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We use in this paper a new concept of a directional subdifferential, namely the Dini-Hadamard-like e-subdifferential, recently introduced in [29], in order to provide a subdifferential formula for the difference of two directionally approximately starshaped functions (a valuable class of nonsmooth functions, see for instance [32]), under weaker conditions than those presented in [7]. As a consequence, we furnish necessary and sufficient optimality conditions for a nonsmooth optimization problem having the difference of two functions as objective.

Mathematics Subject Classification (2010): 26B25, 49J52, 90C56.

Keywords: Fréchet e-subdifferential, Dini-Hadamard e-subdifferential, Dini-Hadamard-like e-subdifferential, directionally convergent sequence, sponge, directional lower limit, approximately starshaped functions, directionally approximately starshaped functions, spongiously local blunt minimizer.