向量优化及其相关问题解的存在性和适定性研究
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摘要
本文研究了向量优化及其相关问题解的存在性和适定性。
第一章,介绍了向量优化及其相关问题解的存在性和适定性在国内外的研究现状,并且阐述了本文的选题动机和主要工作。
第二章,主要介绍了本文涉及的一些基本符号、概念及其性质。
第三章,首先在完备度量空间上建立了集值映射广义Ekeland变分原理,举例说明其与文献[3,4,10]中变分原理不同。然后利用此变分原理在定义域非凸紧,以及定义域非凸非紧的条件下,讨论了一类广义向量平衡问题解的存在性。
第四章,在广义不变凸的条件下,研究了集值优化问题与向量似变分不等式解之间的关系;在C-预不变凸的条件下,得到了约束集值优化问题的广义Lagrange乘子。
第五章,在向量值优化问题序列P.K.收敛于目标向量值优化问题的条件下,讨论了向量值优化问题有效点的适定性和稳定性结果,此结果推广了Lucchetti和Miglierina在文献[11]第三章中的相应结果。
第六章,介绍一类集值优化问题的扩展Hadamard型适定性。利用一种标量化函数,建立此集值优化问题同一种标量优化问题解之间的等价关系,并得到一个关集值映射序列P.K.收敛的标量化定理。基于解的等价关系和标量化定理,得到了集值优化问题扩展Hadamard型适定性的充分性条件。
第七章,介绍一类向量平衡问题的扩展Hadamard型适定性,并利用一种非线性标量化函数得到了此类扩展Hadamard型适定性的充分性条件。
第八章,在度量空间中得到了一些参数向量拟平衡问题近似解的误差分析结果,这些误差分析结果在一些特殊情况下等价于集值解映射在某点处的H?lder稳定性或者Lipschitz稳定性。最后,将这些结果应用到了变分不等式问题中。
在第九章里,我们作了一个简要的总结和讨论。
In this thesis, we establish some solution existence results and well-posedness for vector optimization problems and some related problems. This thesis is divided into nine chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of existence results and well-posedness for vector optimization problems. We also give the motivation and the main research work.
In Chapter 2, we introduce some basic notions, definitions and propositions, which will be used in the sequel.
In Chapter 3, we obtain a general Ekeland's variational principle for set-valued mappings in a complete metric space, which is different from those in [3, 4, 10]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex compact and nonconvex noncompact assumptions of its domain, respective-ly.
In Chapter 4, some solution relationships between set-valued optimization prob-lems and vector variational-like inequalities are established under generalized invexities. In addition, a generalized Lagrange multiplier rule for a constrained set-valued optimi-zation problem is obtained under C-preinvexity.
In Chapter 5, we discuss the wellposedness and stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate prob-lems converges to the data of the original problem in the sense of Painlevé–Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina [11, Section 3].
In Chapter 6, we introduce a kind of extended Hadamard-type well-posedness for set-valued optimization problems. By virtue of a scalarization function, we obtain some solution relationships between the set-valued optimization problem and a scalar optimi-zation problem. Then, we derive a scalarization theorem of P.K. convergence for se-quences of set-valued mappings. Based on these results, we also establish a sufficient condition of extended Hadamard-type well-posedness for the set-valued optimization problems.
In Chapter 7, we introduce a kind of extended Hadamard-type well-posedness for vector equilibrium problems. By virtue of a scalarization function, we also establish a sufficient condition of extended Hadamard-type well-posedness for the vector equili-brium problems.
In Chapter 8, we obtain some results on error estimates of approximate solutions to parametric vector quasiequilibrium problems in metric spaces. Under some special cas-es, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point. An application to variational inequalities is al-so presented.
In Chapter 9, we summarize the results of this thesis and make some discussions.
第一章,介绍了向量优化及其相关问题解的存在性和适定性在国内外的研究现状,并且阐述了本文的选题动机和主要工作。
第二章,主要介绍了本文涉及的一些基本符号、概念及其性质。
第三章,首先在完备度量空间上建立了集值映射广义Ekeland变分原理,举例说明其与文献[3,4,10]中变分原理不同。然后利用此变分原理在定义域非凸紧,以及定义域非凸非紧的条件下,讨论了一类广义向量平衡问题解的存在性。
第四章,在广义不变凸的条件下,研究了集值优化问题与向量似变分不等式解之间的关系;在C-预不变凸的条件下,得到了约束集值优化问题的广义Lagrange乘子。
第五章,在向量值优化问题序列P.K.收敛于目标向量值优化问题的条件下,讨论了向量值优化问题有效点的适定性和稳定性结果,此结果推广了Lucchetti和Miglierina在文献[11]第三章中的相应结果。
第六章,介绍一类集值优化问题的扩展Hadamard型适定性。利用一种标量化函数,建立此集值优化问题同一种标量优化问题解之间的等价关系,并得到一个关集值映射序列P.K.收敛的标量化定理。基于解的等价关系和标量化定理,得到了集值优化问题扩展Hadamard型适定性的充分性条件。
第七章,介绍一类向量平衡问题的扩展Hadamard型适定性,并利用一种非线性标量化函数得到了此类扩展Hadamard型适定性的充分性条件。
第八章,在度量空间中得到了一些参数向量拟平衡问题近似解的误差分析结果,这些误差分析结果在一些特殊情况下等价于集值解映射在某点处的H?lder稳定性或者Lipschitz稳定性。最后,将这些结果应用到了变分不等式问题中。
在第九章里,我们作了一个简要的总结和讨论。
In this thesis, we establish some solution existence results and well-posedness for vector optimization problems and some related problems. This thesis is divided into nine chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of existence results and well-posedness for vector optimization problems. We also give the motivation and the main research work.
In Chapter 2, we introduce some basic notions, definitions and propositions, which will be used in the sequel.
In Chapter 3, we obtain a general Ekeland's variational principle for set-valued mappings in a complete metric space, which is different from those in [3, 4, 10]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex compact and nonconvex noncompact assumptions of its domain, respective-ly.
In Chapter 4, some solution relationships between set-valued optimization prob-lems and vector variational-like inequalities are established under generalized invexities. In addition, a generalized Lagrange multiplier rule for a constrained set-valued optimi-zation problem is obtained under C-preinvexity.
In Chapter 5, we discuss the wellposedness and stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate prob-lems converges to the data of the original problem in the sense of Painlevé–Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina [11, Section 3].
In Chapter 6, we introduce a kind of extended Hadamard-type well-posedness for set-valued optimization problems. By virtue of a scalarization function, we obtain some solution relationships between the set-valued optimization problem and a scalar optimi-zation problem. Then, we derive a scalarization theorem of P.K. convergence for se-quences of set-valued mappings. Based on these results, we also establish a sufficient condition of extended Hadamard-type well-posedness for the set-valued optimization problems.
In Chapter 7, we introduce a kind of extended Hadamard-type well-posedness for vector equilibrium problems. By virtue of a scalarization function, we also establish a sufficient condition of extended Hadamard-type well-posedness for the vector equili-brium problems.
In Chapter 8, we obtain some results on error estimates of approximate solutions to parametric vector quasiequilibrium problems in metric spaces. Under some special cas-es, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point. An application to variational inequalities is al-so presented.
In Chapter 9, we summarize the results of this thesis and make some discussions.
引文
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