  In this article, we study $gH$-subdifferential calculus of convex
interval-valued functions (IVFs) and apply it in a nonconvex composite model of
interval optimization problems (IOPs). It is found that the $gH$-directional
derivative of maximum of finitely many comparable IVFs is the maximum of their
$gH$-directional derivative. Proposed concepts of $gH$-subdifferential are
observed to be useful to derive Fritz-John-type and KKT-type efficiency
conditions for weak efficient solutions of IOPs. Further, we extract a
necessary and sufficient condition to characterize the weak efficient solutions
of nonconvex composite IOPs by applying the proposed concepts. To derive the
results on $gH$-subdifferentials, the concepts of limit supremum and limit
infimum with certain properties for IVFs are defined in the sequel. The whole
analysis is supported by appropriate expository examples.
