集值映射的Henig有效次微分及其稳定性

该文在赋范线性空间中对集值映射引入锥- Henig有效次梯度和锥- Henig有效次 微分的概念. 借助凸集分离定理证明了锥- Henig有效次微分的存在性, 并且建立了线性泛函为锥- Henig有效次梯度的充要条件. 最后, 对于一类参数 扰动集值优化问题讨论了其在Henig有效意义下的稳定性.     

集值映射全局真有效次梯度下的Moreau-Rockafellar定理

本文研究了集值映射的Moreau-Rockafellar型定理的问题.利用集值映射弱次梯度的Moreau-Rockafellar定理,在内部(锥)-凸条件下,获得了集值映射关于全局真有效性的Moreau-Rockafellar型定理结果,推广了集值映射在锥-凸假设下的Moreau-Rockafellar型定理的结果,所得结论深化和丰富了最优化理论的内容.     

生成锥内部凸-锥-类凸集值优化问题的Henig真有效性

该文讨论局部凸空间中的约束集值优化问题. 首先, 在生成锥内部凸-锥-类凸假设下, 建立了Henig真有效解在标量化和Lagrange乘子意义下的最优性条件. 其次, 对集值Lagrange映射引入Henig真鞍点的概念, 并用这一概念刻画了Henig真有效解. 最后, 引入了一个标量Lagrange对偶模型, 并得到了关于Henig真有效解的对偶定理. 另外, 该文所得结果均不需要约束序锥有非空的内部.     

一类集值映射向量优化问题的ε-有效解

集值映射向量优化问题是最优化理论中的一个重要方向.在集值映射为生成锥内部-锥一类凸(简记为ic-锥类凸)的假设条件下,利用择一定理,给出了集值映射向量优化问题ε-弱有效解和ε-有效解的最优性条件和ε-Lagrange乘子定理,是弱有效解和有效解相应结果的推广.     

生成锥内部凸-锥-类凸集值优化问题的超有效性

本文讨论生成锥内部凸-锥-类凸集值向量优化问题的超有效解.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超有效意义下的标量化、Lagrangian乘子和鞍点定理     

向量优化的Henig真有效点（解）集的连通性

在局部凸的Hausdorff空间中,首先给出Henig真有效点的等价形式,由此得到了Henig真有效点的纯量化形式.借助纯量化形式,证明了Henig真有效点集的连通性.其次,对集值优化问题,当目标集值映射足锥凸时,给出其象集的Henig真有效点集与其象集凸包的Henig真有效点集是相等的结论,并给出了集值映射象集的Henig真有效点的纯量化形式.最后,证明了集值优化问题的Hcnig真有效解集的连通性.     

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.     

集值优化强有效解的广义二阶锥方向导数刻画

在实赋范线性空间中考虑集值优化问题的强有效性．借助Henig扩张锥和基泛函的性质,利用广义二阶锥方向相依导数,得到受约束于集值映射的优化问题,取得强有效元的二阶最优性必要条件．当目标函数为近似锥一次类凸映射时,利用强有效点的标量化定理,得到集值优化问题,取得强有效元的二阶充分条件．     

广义凸拓扑线性空间集值优化的ε-强有效解

在Hausdorff局部凸拓扑线性空间中考虑约束集值优化问题(VP)的ε-强有效性.在内部锥类凸假设下,利用凸集分离定理,分别建立了关于ε-强有效解的标量化定理和ε-Lagrange乘子定理.     

集值映射ε-严有效点集的次微分及应用

本文研究了在局部凸Hausdorff拓扑向量空间中的集值映射ε-严有效次梯度和ε-严有效次微分的问题.利用凸集分离定理的方法,获得了该次微分(次梯度)的存在性及它的一些性质,推广了一类参数扰动集值优化问题在ε-严有效意义下的稳定性的结果.     

Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps

In this paper, we introduce the notion of (Benson) proper subgradient of a set-valued map and prove that, for some class of nonconvex set-valued maps, a proper subgradient of the sum of two set-valued maps can be expressed as the sum of two proper subgradients of these maps. This property is also established for weak subgradients. A result in Ref. [Lin, L.J.: J. Math. Anal. Appl. 186, 30–51 (1994)], obtained under some convexity assumption, is included as a special case of the corresponding result of this paper. The author thanks the anonymous referees for their valuable remarks.     

ε-Optimality Conditions for Vector Optimization Problems with Set-Valued Maps

In this paper, I introduce two kinds of ε-subgradients—ε-Pareto subgradients and ε-Benson proper subgradients—for set-valued maps. I give sufficient conditions for the existence of such ε-subgradients. A generalized ε-Moreau–Rockafellar type theorem for ε-Benson proper subdifferentials of set-valued maps is formulated and applied to establish ε-optimality conditions for ε-Benson proper efficient solutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.     

集值映射的次微分以及最优性条件

基于已有的集值映射的弱次微分的概念,定义了集值映射的Henig全局次微分,研究了它的存在性条件以及运算性质．利用这一概念,分别给出了具约束向量集值最优化问题的Henig全局有效解对的必要性条件和充分性条件．     

集值优化问题在Henig真有效解意义下的导数型最优性条件

给出实的赋范空间中集值映射的Henig真有效解集的一些性质,并利用集值映射的相依上图导数和集值映射的次微分给出了集值优化问题Henig真有效解的最优性条件的充要条件.     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

带约束条件集值优化问题近似Henig有效解集的连通性

研究了带约束条件集值优化问题近似Henig有效解集的连通性.在实局部凸Hausdorff空间中,讨论了可行域为弧连通紧的,目标函数为C-弧连通的条件下,带约束条件集值优化问题近似Henig有效解集的存在性和连通性.并给出了带约束条件集值优化问题近似Henig有效解集的连通性定理.     

Higher order optimality conditions for Henig efficient solutions in set-valued optimization

In this paper, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued maps are introduced. Necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.     

Henig proper generalized vector quasiequilibrium problems

In this paper, we give sufficient conditions for the existence of a Henig proper efficient solution of a general model in the theory of set-valued vector quasiequilibrium problems with moving cones. The main result of this paper is new, and is established under assumptions of existence of open lower sections and some properties of cone-semicontinuity and cone-concavity of set-valued maps. The moving cones are assumed to have bases which are Hausdorff lower semicontinuous and satisfy an additional suitable property.     

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.