集优化问题的广义适定性与解的稳定性

本文研究了集优化问题的适定性与解的稳定性. 首次利用嵌入技术引入了集优化问题的广义适定性概念, 得到了此类适定性的一些判定准则和特征, 并给出其充分条件. 此外, 借助一类广义Gerstewitz 函数, 建立了此类适定性与一类标量优化问题广义适定性之间的等价关系. 最后, 在适当条件下研究了含参集优化问题弱有效解映射的上半连续性和下半连续性.     

向量值映射的恰当锥拟凸性和锥半连续性的刻画北大核心CSCD

首先讨论了一个非线性标量化函数的基本性质并给出了其对偶形式.在此基础上建立了对向量值映射的恰当锥拟凸性的刻画.然后提出了锥形邻域的概念并给出了向量值映射的一类新的锥半连续性的统一定义.最后通过两个非线性标量化函数得到了对向量值映射的锥半连续性的完整刻画.     

变动控制结构下几乎-锥-凸映射的标量函数刻画

本文讨论变动控制结构下广义锥凸映射的线性和非线性标量函数的刻画问题.首先在变动序拓扑向量空间中证明了由正极锥的极方向所刻画的向量值映射的几乎-锥-凸性;其次,对变动控制结构引入了一种非线性标量函数,并利用这种非线性标量函数,得到了几乎-锥-凸向量值映射的标量刻画.     

有限理性与广义向量变分不等式问题的良定性

利用非线性标量化的技巧定义了广义向量变分不等式问题的理性函数,利用有限理性模型对广义向量变分不等式问题引入了一种新的良定性,这种良定性统一了广义向量变分不等式问题的Levitin-Polyak良定性与Hadamard良定性,且进一步的给出了广义向量变分不等式问题的各种良定性的充分条件.     

向量值Laplace变换与右连续积分半群

该文利用向量值Ｌａｐｌａｃｅ变换给出一类向量值函数的表示，并将它应用于Ｂａｎａｃｈ空间的Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ性质的刻划，引进了右连续积分半群，证明了一类非稠定且指数增长的算子对应的Ｃａｕｃｈｙ问题是适定的．     

两个向量值映射的极大极小不等式

在不利用非线性标量函数和线性标量函数的情况下,获得了一类两个向量值映射的极小极大定理和广义的向量Ky Fan极小极大不等式.     

改进集映射下参数广义向量拟平衡问题解映射的Berge下半连续性

该文主要讨论了一类新的参数广义向量拟平衡问题解映射的稳定性.首先,定义了改进集映射,基于改进集映射,将序结构进行推广并应用于拟平衡问题的研究,得到了改进集映射下参数广义向量拟平衡问题(IPGVQEP).然后,给出了一类与改进集映射相关的非线性标量化函数Ψ,利用非线性标量化函数Ψ得到了与原问题(IPGVQEP)对应的标量化问题(IPGVQEP)_Ψ,并获得了原问题与标量化问题解之间的关系.最后,引入了一个关键假设H_Ψ,借助关键假设H_Ψ及原问题与标量化问题间解的关系,获得了IPGVQEP解映射Berge下半连续性的充分必要条件,并举例验证了所得结果.     

向量优化问题的线性标量化方法和Lagrange乘子研究

向量优化是数学规划一个重要分支,其理论与方法不仅与很多学科有密切联系,而且在新兴的多学科交叉领域中有着广泛的应用.本文从向量值广义凸映射、择一定理、线性标量化方法和Lagrange乘子存在性定理等4个方面对这一领域的研究进展情况及所用方法作了较为系统的总结.首先,介绍基于像空间方法的一类广义凸向量值映射和集值映射,总结已有的广义凸映射之间的关系.其次,介绍线性系统下择一定理到非线性系统下择一定理的发展,重点总结凸性或广义凸性条件下的择一定理研究.同时,针对择一定理的应用,给出向量优化问题各种解在凸或广义凸性条件下的线性标量化方法,进而总结向量优化问题的解,特别是真有效解的Lagrange乘子存在性结果.     

