集值映射的次微分和最优性条件
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摘要
本文研究了几类集值映射的次微分的存在性、性质以及计算两个集值映射和、复合、交的次微分运算法则，建立了锥凸向量优化问题和D.C.向量优化问题的最优性条件，引入了非凸集值映射的广义ε次微分的概念，讨论了广义ε次微分的存在性、性质、和、差运算法则，并作为应用建立了向量优化问题的最优性条件。具体内容如下：
第一章，首先回顾了向量优化各问题的研究现状，然后分别介绍了次微分和D.C.优化问题的发展和研究现状，最后阐述了本文的选题动机和主要工作。
第二章，介绍了本文涉及的一些基本符号和一些基本概念，也回顾了向量优化问题中的相依导数，相依epi导数，各种次微分，几类有效解等概念。
第三章，首先证明了文献(Yang,1992)和文献(Chen and Jahn,1998)中引入的集值映射的两种弱次梯度的存在性定理，并推广了文献(Chen and Jahn,1998)和文献(Peng et al,2005)中相对应的结论。然后给出了文献(Borwein,1981)中引入的集值映射的强次微分的存在性定理，并得到了强次微分和文献(Baier and Jahn,1998)中用导数所定义的次微分之间的关系。
第四章，给出了文献(Chen and Jahn,1998)中引入的集值映射的CJ-弱次微分的几种性质，利用Sandwich定理得到了两个集值映射和的CJ-弱次微分运算法则。而且根据相依导数的性质，得到了用相依导数所引入的弱次微分的性质以及计算两个集值映射的和、复合、交的次微分运算法则。
第五章，利用集值映射的CJ-弱次微分和文献(李声杰,1998)中用相依导数引入的L-次微分分别讨论了约束集为给定集合和约束集由锥凸映射所决定的两类集值优化问题的充分和必要最优性条件，并比较了一些结论和相应文献中的结果。
第六章，首先利用集值映射的强次微分建立了D.C.集值优化问题的弱有效解的充分最优性条件和必要最优性条件，并且比较了必要最优性条件和相关文献中的结论。而且，借助于一个特殊的标量化函数证明了D.C.优化问题和一个实集值优化问题的解的等价关系。然后利用向量值映射的强次微分和ε-次微分，建立了D.C.向量优化问题的ε-Pareto弱有效解和ε-Pareto真有效解的最优性条件。最后，作为应用，建立了分式规划的ε-Pareto弱有效解和ε-Pareto真有效解的最优性条件。
第七章，利用范数引入了非凸向量映射的广义ε次微分的概念，讨论了广义ε次微分的存在性和性质，刻画了广义ε次微分和方向导数之间的关系，然后给出了两个向量映射的和与差的广义ε次微分运算法则，也给出了广义ε次微分的正齐次性。最后作为应用，建立了向量优化问题的充分和必要最优性条件。
第八章，简要总结了本文中的内容，并提出了一些遗留的问题和今后准备思考的问题。
In this thesis, the existence theorems, some properties and some calculus rules forcalculating the subdifferential of the sum, composition and intersection of twoset-valued mappings for the several kinds of subdiferential for set-valued mappings arestudied. Optimality conditions of cone-convex vector optimization problems and specialnonconvex vector optimization problems, which are D.C. vector optimization problemsare established. A generalized ε subdifferential is introduced for a nonconvex vectorvalued mapping. And the existence theorems, the properties and the calculus rules of thegeneralized ε subdifferential for the sum and the difference of two vector valuedmappings were disscussed. And, as applications, optimality conditions are establishedfor vector optimization problems. It is organized as follows:
In Chapter1, the development and current researches on the topic of vectoroptimization problems are firstly recalled. Then, the development and current researchesfor subdifferential and D.C. multiobjective optimization problems are reviewed,respectively. Finally, the motivations and the main research work are also given.
In Chapter2, some basic notions and definitions of vector optimization problemsare recalled. These definitions mainly refer to contingent derivative, contingentepiderivative, different kinds of subdifferential and kinds of efficient solutions.
In Chapter3, firstly, the existence theorems of two kinds of weak subgradients forset-valued mappings introduced in (Yang,1992) and (Chen and Jahn,1998), which arethe generalizations of Theorem7in (Chen and Jahn,1998) and Theorem4.1in (Peng etal,2005) respectively, are proven. Then, an existence theorem of the subgradients forset-valued mappings, which introduced by Borwein in (Borwein,1981), and therelations between this subdifferential and the subdifferential introduced by Baier andJahn in (Baier and Jahn,1998), are obtained.
