  In this paper we first provide a general formula of inclusion for the
Dini-Hadamard epsilon-subdifferential of the difference of two functions and
show that it becomes equality in case the functions are directionally
approximately starshaped at a given point and a weak topological assumption is
fulfilled. To this end we give a useful characterization of the Dini-Hadamard
epsilon-subdifferential by means of sponges. The achieved results are employed
in the formulation of optimality conditions via the Dini-Hadamard
subdifferential for cone-constrained optimization problems having the
difference of two functions as objective.
