Journal of Nonlinear and Variational Analysis

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DOI: 10.23952/jnva.8.2024.2.09

Volume 8, Issue 2, 1 April 2024, Pages 333-368

 

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.

 

How to Cite this Article:
S. Ghosh, D. Ghosh, A. Petruşel, X. Zhao, Generalized Hukuhara weak subdifferential and its application on identifying optimality conditions for nonsmooth interval-valued functions, J. Nonlinear Var. Anal. 8 (2024), 333-368.