  In this paper, we introduce a new second-order directional derivative and a
second-order subdifferential of Hadamard type for an arbitrary
nondifferentiable function. We derive several second-order optimality
conditions for a local and a global minimum and an isolated local minimum of
second-order in unconstrained optimization. In particular, we obtain two types
results with generalized convex functions. We also compare our conditions with
the results of the recently published paper [Bednavrik, D., Pastor, K.: On
second-order conditions in unconstrained optimization. Math. Program. Ser A,
{\bf 113}, 283--291 (2008)] and a lot of other works, published in high level
journals, and prove that they are particular cases of our necessary and
sufficient ones. We prove that the necessary optimality conditions concern more
functions than the lower Dini directional derivative, even the optimality
conditions with the last derivative can be applied to a function, which does
not belong to some special class. At last, we apply our optimality criteria for
unconstrained problems to derive necessary and sufficient optimality conditions
for the cone-constrained vector problems.
