摘要
凸化集的概念最初是针对正齐次函数引入的,并规定其是一个凸紧集。它能描述一个正齐次函数的上凸和下凹近似,由于方向导数是正齐次函数,故在实际应用中,我们一般用凸化集来讨论方向可微函数的方向导数。随着认识的不断深入,人们将凸化集的概念加以推广,将其应用到连续函数中,只限定其是一个闭集即可,而不一定是凸集或紧集。由于凸化集越小就越能很好地描述一个函数,故又引入了极小凸化集的概念,但寻找凸化集的极小问题和唯一性问题至今仍然未得到很好的解决。本论文取得的主要结果可概括如下:
1.第2章由于拟可微函数是一类重要的非光滑函数,本文将凸化集的概念引入到拟可微函数中进行讨论,得出了拟可微函数的凸化集关于线性运算是封闭的结论;构造了拟可微函数的两个凸化集(其中的一个比另一个要小);拟可微函数取极大极小运算仍然有凸化集,并给出了其运算公式。
2.第3章对于推广概念的凸化集,我们用其来研究拟凸函数与伪凸函数;各种运算法则与极值性质被给出;具有等式约束与不等式约束的K-T充分条件被给出。
3.第4章基于实值函数凸化集的思想,对于方向可微函数又引入了近似广义海森阵的概念,它是近似海森阵概念的一种推广,我们利用它推导出了连续方向可微函数的二阶泰勒展式;关于近似广义海森阵的极小利用正则性条件给出。
The notion of convexificator is originally given for the positively homogeneous functions, and it is defined as a convex and compact set. It can describe the upper convex approximation and lower concave approximation of the positively homogeneous functions. Because directional derivative is the positively homogeneous function, we apply the convexificator to the directional derivative of the directionally differentiable function in the application. With the deeper realization the notion of convexificator is extended for a continuous function and it is defined as a closed set and it is not necessarily convex or compact set. As the smaller convexificator can describe the function better, so the notion of minimal convexificator is introduced. But the question of finding conditions for minimal convexificators of a continuous function and also of guaranteeing uniqueness of minimal convexificators has reminded so far open. The main results, obtained in this dissertation, may be summarized as follows:1. Chapter 2, because the quasidifferentiable function is an important class of the non-smooth analysis. This dissertation introduces the convexificator to the quasidifferentiable function. The conclusion we draw is that the convexificators of quasidifferentiable functions is closed about linear operation; we construct two convexificators for quasidifferentiable functions(one is smaller than the other); minimizing and maximizing quasidifferentiable functions still admit a convexificator, and their operational formulas are given.2. Chapter 3, in terms of the generalized convexificator, we present the characterization of pseudoconvexity and quasiconvexity; K-T sufficient condition with equality and inequality constraints is presented, various calculus rules and extremality about convexificators are given.3. Chapter 4, the definitions of the generalized second-order directional derivative and the upper and lower approximate generalized Hessian matrices are introduced; generalized Taylor's expansions for directionally differentiable functions are presented by using approximate generalized Hessian; we present conditions in terms of the set of extreme points for minimal and the unique minimal approximate generalized Hessians; Several calculus rules are given for directionally differentiable functions in terms of approximate generalized Hessian.
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