF OPTIMIZATION PROBLEM OF SET－VALUED MAPPING

ＴＯＰＯＬＯＧＩＣＡＬ ＳＴＲＵＣＴＵＲＥ ＯＦ ＥＦＦＩＣＩＥＮＴ ＳＥＴ ＯＦ ＯＰＴＩＭＩＺＡＴＩＯＮ ＰＲＯＢＬＥＭ ＯＦ ＳＥＴ－ＶＡＬＵＥＤ ＭＡＰＰＩＮＧ ￥ＬＩＹＵＡＮＸＩＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉ...     

Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied.     

Some useful set-valued maps in set optimization

In this article, we introduce several classes of set-valued maps which can be useful in set optimization due to their applications. Exactly, we present some set-valued maps defined by scalar and vector functions and study their properties such as continuity and convexity among others. In addition, we compute their asymptotic maps which can be employed to establish coercivity and existence results in the framework of set optimization problems. Finally, we expose some possible directions for further research.     

Generalized convex set-valued maps

The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x,y in the domain and for all t in ]0,1[…” by the corresponding conditions of type “for all x,y in the domain there exists t in ]0,1[….”