Abstract
ABSTRACTIn this article, we first examine some properties and calculus rules for the limiting subdifferential with respect to a set for functions. To do this, we propose corresponding derivatives with respect to a set for functions and study the relationships between these derivatives and the limiting subdifferential with respect to a set. We then find applications of these constructions in establishing optimality criteria for nonsmooth optimization problems. In particular, sufficient/necessary conditions for directionally optimal (strictly) solutions of both unconstrained and constrained nonsmooth optimization problems are provided and some of them are shown to be more useful in comparison with several existing results in the literature.
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