非可微二层凸规划的最优性条件

本文考虑的是构成函数为非可微凸函数的二层规划问题(NDBP),得到了下层极值函数和上层复合目标函数的方向导数和次微分的估计式,给出非可微二层凸规划(NDBP)最优解的几种最优性条件。     

拟凸函数的一个充分条件

十多年来,广义凸函数的研究构成了数学规划研究的趋向之一。拟凸函数是数学规划中常见的一种广义凸函数类。然而,对于给定的一个函数,如何用二阶导数判别其广义凸性。现有文献中,见之不多。Ferland[1]等人在七十年代初用K阶加边行列式的方法曾经讨论过一个二次可微函数是拟凸函数的必要条件和充分条件。熟知,对于一个二次可微函数,二阶导数可以表征函数的凸性。本文根据微分方程的极值原理,给出一个二次可微函数是拟     

非可微分二层Lipschitz规划的广义微分和广义方向导数

本文对构成函数为Ｌｉｐｓｃｈｉｔｚ函数的二层规划问题,利用非光滑分析工具,讨论了下层极值函数和上层复合目标函数的Ｌｉｐｓｃｈｉｔｚ连续性,给出了这些函数的广义微分和广义方向导数的估计式。本文得到的结果为进一步研究非可微二层Ｌｉｐｓｃｈｉｔｚ规划的最优性条件和有效算法等理论和方法问题奠定了基础。     

一类不可微优化的Fritz-John条件

基于星形集空间的性质,定义一类星形可微函数.这类函数是方向可微的,其方向导数可以表示成两个正齐次非负连续函数之差,其星形微分为一星形集对.对于含有不等式约束条件的星形可微优化问题,给出一个Fritz-John形式的最优性必要条件.     

无约束拟可微优化的信赖域方法的全局收敛性

设f：R^n→R为实值拟可微函数，x∈domf，则f在x点沿方向d∈R^n的方向导数表示为     

方向导数的应用

本文将一元函数的高阶导数对应为多元函数的高阶导数，用方向导数表达泰勒公式，使之与一元函数的泰勒公式有统，的形式。又引入方向单调性、方向极值等概念，使多元函数的极值判别法基于一元函数极值判别法，此法不但直观又解法了判别式Δ＝０时的不确定性。限于篇幅只讨论二元函数。     

参数非光滑优化的微分稳定性

本文研究了含有向量参数的非光滑优化问题的极值函数或叫做边缘函数的连续性及某种意义下的微分性质。给出了目标函数及不等式约束为李普希兹函数,等式约束为连续可微函数,并且带有闭凸约束集C的非凸非光滑问题的最优值函数的几种方向导数的界,把[4],[1]中关于一个参数的单边扰动推广到向量参数的扰动,亦可认为是把[2]由光滑函数类推广到李普希兹函数类。     

高考中导数应用的新变化——用导数探讨函数图象的交点问题

观察近几年高考试题,其中导数命题的方向基本没变,主要从五个方面(①与切线有关的问题;②函数的单调性和单调区间问题;③函数的极值和最值问题;④不等式证明问题;⑤与函数的单调性、极值、最值有关的参数问题)考查了学生对导数的掌握水平,但在方向基本没变的情况下,又有所创新,导数命题创新有两个方面:一是研究对象的多元化,由研究单一函数转向研究两个函数或多个函数;二是研究内容的多元化,由用导数研究函数的性质(单调性、最值、极值)转向运用导数进行函数的性质、函数图象的交点和方程根的分布等综合研究,实际上就是导数考查函数图象的交…     

Hilbert空间中的一类双层规划问题的一阶与二阶最优性条件

本文考虑Hilbert空间中的,上层为有限个不等式约束,下层是一锥约束参数规划的双层规划问题的最优性条件.首先,利用下层问题最优值函数的方向导数的上下界的性质给出一阶最优性条件.之后,在使下层问题的最优值函数是二阶方向可微的条件下,证明了二阶必要性条件.     

例析设而不求思想在函数求值中的应用

我们知道,对于可导函数求极值问题。首先求导,让导数为0,求出可疑极值点．但有些函数的导函数为超越函数,其零点（可疑极值点）很难求出,     

A note on sensitivity of value functions of mathematical programs with complementarity constraints

Using standard nonlinear programming (NLP) theory, we establish formulas for first and second order directional derivatives of optimal value functions of parametric mathematical programs with complementarity constraints (MPCCs). The main point is that under a linear independence condition on the active constraint gradients, optimal value sensitivity of MPCCs is essentially the same as for nonlinear programs, in spite of the combinatorial nature of the MPCC feasible set. Unlike NLP however, second order directional derivatives of the MPCC optimal value function show combinatorial structure. Received: October 31, 2000 / Accepted: March 8, 2002?Published online June 25, 2002     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

最优值函数的方向导数

本文详细介绍了参数非线性规划问题最优值函数的方向导数的存在性、表示形式和有关的发展历史,本文不仅讨论了这一问题的经典结论,还叙述了最近二十年来的新结果。本文最后指出进一步研究的方向。     

非线性参数规划问题$\varepsilon$-最优解集集值映射的连续性

对非线性参数规划问题$\varepsilon$-最优解集集值映射的连续性条件进行了研究.首先在可行集集值映射局部有界且正则的条件下,讨论了非线性参数规划问题最优值函数的连续性,然后针对$\varepsilon$-最优解集集值映射的结构特征并利用此结果和集值分析理论,给出了非线性参数规划问题$\varepsilon$-最优解集集值映射连续的一个充分条件.     

AN OUTER APPROXIMATION ALGORITHM FOR SEMI-INFINITE NONDIFFERENTIABLE PROGRAMMINGS

This paper presents a conceptual outer approximation algorithm for dealing with the semi-infinite nondifferentiable programming in which functions are locally Lipschitzian. By weakening the restriction on the family of functions for a parametric programming, we answer a question proposed in [1].     

A practicable way for computing the directional derivative of the optimal value function in convex programming

Formulas for computing the directional derivative of the optimal value function or of lower or upper bounds of it are well-known from literature. Because they have as a rule a minmax structure, methods from nondifferentiable optimization are required.

Considering a fully parametrized convex problem, in the paper the mentioned minmax formulas are transformed into usual programming problems. Although they are nonconvex in general, the computational effort is much lower than that for minmax problems. In several special cases, for instance, for linear least squares problems, linear programming problems arise.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.     

参数非线性规划中最优值函数的预不变凸凹性

本文给出了预拟不变单调函数的定义,提出了不变凸点到集映射、不变凹点到集映射的新概念,得到了不变凸点到集映射的一个充分必要条件,研究了参数非线性规划中最优值函数的预不变凸凹性。