  In this article, we introduce the idea of $gH$-weak subdifferential for
interval-valued functions (IVFs) and show how to calculate $gH$-weak
subgradients. It is observed that a nonempty $gH$-weak subdifferential set is
closed and convex. In characterizing the class of functions for which the
$gH$-weak subdifferential set is nonempty, it is identified that this class is
the collection of $gH$-lower Lipschitz IVFs. In checking the validity of sum
rule of $gH$-weak subdifferential for a pair of IVFs, a counterexample is
obtained, which reflects that the sum rule does not hold. However, under a mild
restriction on one of the IVFs, one-sided inclusion for the sum rule holds.
Next, as applications, we employ $gH$-weak subdifferential to provide a few
optimality conditions for nonsmooth IVFs. Further, a necessary optimality
condition for interval optimization problems with difference of two nonsmooth
IVFs as objective is established. Lastly, a necessary and sufficient condition
via augmented normal cone and $gH$-weak subdifferential of IVFs for finding
weak efficient point is presented.
