Converting exhausters and coexhausters

TL;DR

The paper tackles converting between minmax (upper exhauster) and maxmin (lower exhauster) representations of the directional derivative in constructive nonsmooth analysis, formalized as $h(Δ)=min_{C\in E^*}max_{v\in C}⟨v,Δ⟩$ or $h(Δ)=max_{C\in E_*}min_{w\in C}⟨w,Δ⟩$. It develops a general converting procedure applicable to finite polytopal families, extending prior 2D methods. It proves discrete minimax-type theorems (Abbasov-type and Fan-type) that give sufficient conditions for the equality of minmax and maxmin representations and applies them to exhauster/coexhauster conversion. The results address stability concerns via coexhauster formulations and include practical reduction steps to prune redundant sets, thereby enabling robust nonsmooth optimization algorithms. The work broadens the scope beyond two dimensions and provides a concrete toolkit for constructive nonsmooth analysis.

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.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3