On Reduction of Exhausters via a Support Function Representation

TL;DR

This paper introduces geometric techniques based on support functions to reduce upper exhausters of positively homogeneous functions in two dimensions, ensuring minimality and simplifying their representation.

Contribution

It proposes new reduction methods for upper exhausters using support function representations, providing geometric insights and a necessary and sufficient condition for minimality.

Findings

Reduction techniques are concretely characterized geometrically.

Techniques are applicable to functions from R^2 to R.

Examples demonstrate the effectiveness of the methods.

Abstract

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.