In this paper is firstly pointed out the strict relationship existing between the optimality of a point x0, belonging to a feasible region S, and the behaviour of the objective function along the directions of the cone of tangents to S at x0. Such a relationship is established either for a directionally differentiable and locally lipschitz continuous function at x0 or for a differentiable function at x0 and allows finding first and second order optimality conditions in the case where xo is the vertex of a star shaped set S or, as a special case, is the vertex of a polyhedral cone. Finally, relationships between optimality at the vertex x0 of a star shaped set S and optimailty at x0 along the directions of S are investigated by means of generalized concavity.