  In this paper we derive new second-order optimality conditions for a very
general set-constrained optimization problem where the underlying set may be
nononvex. We consider local optimality in specific directions (i.e., optimal in
a directional neighborhood) in pursuit of developing these new optimality
conditions. First-order necessary conditions for local optimality in given
directions are provided by virtue of the corresponding directional normal
cones. Utilizing the classical and/or the lower generalized support function,
we obtain new second-order necessary and sufficient conditions for local
optimality of general nonconvex constrained optimization problem in given
directions via both the corresponding asymptotic second-order tangent cone and
outer second-order tangent set. Our results do not require convexity and/or
nonemptyness of the outer second-order tangent set. This is an important
improvement to other results in the literature since the outer second-order
tangent set can be nonconvex and empty even when the set is convex.
