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.     

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.     

关于Henig真有效性

在局部凸空间中,获得了Henig真有效点的一些等价条件,讨论了Henig真有效点与Benson真有效点之间的关系.     

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。     

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.     

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.     

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．     

多目标规划强较多有效解和弱较多有效解的标量化和对偶性

利用n维Euclid空间中较多锥的闭包,定义了多目标规划强较多有效解.利用较多锥闭包的Minkowski泛函,给出了强较多有效解和弱较多有效解的标量化结果,并讨论了强较多有效解和弱较多有效解的标量对偶问题及其应用.     

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.     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Superefficiency in Local Convex Spaces

In the framework of normed spaces, Borwein and Zhuang introduced superefficiency and gave its concise dual form when the underlying decision problem is convex. In this paper, we consider four different generalizations of the Borwein and Zhuang superefficiency in locally convex spaces and give their concise dual forms for convex vector optimization. When the ordering cone has a base, we clarify the relationship between Henig efficiency and the various kinds of superefficiency. Finally, we show that whether the four kinds of superefficiency are equivalent to each other depends on the normability of the underlying locally convex spaces.     

Integral type linear functionals on ordered cones

We introduce linear functionals on an ordered cone that are minimal with respect to a given subcone. Using concepts developed for Choquet theory we observe that the properties of these functionals resemble those of positive Radon measures on locally compact spaces. Other applications include monotone functionals on cones of convex sets, H-integrals on H-cones in abstract potential theory, and classical Choquet theory itself.

     

SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.     

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.     

Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior

In this paper, we consider the set-valued vector optimization problems with constraint in locally convex spaces. We present the necessary and sufficient conditions for Henig efficient solution pair, globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior.     

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.