三类集值映射的（方向）导数及在优化中的应用
摘要
本文针对研究非光滑函数的高阶算法的理论基础和集值映射的微分的计算的课题，主要研究几类特殊类型的集值映射的(方向)导数的计算与近似，并将得到的结果应用到优化的最优性理论中。本文取得的主要结果可概括如下：
1．在第2章中，建立了一类基于凸集对空间的理论在Tyurin(1965)和Banks & Jacobs(1970)意义下集值映射的导数的计算公式和凸函数的次微分映射的微分。
2．在第3章中，将第2章的结果应用到参数规划的解集的估计，得到了参数线性规划和参数二次规划的稳定性结果，同时也给出参数线性规划的解集的界的估计，在相同的假设条件下得到的结果比目前已有的结果好(即，更sharper)。
3．在第4章中，研究了拟可微分析中的微分结构—拟微分结构。在这一章里，首先给出核拟微分，星核与星微分的定义及其它们的运算性质；然后证明了拟核微分的一个充分条件定理及一个充要条件定理；最后讨论拟可微函数星核的存在性及方向可微函数星微分的存在性以及Penot-微分与上下导数之间的关系。
4．在第5章中，针对近几年发展起来的集值优化，基于Clarke切锥利用Epigraph建立了一类集值映射的Epi-导数并讨论它的一些性质，同时给出集值优化的充分(或必要)的最优性条件。
This dissertation studies mainly approximations to special classes of set-valued maps and their applications, in order to compute differentials of some class of set-valued maps and to solve basis theories of constructing high-ordered methods of nonsmooth functions. Then results obtained in this dissertation are applied to optimality theories in optimization. The main results obtained in this dissertation are summarized as follows:
1. Chapter 2 established derivatives of a class of set-valued maps and differentials of subdifferential maps of convex functions in the sense of Tyurin (1965) and Banks & Jacobs (1970) based on theories of convex pairs space.
2. Chapter 3 applied the results obtained in Chapter 2 to estimate to solution-set of a parametric mathematical programming. Results about stability of a parametric linear programming and estimate to bound of solution-set of a parametric linear programming is established; under the assumptions in this Chapter, results obtained is more sharper than ones obtained in last.
3. Chapter 4 is devoted to the study of diferential structure in Quasidiferential analysis-quasidiferential structure. This chapter proposes three conceptions, i.e., Kernelled quasidiferential, star-kernel and star-diferential, and establishes their operational properties. A sufficient theorem and a sufficent and necessity theorem for a quasi-kernel being a kernelled quasidiferential are proven. Both the existence of star-kernel for a quasidiferentiable function and the existence of star-differential for a direnction-ally diferentiable function are established. The relationships between sub- and super-derivatives and Penot diferentials are dicussed as well.
4. In recent years, set-valued optimization make much progress. In Chapter 5, based on Clarke tangent cone, we establish epiderivative of a class of set-valued maps and its properties. And furthermore, sufficiency (or neccessity) optimization conditions of set-valued optimization are also obtained.
1．在第2章中，建立了一类基于凸集对空间的理论在Tyurin(1965)和Banks & Jacobs(1970)意义下集值映射的导数的计算公式和凸函数的次微分映射的微分。
2．在第3章中，将第2章的结果应用到参数规划的解集的估计，得到了参数线性规划和参数二次规划的稳定性结果，同时也给出参数线性规划的解集的界的估计，在相同的假设条件下得到的结果比目前已有的结果好(即，更sharper)。
3．在第4章中，研究了拟可微分析中的微分结构—拟微分结构。在这一章里，首先给出核拟微分，星核与星微分的定义及其它们的运算性质；然后证明了拟核微分的一个充分条件定理及一个充要条件定理；最后讨论拟可微函数星核的存在性及方向可微函数星微分的存在性以及Penot-微分与上下导数之间的关系。
4．在第5章中，针对近几年发展起来的集值优化，基于Clarke切锥利用Epigraph建立了一类集值映射的Epi-导数并讨论它的一些性质，同时给出集值优化的充分(或必要)的最优性条件。
This dissertation studies mainly approximations to special classes of set-valued maps and their applications, in order to compute differentials of some class of set-valued maps and to solve basis theories of constructing high-ordered methods of nonsmooth functions. Then results obtained in this dissertation are applied to optimality theories in optimization. The main results obtained in this dissertation are summarized as follows:
1. Chapter 2 established derivatives of a class of set-valued maps and differentials of subdifferential maps of convex functions in the sense of Tyurin (1965) and Banks & Jacobs (1970) based on theories of convex pairs space.
2. Chapter 3 applied the results obtained in Chapter 2 to estimate to solution-set of a parametric mathematical programming. Results about stability of a parametric linear programming and estimate to bound of solution-set of a parametric linear programming is established; under the assumptions in this Chapter, results obtained is more sharper than ones obtained in last.
3. Chapter 4 is devoted to the study of diferential structure in Quasidiferential analysis-quasidiferential structure. This chapter proposes three conceptions, i.e., Kernelled quasidiferential, star-kernel and star-diferential, and establishes their operational properties. A sufficient theorem and a sufficent and necessity theorem for a quasi-kernel being a kernelled quasidiferential are proven. Both the existence of star-kernel for a quasidiferentiable function and the existence of star-differential for a direnction-ally diferentiable function are established. The relationships between sub- and super-derivatives and Penot diferentials are dicussed as well.
4. In recent years, set-valued optimization make much progress. In Chapter 5, based on Clarke tangent cone, we establish epiderivative of a class of set-valued maps and its properties. And furthermore, sufficiency (or neccessity) optimization conditions of set-valued optimization are also obtained.
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