基于支撑函数表示的排气口约简
On Reduction of Exhausters via a Support Function Representation
摘要：
穷举器是一类紧致凸集，它提供了正齐次函数的极大极小表示，是研究非光滑函数的有效工具。正齐次函数的上、下穷举器被用来用几何术语描述最优性条件，也用来寻找最陡下降最速下降或上升方向。由于一个上/下凸集可能包含无穷多个或无穷多个紧凸集，因此自然产生了极小性和约简穷举集的问题。有几种方法可以减少排气装置。在包含极小意义下，利用支撑函数的表示，给出了定义在$\mathbb R^2$到$\mathbb R$之间的正齐次函数的上穷举约简技术.这些技术有具体的几何意义，他们形成了一个充要条件的基础上，包括最小的排气。一些例子来说明每个减少技术。
英文摘要：
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.
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