Generalized Hukuhara Weak Subdifferential and its Application on Identifying Optimality Conditions for Nonsmooth Interval-valued Functions

TL;DR

This paper introduces the $gH$-weak subdifferential for interval-valued functions, explores its properties, and applies it to derive optimality conditions in nonsmooth interval optimization problems.

Contribution

It defines the $gH$-weak subdifferential, investigates its properties, and applies it to establish optimality conditions for nonsmooth interval-valued functions.

Findings

The $gH$-weak subdifferential set is nonempty, closed, and convex.

The sum rule for $gH$-weak subdifferential generally does not hold, but one-sided inclusion can be established.

Necessary and sufficient optimality conditions are derived for interval optimization problems.

Abstract

In this article, we introduce the idea of $gH$-weak subdifferential for interval-valued functions (IVFs) and show how to calculate $gH$-weak subgradients. It is observed that a nonempty $gH$-weak subdifferential set is closed and convex. In characterizing the class of functions for which the $gH$-weak subdifferential set is nonempty, it is identified that this class is the collection of $gH$-lower Lipschitz IVFs. In checking the validity of sum rule of $gH$-weak subdifferential for a pair of IVFs, a counterexample is obtained, which reflects that the sum rule does not hold. However, under a mild restriction on one of the IVFs, one-sided inclusion for the sum rule holds. Next, as applications, we employ $gH$-weak subdifferential to provide a few optimality conditions for nonsmooth IVFs. Further, a necessary optimality condition for interval optimization problems with difference of two nonsmooth IVFs as objective is established. Lastly, a necessary and sufficient condition via augmented normal cone and $gH$-weak subdifferential of IVFs for finding weak efficient point is presented.