Generalized-Hukuhara subgradient and its application in optimization problem with interval-valued functions

Abstract

In this article, the concepts of gH-subgradient and gH-subdifferential 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 contains only the gH-gradient. It is observed that the directional gH-derivative of a convex interval-valued function is the maximum of all the products between gH-subgradients and the direction. Importantly, we prove that a convex interval-valued function is gH-Lipschitz continuous if it has gH-subgradients at each point in its domain. Furthermore, relations between efficient solutions of an optimization problem with interval-valued function and its gH-subgradients are derived.