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