Second-order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization

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 second-order necessary and sufficient 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 results with strongly pseudoconvex functions. We also compare our conditions with the results of the recently published paper (Bednarik and Pastor, 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 conditions in terms of 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 scalar problems to derive necessary and sufficient optimality conditions in the cone-constrained vector optimization.