非凸多目标优化中真有效解的一个非线性标量化性质

基于Pascoletti-Serafini标量化方法,利用罚函数思想提出了一类新的标量化函数,进而获得非凸多目标优化问题真有效解的充分条件和必要条件.该结果的建立不需要目标函数的像集有界这一条件,故文章是对Akbari等人[J.Optim.Theory Appl.,2018,178(2):560-590]建立的相应标量化结果的改进.     

广义混合变分不等式的Levitin-Polyak适定性

首先给出广义混合变分不等式的Levitin-Polyak-α-近似序列以及适定性的定义.然后,定义广义混合变分不等式的gap函数并证明广义混合变分不等式的Levitin-Polyak适定性与其相应的gap函数的极小化问题的Levitin-Polyak适定性之间的等价性.最后,研究广义混合变分不等式的(广义)Levitin-Polyak-α-适定性的Furi-Vignoli型度量性质.     

Scalarization and pointwise well-posedness in vector optimization problems

The aim of this paper is applying the scalarization technique to study some properties of the vector optimization problems under variable domination structure. We first introduce a nonlinear scalarization function of the vector-valued map and then study the relationships between the vector optimization problems under variable domination structure and its scalarized optimization problems. Moreover, we give the notions of DH-well-posedness and B-well-posedness under variable domination structure and prove that there exists a class of scalar problems whose well-posedness properties are equivalent to that of the original vector optimization problem.     

Scalarization of Levitin–Polyak well-posed set optimization problems

The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively.     

Hadamard well-posed vector optimization problems

In this paper, two kinds of Hadamard well-posedness for vector-valued optimization problems are introduced. By virtue of scalarization functions, the scalarization theorems of convergence for sequences of vector-valued functions are established. Then, sufficient conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems.     

Pointwise Well-Posedness of Perturbed Vector Optimization Problems in a Vector-Valued Variational Principle

In this note, we point out and correct some errors in Ref. 1. Another type of pointwise well-posedness and strong pointwise well-posedness of vector optimization problems is introduced. Sufficient conditions to guarantee this type of well-posedness are provided for perturbed vector optimization problems in connection with the vector-valued Ekeland variational principle.     

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.     

Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints

In this paper we consider strong bilevel vector equilibrium problems and introduce the concepts of Levitin–Polyak well-posedness and Levitin–Polyak well-posedness in the generalized sense for such problems. The notions of upper/lower semicontinuity involving variable cones for vector-valued mappings and their properties are proposed and studied. Using these generalized semicontinuity notions, we investigate sufficient and/or necessary conditions of the Levitin–Polyak well-posedness for the reference problems. Some metric characterizations of these Levitin–Polyak well-posedness concepts in the behavior of approximate solution sets are also discussed. As an application, we consider the special case of traffic network problems with equilibrium constraints.     

Well-posedness for parametric optimization problems with variational inclusion constraint

In this paper, we study the well-posedness for the parametric optimization problems with variational inclusion problems as constraint (or the perturbed problem of optimization problems with constraint). Furthermore, we consider the relation between the well-posedness for the parametric optimization problems with variational inclusion problems as constraint and the well-posedness in the generalized sense for variational inclusion problems.     

多目标广义对策的良定性

By extending the concept of asymptotic weakly Pareto-Nash equilibrium point to vector-valued case, Tikhonov well-posedness and Hadamard well-posedness results of the multiobjective generalized games are established in this paper.     

Levitin–Polyak well-posedness for parametric quasivariational inequality problem of the Minty type

In this paper, we introduce the notions of Levitin?CPolyak (LP) well-posedness and Levitin?CPolyak well-posedness in the generalized sense, for a parametric quasivariational inequality problem of the Minty type. Metric characterizations of LP well-posedness and generalized LP well-posedness, in terms of the approximate solution sets are presented. A parametric gap function for the quasivariational inequality problem is introduced and an equivalence relation between LP well-posedness of the parametric quasivariational inequality problem and that of the related optimization problem is obtained.