拟可微分析与优化中某些问题的研究结果：核·凸化集·最优性条件
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摘要
本文针对拟可微函数的微分理论,分别从拟微分核和凸化集两个方面作了一些工作,
并研究了约束拟可微优化的最优性条件.主要结果可概括如下:
1. 第3章,在正交互补的假设条件下,给出了凸紧集的Demyanov差的两个运算公式(等
式),该公式可用于求解和函数与极大(小)值函数的次微分与负的超微分的Demyanov
差(拟微分的Demyanov和);针对拟可微函数拟微分的不唯一性,在次微分与负的超
微分的Demyanov差与Minkowski差相等的条件下,研究了高维空间上的拟可微函
数的拟微分核的性质,给出了一个特殊的具有高维核(高岩意义下)的拟可微函数类
一次超可微函数;对于一类特殊的拟可微优化-D.C.优化,给出了其最速下降算法
的收敛性分析.
2. 第4章,对一般的拟可微函数引入了凸化核的概念,给出了判定一般拟可微函数为
次可微的一个充分条件;利用一般正齐次函数的回收函数给出了拟可微函数的一个
具体的凸化集,并证明了该凸化集即为拟微分的Demyanov和.
3. 第5章,对于一般的带有等式、不等式及抽象约束的拟可微优化问题,在较弱的假
设条件(存在一对拟微分,使得次微分与超微分分别上半连续)下,利用Ekeland原
理给出了用Demyanov差表示的Fritz John型的必要最优性条件,并在具有极大秩
的假设条件下,给出了KKT型的必要最优性条件;作为应用,给出了用Demyanov
和表述的双层优化的最优性条件;在没有任何假设条件下,给出了用次线性泛函表
述的Fritz John型的必要最优性条件.两个结果的共同点是最优性条件与超梯度的
特殊选取无关.
This dissertation studies the calculus theory of quadifferentiable functions in two ways, one of which is from the way of quasidifferential kernels, the other one is from the way of convexificators, and the necessary optimality conditions for constrained quasidif-ferentiable optimization are discussed. The main results may be summarized as follows:1. Chapter 3 establishes two formulae (equality) of Demyanov difference of convex compact sets under the assumption of orthogonal complementarity, which are useful to compute Demyanov difference of subdifferential and negative superdifferential (Demyanov sum of quasidifferential) for the summation functions and the maximal (minimal) functions. In order to solve the quasidifferential uniqueness for quasidifferen-tiable functions, some properties of high dimensional kernels are given under the assumption that Demyanov difference of subdifferential and negative superdifferential is consistent with Minkowski difference of those, and a special class of quasidiffer-entiable functions with high demensional kernels (in the sense of Yan Gao), sub-superdifferentiable functions, is presented. For a special class of quasidifferentiable optimization-D. C. optimization, the convergence analysis of the steepest descent algorithm is obtained.2. Chapter 4 proposes the concept of convexificator kernels for quasidifferentiable functions, which can be used to tell whether a quasidifferentiable function is subdiffer-entiable. Concrete convexificators of quasidifferentiable functions are given by the recession functions of positively homogeneous functions. We can show that the convexificators are consistent with Demyanov sum of quasidifferentials.3. Chapter 5 is devoted to the study of constrained quasidifferentiable optimization with equality, inequality and abstract constraints. Under the mild assumption (existing a pair of quasidifferential such that the subdifferential and the superdifferential are upper semi-continuous), Fritz John necessary optimality conditions via Demyanov difference are obtained by Ekeland principle. As an application, necessary optimality conditions for bilevel programming are given. If the condition of being of maximal rank is added, KKT necessary optimality conditions can be established. Moreover, Fritz John necessary optimality conditions via sublinear functional are given without any assmuption. The above two results are both independent of the choose of the supergradients.
并研究了约束拟可微优化的最优性条件.主要结果可概括如下:
1. 第3章,在正交互补的假设条件下,给出了凸紧集的Demyanov差的两个运算公式(等
式),该公式可用于求解和函数与极大(小)值函数的次微分与负的超微分的Demyanov
差(拟微分的Demyanov和);针对拟可微函数拟微分的不唯一性,在次微分与负的超
微分的Demyanov差与Minkowski差相等的条件下,研究了高维空间上的拟可微函
数的拟微分核的性质,给出了一个特殊的具有高维核(高岩意义下)的拟可微函数类
一次超可微函数;对于一类特殊的拟可微优化-D.C.优化,给出了其最速下降算法
的收敛性分析.
2. 第4章,对一般的拟可微函数引入了凸化核的概念,给出了判定一般拟可微函数为
次可微的一个充分条件;利用一般正齐次函数的回收函数给出了拟可微函数的一个
具体的凸化集,并证明了该凸化集即为拟微分的Demyanov和.
3. 第5章,对于一般的带有等式、不等式及抽象约束的拟可微优化问题,在较弱的假
设条件(存在一对拟微分,使得次微分与超微分分别上半连续)下,利用Ekeland原
理给出了用Demyanov差表示的Fritz John型的必要最优性条件,并在具有极大秩
的假设条件下,给出了KKT型的必要最优性条件;作为应用,给出了用Demyanov
和表述的双层优化的最优性条件;在没有任何假设条件下,给出了用次线性泛函表
述的Fritz John型的必要最优性条件.两个结果的共同点是最优性条件与超梯度的
特殊选取无关.
This dissertation studies the calculus theory of quadifferentiable functions in two ways, one of which is from the way of quasidifferential kernels, the other one is from the way of convexificators, and the necessary optimality conditions for constrained quasidif-ferentiable optimization are discussed. The main results may be summarized as follows:1. Chapter 3 establishes two formulae (equality) of Demyanov difference of convex compact sets under the assumption of orthogonal complementarity, which are useful to compute Demyanov difference of subdifferential and negative superdifferential (Demyanov sum of quasidifferential) for the summation functions and the maximal (minimal) functions. In order to solve the quasidifferential uniqueness for quasidifferen-tiable functions, some properties of high dimensional kernels are given under the assumption that Demyanov difference of subdifferential and negative superdifferential is consistent with Minkowski difference of those, and a special class of quasidiffer-entiable functions with high demensional kernels (in the sense of Yan Gao), sub-superdifferentiable functions, is presented. For a special class of quasidifferentiable optimization-D. C. optimization, the convergence analysis of the steepest descent algorithm is obtained.2. Chapter 4 proposes the concept of convexificator kernels for quasidifferentiable functions, which can be used to tell whether a quasidifferentiable function is subdiffer-entiable. Concrete convexificators of quasidifferentiable functions are given by the recession functions of positively homogeneous functions. We can show that the convexificators are consistent with Demyanov sum of quasidifferentials.3. Chapter 5 is devoted to the study of constrained quasidifferentiable optimization with equality, inequality and abstract constraints. Under the mild assumption (existing a pair of quasidifferential such that the subdifferential and the superdifferential are upper semi-continuous), Fritz John necessary optimality conditions via Demyanov difference are obtained by Ekeland principle. As an application, necessary optimality conditions for bilevel programming are given. If the condition of being of maximal rank is added, KKT necessary optimality conditions can be established. Moreover, Fritz John necessary optimality conditions via sublinear functional are given without any assmuption. The above two results are both independent of the choose of the supergradients.
引文
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