Converting exhausters and coexhausters

TL;DR

This paper develops a new method for converting between upper and lower exhausters and coexhausters, facilitating the analysis of nonsmooth functions by switching between $ ext{min} ext{max}$ and $ ext{max} ext{min}$ representations.

Contribution

The paper introduces a novel approach to transform exhausters and coexhausters, enabling easier transition between different approximation forms in nonsmooth analysis.

Findings

Proposed a new method for converting exhausters and coexhausters.

Facilitates switching between $ ext{min} ext{max}$ and $ ext{max} ext{min}$ representations.

Enhances the analysis of extremal properties of nonsmooth functions.

Abstract

Exhausters and coexhausters are notions of constructive nonsmooth analysis which are used to study extremal properties of functions. An upper exhauster (coexhauster) is used to get an approximation of a considered function in the neighborhood of a point in the form of $\min\max$ of linear (affine) functions. A lower exhauster (coexhauster) is used to represent the approximation in the form of $\max\min$ of linear (affine) functions. Conditions for a minimum in a most simple way are expressed by means of upper exhausters and coexhausters, while conditions for a maximum are described in terms of lower exhausters and coexhausters. Thus the problem of obtaining an upper exhauster or coexhauster when the lower one is given and vice verse arises. We study this problem in the paper and propose new method for its solution which allows one to pass easily between $\min\max$ and $\max\min$ representations.