【作者】 张文燕；

【导师】 李声杰；

【作者基本信息】 重庆大学 ， 计算数学， 2010， 博士

【摘要】 本文主要研究了集优化问题、对称向量拟平衡问题及广义向量拟平衡问题的适定性,向量值优化问题的Hadamard适定性,以及凸向量值优化问题和凸集值优化问题的适定性和稳定性。全文共分八章,具体内容如下:在第一章,我们介绍了标量优化问题、向量值优化问题、集值优化问题、变分不等式问题以及变分平衡问题适定性的研究概况,向量值优化稳定性的研究概况以及稳定性和适定性的关系,并且阐述了本文的选题动机和主要工作。在第二章,介绍本文中频繁使用的一些基本概念。在第三章,对集优化问题引入了三种类型的适定性。利用广义的Gerstewitz’s函数,分别建立了这三种类型的适定性和对应的标量优化问题之间的等价关系。利用广义压迫函数得出了集优化问题适定性的充要条件。最后,给出了集优化适定性的判别准则。在第四章,定义了关于对称向量拟平衡问题的Levitin-Polyak适定性。利用集值优化问题不动点问题的适定性得出了对称向量拟平衡问题的Levitin-Polyak适定的充分性条件。在第五章,在两类广义向量拟平衡问题中引入Levitin-Polyak适定性的概念,然后在该模型中讨论了一些适定性的经典性质,通过两类广义向量拟平衡问题的间隙函数得到了标量优化问题Levitin-Polyak适定性和广义向量拟平衡问题Levitin-Polyak适定性的关系,最后建立了一种集值的Ekeland变分原理,并利用该原理得到了一类广义向量拟平衡问题Levitin-Polyak适定性的充分条件。在第六章,定义了向量值优化问题的两类Hadamard适定性。利用标量化函数,建立了关于向量值映射序列Gamma-收敛的标量化定理。利用所得的标量化定理,给出了对于向量值优化问题的Hadamard适定性的充分性条件。在第七章,研究了关于映射序列的P.K.收敛的凸向量值优化问题和凸集值优化问题的稳定性和适定性。讨论了向量映射列的Gamma-收敛,P.K.-收敛,逐点收敛和连续收敛之间的包含关系和等价条件。在第八章,我们作了一个简要的总结和讨论。更多还原

【Abstract】 In this thesis, the well-posedness of set optimization problems, symmetric vector quasi-equilibrium problems and generalized vector quasi-equilibrium problems, and Hadamard well-posedness of vector optimization problems are studied. Moreover, the well-posedness and stability properties of convex vector optimization problems and set-valued optimization problems are disscussed, respectively. This thesis is divided into eight chapters. It is organized as follows.In Chapter 1, the development and researches on the topic of wellposedness of scalar-valued problems, vector-valued problems, set-valued problems, variational inequalities and vector equilibrium problems are described. The study on the stability of vector optimization and the relationship between the stability and well-posedness are discussed. Also, the motivation is given and main works are listed.In Chapter 2, some definitions, which will be frequently used, are shown.In Chapter 3, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz’s function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.In Chapter 4, a generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems is defined. By verifying the result of Hadamard well-posedness of set-valued fixed point problems, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are given.In Chapter 5, the definitions of Levitin-Polyak well-posedness for two classes of generalized vector quasi-equilibrium problems are introduced. Then, some classical criteria and characterizations of the Levitin-Polyak well-posedness are investigated. And by virtue of gap functions for the generalized vector quasi-equilibrium problems, some equivalent relations are obtained between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for the generalized vector quasi-equilibrium problems. Finally, a set-valued version of Ekeland’s variational principle is derived and applied to establish a sufficient condition for Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.In Chapter 6, 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. Sufficient conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems.In Chapter 7, well-posedness and stability for convex vector-valued optimization problems and set-valued optimization problems are introduced. The relationship among Gamma-convergence, P.K.convergence, pointwise convergence and continuous convergence of sequences of vector-valued functions are investigated.In Chapter 8, the results of this thesis are summarized and some discussions are made.更多还原