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.     

广义强向量拟均衡问题解集的通有稳定性

研究广义强向量拟均衡问题(GSVQEP)解集的通有稳定性.在约束集值映射满足一定连续与目标映射是锥-真拟凸的集值映射条件下,证明了广义强向量拟均衡问题构成的空间M中,在Baire分类意义下,广义强向量拟均衡问题解集是通有稳定的,且给出了空间M中对每个广义强向量拟均衡问题的解集至少存在一个本质连通区.     

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.     

关于集值目标映射的向量平衡问题的广义Tykhonov适定性

引入集值目标映射的向量平衡问题的两类广义Tykhonov适定性,利用非紧性Kuratowski测度给出它们的度量刻划和讨论这两类适定性的充分性条件.最后证明向量平衡问题的广义Tykhonov适定性与约束极小化问题的广义Tykhonov适定性之间的等价关系.     

广义向量拟平衡问题解集映射的稳定性

我们在局部凸Hausdorff拓扑向量空间中,讨论了广义向量拟平衡问题解集映射的上半连续性以及闭性,并利用扰动间隙函数证明解集的Hausdorff下半连续性.     

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.     

含参集值向量拟均衡问题的适定性

首次对含参集值向量拟均衡问题的适定性进行了研究,并在适当的条件下建立了所研究问题适定性的充分条件.     

强预拟不变凸函数的充要条件

本文引入了一类新的广义凸函数—强预拟不变凸函数.讨论了强预拟不变凸函数与预拟不变凸函数、严格预拟不变凸函数及半严格预拟不变凸函数之间的关系,得到它的三个充要条件:(i)当条件P_1满足时,f是强预拟不变凸函数的充分必要条件是f是预拟不变凸函数且f满足中间点强预拟不变凸性;(ii)当条件P_2满足时,f是强预拟不变凸函数的充分必要条件是f是严格预拟不变凸函数且f满足中间点强顶拟不变凸性;(iii)当条件P_2满足时,f是强预拟不变凸函数的充分必要条件是f是半严格预拟不变凸函数且f满足中间点强预拟不变凸性.     

含参弱向量均衡问题的适定性

在实Hausdorff拓扑线性空间中研究了含参弱向量均衡问题的适定性.证明了在适当条件下由近似网定义的含参适定性等价于近似解映射的上半连续性,并给出了所研究问题各种适定的充分性条件.     

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.     

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.     

On well-posedness for parametric vector quasiequilibrium problems with moving cones

In this paper we consider weak and strong quasiequilibrium problems with moving cones in Hausdorff topological vector spaces. Sufficient conditions for well-posedness of these problems are established under relaxed continuity assumptions. All kinds of wellposedness are studied: (generalized) Hadamard well-posedness, (unique) well-posedness under perturbations. Many examples are provided to illustrate the essentialness of the imposed assumptions. As applications of the main results, sufficient conditions for lower and upper bounded equilibrium problems and elastic traffic network problems to be well-posed are derived.     

The Levitin-Polyak well-posedness by perturbations for systems of general variational inclusion and disclusion problems

In this paper, the notions of the Levitin-Polyak well-posedness by perturbations for system of general variational inclusion and disclusion problems (shortly, (SGVI) and (SGVDI)) are introduced in Hausdorff topological vector spaces. Some sufficient and necessary conditions of the Levitin-Polyak well-posedness by perturbations for (SGVI) (resp., (SGVDI)) are derived under some suitable conditions. We also explore some relations among the Levitin-Polyak well-posedness by perturbations, the existence and uniqueness of solution of (SGVI) and (SGVDI), respectively. Finally, the lower (upper) semicontinuity of the approximate solution mappings of (SGVI) and (SGVDI) are established via the Levitin-Polyak well-posedness by perturbations.     

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.     

Results on Nonlocal Boundary Value Problems

In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting.

We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincaré-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence.     

Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems

We generalize the concept of well-posedness to a mixed variational inequality and give some characterizations of its well-posedness. Under suitable conditions, we prove that the well-posedness of a mixed variational inequality is equivalent to the well-posedness of a corresponding inclusion problem. We also discuss the relations between the well- posedness of a mixed variational inequality and the well-posedness of a fixed point problem. Finally, we derive some conditions under which a mixed variational inequality is well-posed. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005). The research of the third author was partially support by NSC 95-2221-E-110-078.     

Generalized Hadamard well-posedness for infinite vector optimization problems

In this paper, we study the generalized Hadamard well-posedness of infinite vector optimization problems (IVOP). Without the assumption of continuity with respect to the first variable, the upper semicontinuity and closedness of constraint set mappings are established. Under weaker assumptions, sufficient conditions of generalized Hadamard well-posedness for IVOP are obtained under perturbations of both the objective function and the constraint set. We apply our results to the semi-infinite vector optimization problem and the semi-infinite multi-objective optimization problem.     

Well-posedness by perturbations of mixed variational inequalities in Banach spaces

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a mixed variational inequality problem in a Banach space. We establish some metric characterizations of the well-posedness by perturbations. We also show that under suitable conditions, the well-posedness by perturbations of a mixed variational inequality problem is equivalent to the well-posedness by perturbations of a corresponding inclusion problem and a corresponding fixed point problem. Also, we derive some conditions under which the well-posedness by perturbations of a mixed variational inequality is equivalent to the existence and uniqueness of its solution.