Generalized Hukuhara Subdifferentiability for Convex Interval-Valued Functions and Its Applications in Nonsmooth Interval Optimization

Abstract

In this era of data science and machine learning, optimization tools and techniques for handling inexact and uncertain data are of utmost importance. In this chapter, we develop and analyze the notion of subdifferentiability of convex interval-valued functions (IVFs) and apply it to find optimality conditions for interval optimization problems. Using the existing calculus on IVFs, the compactness of gH-subdifferential set of a gH-continuous IVF is studied in the sequel. It has been observed that the support function of a subdifferential set of supremum of a finite index of convex IVFs is identical to the support function of the convex hull of gH-subdifferentials of this finite index of convex IVFs. To propose this result, we recall the concept of a support function of a nonempty subset of $$I(\mathbb {R})^{n}$$ and introduce the concepts of a convex combination of intervals and a convex hull for a set of intervals and further derive some necessary results. Finally, we provide two optimality conditions and related examples that help to obtain efficient solutions of interval optimization problems.