In Chapter4, some properties of the CJ-weak subdifferential of set-valuedmappings introduced in (Chen and Jahn,1998) and the calculus rules of the CJ-weaksubdifferential for the sum of two set-valued mappings are obtained by using a so-calledSandwich theorem. Moreover, in virtue of the property of the contingent derivative,some properties and some exact calculus rules for calculating the subdifferential of thesum, composition and intersection of two set-valued mappings are given.
In Chapter5, by using the concepts of the CJ-weak subdifferential of set-valued mappings and the L-subdifferential introduced by the contingent derivative in (Li,1998),necessary and sufficient optimality conditions are discussed of cone-convex set-valuedoptimization problems, whose constraint sets are determined by a fixed set and aset-valued mapping, respectively. And some results are compared with the relatedresults in the literature.
In Chapter6, by using the concept of the strong subdifferential for set-valuedmappings, the sufficient and necessary optimality conditions for generalized D.C.multiobjective optimization problems are established. And the necessary optimalityconditions are compared with the related results in the literature. Moreover, by using aspecial scalarization function, a real set-valued optimization problem is introduced andthe equivalent relations between the solutions are proved for the real set-valuedoptimization problem and a generalized D.C. multiobjective optimization problem.Then, by virtue of the concepts of the strong subdifferential and the ε-subdifferentialof vector valued mappings, sufficient and necessary optimality conditions areestablished for an ε-weak Pareto minimal point and an ε-proper Pareto minimal pointof a D.C. vector optimization problem. As an application, sufficient and necessaryoptimality conditions are also given for an ε-weak Pareto minimal point and anε-proper Pareto minimal point of a vector fractional mathematical programming.
In Chapter7, a generalized ε subdifferential, which was defined by a norm, isfirst introduced for a nonconvex vector valued mapping. Some existence theorems andthe properties of the generalized ε subdifferential are discussed. A relationshipbetween the generalized ε subdifferential and a directional derivative is investigatedfor a vector valued mapping. Then, the calculus rules of the generalizedε subdifferential for the sum and the difference of two vector valued mappings weregiven. The positive homogeneity of the generalized ε subdifferential is also provided.Finally, as applications, necessary and sufficient optimality conditions are establishedfor vector optimization problems.
In Chapter8, the results of this thesis are briefly summarized. And some problemswhich are remained and thought over in future are put forward.
第一章，首先回顾了向量优化各问题的研究现状，然后分别介绍了次微分和D.C.优化问题的发展和研究现状，最后阐述了本文的选题动机和主要工作。
第二章，介绍了本文涉及的一些基本符号和一些基本概念，也回顾了向量优化问题中的相依导数，相依epi导数，各种次微分，几类有效解等概念。
第三章，首先证明了文献(Yang,1992)和文献(Chen and Jahn,1998)中引入的集值映射的两种弱次梯度的存在性定理，并推广了文献(Chen and Jahn,1998)和文献(Peng et al,2005)中相对应的结论。然后给出了文献(Borwein,1981)中引入的集值映射的强次微分的存在性定理，并得到了强次微分和文献(Baier and Jahn,1998)中用导数所定义的次微分之间的关系。
第四章，给出了文献(Chen and Jahn,1998)中引入的集值映射的CJ-弱次微分的几种性质，利用Sandwich定理得到了两个集值映射和的CJ-弱次微分运算法则。而且根据相依导数的性质，得到了用相依导数所引入的弱次微分的性质以及计算两个集值映射的和、复合、交的次微分运算法则。
第五章，利用集值映射的CJ-弱次微分和文献(李声杰,1998)中用相依导数引入的L-次微分分别讨论了约束集为给定集合和约束集由锥凸映射所决定的两类集值优化问题的充分和必要最优性条件，并比较了一些结论和相应文献中的结果。
第六章，首先利用集值映射的强次微分建立了D.C.集值优化问题的弱有效解的充分最优性条件和必要最优性条件，并且比较了必要最优性条件和相关文献中的结论。而且，借助于一个特殊的标量化函数证明了D.C.优化问题和一个实集值优化问题的解的等价关系。然后利用向量值映射的强次微分和ε-次微分，建立了D.C.向量优化问题的ε-Pareto弱有效解和ε-Pareto真有效解的最优性条件。最后，作为应用，建立了分式规划的ε-Pareto弱有效解和ε-Pareto真有效解的最优性条件。
第七章，利用范数引入了非凸向量映射的广义ε次微分的概念，讨论了广义ε次微分的存在性和性质，刻画了广义ε次微分和方向导数之间的关系，然后给出了两个向量映射的和与差的广义ε次微分运算法则，也给出了广义ε次微分的正齐次性。最后作为应用，建立了向量优化问题的充分和必要最优性条件。
第八章，简要总结了本文中的内容，并提出了一些遗留的问题和今后准备思考的问题。
In this thesis, the existence theorems, some properties and some calculus rules forcalculating the subdifferential of the sum, composition and intersection of twoset-valued mappings for the several kinds of subdiferential for set-valued mappings arestudied. Optimality conditions of cone-convex vector optimization problems and specialnonconvex vector optimization problems, which are D.C. vector optimization problemsare established. A generalized ε subdifferential is introduced for a nonconvex vectorvalued mapping. And the existence theorems, the properties and the calculus rules of thegeneralized ε subdifferential for the sum and the difference of two vector valuedmappings were disscussed. And, as applications, optimality conditions are establishedfor vector optimization problems. It is organized as follows:
