Generalized Hukuhara Hadamard Derivative of Interval-valued Functions and Its Applications to Interval Optimization

TL;DR

This paper introduces the generalized Hukuhara Hadamard derivative for interval-valued functions, explores its properties, and demonstrates its applications in interval optimization, including optimality conditions and convexity analysis.

Contribution

It defines a new derivative concept for interval-valued functions, establishes its properties, and applies it to derive optimality conditions in interval optimization problems.

Findings

Existence of gH-Hadamard derivative implies gH-Frechet derivative and vice versa.

The derivative helps characterize convexity and efficient points in interval optimization.

Extended KKT conditions are derived using the proposed derivative.

Abstract

In this article, we study the notion of gH-Hadamard derivative for interval-valued functions (IVFs) and its applications to interval optimization problems (IOPs). It is shown that the existence of gH-Hadamard derivative implies the existence of gH-Frechet derivative and vise-versa. Further, it is proved that the existence of gH-Hadamard derivative implies the existence of gH-continuity of IVFs. We found that the composition of a Hadamard differentiable real-valued function and a gH-Hadamard differentiable IVF is gHHadamard differentiable. Further, for finite comparable IVF, we prove that the gH-Hadamard derivative of the maximum of all finite comparable IVFs is the maximum of their gH-Hadamard derivative. The proposed derivative is observed to be useful to check the convexity of an IVF and to characterize efficient points of an optimization problem with IVF. For a convex IVF, we prove that if at a point the gH-Hadamard derivative does not dominate to zero, then the point is an efficient point. Further, it is proved that at an efficient point, the gH-Hadamard derivative does not dominate zero and also contains zero. For constraint IOPs, we prove an extended Karush-Kuhn-Tucker condition by using the proposed derivative. The entire study is supported by suitable examples.