向量集值映射的共轭对偶

借助抽象算子将共轭映射的概念的到抽象空间,引入了集值映射的共轭映射和次梯度,据此讨论了集值映射共轭对偶的全局稳定性。     

Conjugate Duality in Constrained Set-Valued Vector Optimization Problems

In this article, under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are proposed for a constrained set-valued vector optimization problem. Weak duality, strong duality, and stability criteria are investigated. The inclusion relations between the image sets of the dual problems are also discussed.     

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.     

非线性参数规划问题$\varepsilon$-最优解集集值映射的连续性

对非线性参数规划问题$\varepsilon$-最优解集集值映射的连续性条件进行了研究.首先在可行集集值映射局部有界且正则的条件下,讨论了非线性参数规划问题最优值函数的连续性,然后针对$\varepsilon$-最优解集集值映射的结构特征并利用此结果和集值分析理论,给出了非线性参数规划问题$\varepsilon$-最优解集集值映射连续的一个充分条件.     

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.     

Conjugate dual problems in constrained set-valued optimization and applications

In this paper, two conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

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.     

Cascade search for preimages and coincidences: Global and local versions

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.     

对称集值向量拟均衡问题解集的稳定性

本文研究了对称集值向量拟均衡问题解集的稳定性.证明了在约束映射满足一定连续性与目标映射是锥-恰当拟凸的集值映射条件下,对称集值向量拟均衡问题的解集是稳定的,还证明了每个对称集值向量拟均衡问题的解集至少存在一个本质连通区.     

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.     

Conjugate Duality for Constrained Vector Optimization in Abstract Convex Frame

This article focuses on a conjugate duality for a constrained vector optimization in the framework of abstract convexity. With the aid of the extension for the notion of infimum to the vector space, a set-valued topical function and the corresponding conjugate map, subdifferentials are presented. Following this, a conjugate dual problem is proposed via this conjugate map. Then, inspired by some ideas in the image space analysis, some equivalent characterizations of the zero duality gap are established by virtue of the subdifferentials.     

Optimality conditions for set-valued maps with set optimization

Problems in set-valued optimization can be solved via set optimization. In this paper optimality conditions are studied for set-valued maps with set optimization. Optimality requirements are established for continuous selections using directional derivatives. Necessary and sufficient conditions for the existence of solutions are shown for set-valued maps under generalized convexity assumptions and with the notion of the contingent derivative.     

Multi-mobs

The study of set-valued multiplications on a set is originated and developed by O. Ore. The development of projective geometry as a multigroup by W. Prenowitz is a cute application to it. On the other hand, the topological observations of set-valued functions have been investigated extensively over the past forty years. No literature concerning topological algebra based on a set-valued multiplication could readily be found. This paper is devoted to the investigation of a basic theory of binary set-valued topological algebra combining the above two algebraic and topological concepts together. The author wishes to express his sincere gratitude to Professor Alexander D. Wallace for his continued guidance and encouragement.     

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.     

A minimal set-valued strong derivative for vector-valued Lipschitz functions

A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here.     

集值局部平方可积鞅

介绍集值类（D）过程和集值局部鞅的概念和有关性质，进而讨论了集值局部平方可积鞅的概念和性质。     

向量均衡问题强有效解的稳定性

在可行集扰动而向量值映射不扰动、可行集与向量值映射均扰动、可行集扰动而集值映射不扰动以及可行集与集值映射均扰动四种情况下,分别讨论了向量均衡问题强有效解的稳定性.     

目标函数扰动的集值优化问题的稳定性

本文研究了由目标函数扰动的集值优化问题的有效点集所定义的集值映射的半连续性.讨论了目标函数扰动的集值优化问题在上半连续意义下的稳定性.特别地,在广义适定性条件下,证明了集值优化问题在上半连续意义下的稳定性.