In Chapter1, the development and current researches on the topic of vectoroptimization problems are firstly recalled. Then, the development and current researchesfor subdifferential and D.C. multiobjective optimization problems are reviewed,respectively. Finally, the motivations and the main research work are also given.
In Chapter2, some basic notions and definitions of vector optimization problemsare recalled. These definitions mainly refer to contingent derivative, contingentepiderivative, different kinds of subdifferential and kinds of efficient solutions.
In Chapter3, firstly, the existence theorems of two kinds of weak subgradients forset-valued mappings introduced in (Yang,1992) and (Chen and Jahn,1998), which arethe generalizations of Theorem7in (Chen and Jahn,1998) and Theorem4.1in (Peng etal,2005) respectively, are proven. Then, an existence theorem of the subgradients forset-valued mappings, which introduced by Borwein in (Borwein,1981), and therelations between this subdifferential and the subdifferential introduced by Baier andJahn in (Baier and Jahn,1998), are obtained.
In Chapter4, some properties of the CJ-weak subdifferential of set-valuedmappings introduced in (Chen and Jahn,1998) and the calculus rules of the CJ-weaksubdifferential for the sum of two set-valued mappings are obtained by using a so-calledSandwich theorem. Moreover, in virtue of the property of the contingent derivative,some properties and some exact calculus rules for calculating the subdifferential of thesum, composition and intersection of two set-valued mappings are given.
In Chapter5, by using the concepts of the CJ-weak subdifferential of set-valued mappings and the L-subdifferential introduced by the contingent derivative in (Li,1998),necessary and sufficient optimality conditions are discussed of cone-convex set-valuedoptimization problems, whose constraint sets are determined by a fixed set and aset-valued mapping, respectively. And some results are compared with the relatedresults in the literature.
In Chapter6, by using the concept of the strong subdifferential for set-valuedmappings, the sufficient and necessary optimality conditions for generalized D.C.multiobjective optimization problems are established. And the necessary optimalityconditions are compared with the related results in the literature. Moreover, by using aspecial scalarization function, a real set-valued optimization problem is introduced andthe equivalent relations between the solutions are proved for the real set-valuedoptimization problem and a generalized D.C. multiobjective optimization problem.Then, by virtue of the concepts of the strong subdifferential and the ε-subdifferentialof vector valued mappings, sufficient and necessary optimality conditions areestablished for an ε-weak Pareto minimal point and an ε-proper Pareto minimal pointof a D.C. vector optimization problem. As an application, sufficient and necessaryoptimality conditions are also given for an ε-weak Pareto minimal point and anε-proper Pareto minimal point of a vector fractional mathematical programming.
In Chapter7, a generalized ε subdifferential, which was defined by a norm, isfirst introduced for a nonconvex vector valued mapping. Some existence theorems andthe properties of the generalized ε subdifferential are discussed. A relationshipbetween the generalized ε subdifferential and a directional derivative is investigatedfor a vector valued mapping. Then, the calculus rules of the generalizedε subdifferential for the sum and the difference of two vector valued mappings weregiven. The positive homogeneity of the generalized ε subdifferential is also provided.Finally, as applications, necessary and sufficient optimality conditions are establishedfor vector optimization problems.
In Chapter8, the results of this thesis are briefly summarized. And some problemswhich are remained and thought over in future are put forward.
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