Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

TL;DR

This paper introduces the gH-Clarke derivative for interval-valued functions, exploring its properties and establishing conditions under which it coincides with directional derivatives, supported by illustrative examples.

Contribution

It defines the gH-Clarke derivative for interval-valued functions and analyzes its properties, including its relation to Lipschitz continuity and convexity.

Findings

Upper gH-Clarke derivative of gH-Lipschitz IVFs is sublinear.

Every gH-Lipschitz continuous function is upper gH-Clarke differentiable.

For convex gH-Lipschitz IVFs, the upper gH-Clarke derivative equals the gH-directional derivative.

Abstract

In this article, the notion of gH-Clarke derivative for interval-valued functions is proposed. To define the concept of gH-Clarke derivatives, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. The upper gH-Clarke derivative of a gH-Lipschitz interval-valued function (IVF) is observed to be a sublinear IVF. It is found that every gH-Lipschitz continuous function is upper gH-Clarke differentiable. For a convex and gH-Lipschitz IVF, it is shown that the upper gH-Clarke derivative coincides with the gH-directional derivative. The entire study is supported by suitable illustrative examples.