GENERALIZED HUKUHARA WEAK SUBDIFFERENTIAL AND ITS APPLICATION ON IDENTIFYING OPTIMALITY CONDITIONS FOR NONSMOOTH INTERVAL-VALUED FUNCTIONS

Abstract

In this paper, 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 convex and closed. 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 the 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. 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 a difference of two nonsmooth IVFs as the objective is established. Next, a necessary and sufficient condition via augmented normal cone and gH-weak subdifferential of IVFs for finding weak efficient points is presented. Lastly, in investigating a ‘sup-relation’ between gH-direction derivative and gH-weak subgradients, we approximately compute gH-weak subgradient at each iterative step. In the sequel, we propose W -gH-weak subgradient method to identify a weak efficient solution of an unconstrained nonsmooth IOP. We apply the proposed method to solve an interval optimization problem by taking a test example. We present a convergence analysis of the proposed method for constant and diminishing step sizes. ©2024 Journal of Nonlinear and Variational Analysis.