论文标题
通过支持函数表示减少呼气器
On Reduction of Exhausters via a Support Function Representation
论文作者
论文摘要
呼气是具有紧凑的凸组的家族，可提供正均匀函数的Minmax或Maxmin表示，它们是研究非平滑功能的有效工具。呈正同质函数的上和下排气器以几何术语来描述最佳条件，并找到最陡峭下降或上升的方向。由于上/下呼吸器可能包含有限或无限的许多紧凑型凸组，因此自然会出现呼气器的最小和减少问题。有几种减少呼气的方法。在这项研究中，从包容性最小性的意义上，通过代表支持函数的代表，提出了从$ \ mathbb r^2 $到$ \ mathbb r $定义的积极均匀函数的一些还原技术。这些技术具有具体的几何含义，它们构成了呼气器最小程度的必要条件的基础。提出了一些示例，以说明每种还原技术。
Exhausters are families of compact, convex sets which provide minmax or maxmin representations of positively homogeneous functions and they are efficient tools for the study of nonsmooth function. Upper and lower exhausters of positively homogeneous functions are employed to describe optimality conditions in geometric terms and also to find directions of steepest descent or ascent. Since an upper/lower exhauster may contain finitely or infinitely many compact convex sets, the problem of minimality and reduction of exhausters naturally arise. There are several approaches to reduce exhausters. In this study, in the sense of inclusion-minimality, some reduction techniques for upper exhausters of positively homogeneous functions defined from $\mathbb R^2$ to $\mathbb R$ is proposed by means of a representation of support functions. These techniques have concrete geometric meanings and they form a basis for a necessary and sufficient condition for inclusion-minimality of exhausters. Some examples are presented to illustrate each reduction technique.
