Generalized-Hukuhara Subgradient and its Application in Optimization Problem with Interval-valued Functions

TL;DR

This paper introduces the concepts of gH-subgradients and gH-subdifferentials for interval-valued functions, exploring their properties and applications in optimization, including optimality conditions and Lipschitz continuity.

Contribution

It develops the theory of gH-subgradients for convex interval-valued functions and applies it to optimization problems, providing new insights and tools for interval analysis.

Findings

gH-subdifferential properties like closeness and boundedness

gH-gradient is the only element in the gH-subdifferential for differentiable functions

gH-Lipschitz continuity characterized by subgradients

Abstract

In this article, the concepts of gH-subgradients and gH-subdifferentials of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex interval-valued function only contains gH-gradient of that interval-valued function. It is observed that the gH-directional derivative of a convex interval-valued function in each direction is maximum of all the products of gH-subgradients and the direction. Importantly, we show that a convex interval-valued function is gH-Lipschitz continuous if it has gH-subgradients at each point in its domain. Furthermore, the relations between efficient solutions of an optimization problem with interval-valued function and its gH-subgradients are derived.