Epsilon-subdifferentiability for interval-valued functions and its application in interval optimization problems

Abstract

In this paper, we propose the notion of ϵ-subdifferentiability or gHϵ-subdifferentiability for convex interval-valued functions. Several important characteristics of gHϵ-subdifferential set, e.g., nonemptiness, closedness, convexity, boundedness, etc. are studied. To prove the convexity of gHϵ-subdifferential set, we define the concept of gHϵ-directional derivative for convex interval-valued functions. We show the boundedness of gHϵ-subdifferential set at an interior point of the effective domain by a weighted mapping for intervals. Subsequently, it is found that the gHϵ-subdifferential set can be unbounded on the boundary of the effective domain. Furthermore, a new solution concept, namely approximate solution or ϵ-solution, for an interval optimization problem is introduced. Using the proposed gHϵ-subdifferentiability, we develop two necessary and sufficient optimality conditions to find an ϵ-solution for an unconstrained interval optimization problem. Lastly, a theorem has been proved to solve interval minimax optimization problems using gHϵ-subdifferentiability. Numerical examples illustrate the whole study. © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2025.