摘要
本文分别研究了(向量)平衡问题和(向量)拟平衡问题的扰动集值解映射的H?lder连续性,近似集值解映射的上下半连续性、Lipshitz/H?lder连续性,字典序下的向量极大极小问题、向量鞍点问题以及多目标拉格朗日对偶问题。全文共分八章,具体内容如下:
在第一章里,介绍本文中频繁使用的定义及其性质。
在第二章里,我们介绍了向量优化及其相关问题的解映射的半连续性,Lipshitz/H?lder连续性等、极大极小问题和拉格朗日对偶理论、优化问题的发展概况,并且阐述了本文的选题动机和主要工作。
在第三章里,我们引入集值映射的弱伪单调性假设。基于此假设,研究了带集值映射的广义向量拟平衡问题的扰动集值解映射的H?lder连续性。把此结论分别应用到向量变分不等式和向量平衡问题中,得到了扰动解映射和精确解映射之间的误差估计。
在第四章里,在度量空间中得到了一些参数向量拟平衡问题扰动解的误差分析结果,这些误差估计结果在一些特殊情况下等价于解集映射在某点处的H?lder稳定性或者Lipschitz稳定性。
在第五章里,我们引入向量值函数的强单调概念,并讨论了此概念的一些性质。基于此概念,讨论一类弱向量平衡问题的集值解映射的H?lder连续性。由于向量值函数的强单调并不蕴含弱向量平衡问题的解是唯一的,因此本文的主要结果改正了文献(Anh and Khanh, 2008c)中的错误。
在第六章里,利用平衡问题的近似解集的包含关系,首先研究了平衡问题的近似解映射的下半连续性和连续性。基于此结果、标量化技巧和一簇下半连续集值映射的并仍是下半连续的性质,研究了向量平衡问题近似解映射的下半连续性和连续性。我们还进一步研究了平衡问题的近似解映射的Lipshitz/H?lder,显然,这一结果是比该问题的近似解的连续性更强的结果。我们将这些主要结果应用到变不等式和优化问题中,得到这些问题的近似解映射的Lipshitz/H?lder连续的充分条件。
在第七章里,我们首先讨论字典有效点的一些性质,然后在字典序下引入一类新的向量值函数。基于这些概念和性质,研究了字典序下的极大极小问题和鞍点问题,以及二者之间的关系。然后,我们利用字典有效点和向量值的拉格朗日函数,研究了字典序下的多目标规划及其拉格朗日对偶问题,建立了强对偶和弱对偶定理,并得到了一系列的拉格朗日乘子法则和鞍点定理。最后,讨论了字典鞍点与多目标规划问题的字典有效解之间的关系。
在第八章里,我们作了一个简要的总结和讨论。
In this thesis, we investigated the H?lder continuity of perturbed set-valued solution mapping, the lower and upper, Lipshitz/H?lder continuity of approximate set-valued solution mapping for (vector) equilibrium problems or (vector) quasiequilibrium problems, vector minimax problems, vector saddle point problems and multiobjective Lagrangian duality problems in the lexicographic order, respectively. This thesis is divided into eight chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of vector optimization, including the semicontinuity, Lipshitz/H?lder continuity of solution mappings to the vector optimization and other problems, and the vector optimization in lexicographic order. We also give the motivation and the main research work.
In Chapter 2, some definitions and their properties, which are frequently used, are shown.
In Chapter 3, we introduce the concept of weak pseudomontone for set-valued mapping. Based on this assumption, we discuss the H?lder continuity of perturbed solution mapping for generalized vector quasiequilibrium problems with set-valued mapping. As an application of our main results, we establish the error estimates of the distance between perturbed solution mapping and exact solution mapping for vector variation inequalities and vector equilibrium problems, respectively.
In Chapter 4,we obtain some results on the error estimates of perturbed solutions to parametric vector quasiequilibrium problems in metric spaces. Under some special cases, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point.
In Chapter 5, we introduce the concept of strongly monotone for vector-valued mapping. Then, the various properties of this mapping are discussed in this chapter. Based on the assumptions of strongly monotone, we investigate the H?lder continuity of the perturbed solution mapping, which is set-valued one, for a weak vector equilibrium problem. Since the strongly monotone mapping for vector-valued mapping does not imply the singleton of the solution mapping of the weak vector equilibrium problem, our main results in this chapter are a correction of the corresponding ones of Anh and Khanh (2008c).
In Chapter 6, by virtue of the set inclusion relation, we investigate the lower semicontinuity and continuity of the approximate solution mapping for equilibrium problems. Based on these main results, a scalarization representation of the solution mapping and a property involving the union of a family of lower semicontinuous set-valued mappings, we also discuss the lower semicontinuity and continuity of the approximate solution mapping for vector equilibrium problems. Futhermore, we investigate the Lipshitz/H?lder continuity of the approximate solution mapping for equilibrium problems, which strengthenes the results of continuity of the approximate solution mapping for vector equilibrium problems. Appling our main results to variational inequalties and optimization problems, we establish the Lipshitz/H?lder continuity of their approximate solution mapping, respectively.
In Chapter 7, we first study the properties of the efficient points in lexicographic order, and introduce a new vector-valued mapping in lexicographic order. Based on these concepts and their properties, we investigate the vector minimax and saddle point problems. The relations between the minimax theorem and the saddle point theorem are also discussed in this chapter. Then, based on the efficient point and vector-valued Lagrangian function, we study the multiobjective programming and Lagrangian duality problems, and establish the weak (strong) duality theorems and Lagrangian multiplier rules. We furthermore discuss the relations between the efficient point of multiobjecive programming problems and saddle point problems in lexicographic order.
In Chapter 8, we summarize the results of this thesis and make some discussions.
引文
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