Generalized Hukuhara directional differentiability of interval-valued functions on Riemannian manifolds
Hilal Ahmad Bhat, Akhlad Iqbal
TL;DR
The paper investigates generalized Hukuhara directional differentiability (gH-directional differentiability) of interval-valued functions on Riemannian manifolds and demonstrates that, in general, gH-directional differentiability is not equivalent to the directional differentiability of the center or half-width or of endpoint functions, countering claims valid only in special Hadamard contexts. It provides counterexamples showing convexity-at-a-point is insufficient for standard directional derivatives and clarifies when gH-differentiability can be expressed via component derivatives, including a condition of nondecreasing radius along geodesics. The work derives both necessary and sufficient results under cw-convexity and related assumptions, and highlights the limitations due to incompleteness of interval order. The findings have implications for developing KKT-type optimality conditions and numerical methods for interval-valued optimization in fuzzy or uncertain environments on manifolds.
Abstract
In this paper, we show that generalized Hukuhara directional differentiability of an interval-valued function (IVF) defined on Riemannian manifolds is not equivalent to the directional differentiability of its center and half-width functions and hence not to its end point functions. This contrasts with S.-L. Chen's \cite{chen} assertion which says the equivalence holds in terms of endpoint functions of an IVF which is defined on a Hadamard manifold. Additionally, the paper addresses some other inaccuracies which arise when assuming the convexity of a function at a single point in its domain. In light of these arguments, the paper presents some basic results that relate to both the convexity and directional differentiability of an IVF.