A general inequality-constrained minimization program is considered on a real normed space E. The constraints are given by a finite number of inequalities and an explicit set C. The paper deals with necessary conditions of Kuhn-Tucker type for a local minimum. The essential classical cases are mentioned and well-known constraints qualifications are compared. The authors present nonsmooth analogs of the Cottle constraint qualification in which gradients are replaced by directional derivatives associated with closed tangent cones. Using these constraint qualifications, they get the optimality conditions involving subgradients associated with closed convex tangent cones. The latter may be Clarke tangent or Penot's prototangent or some other cones. The authors define the class of directionally Lipschitzian functions containing the known class of locally Lipschitzian ones. This allows to improve results of some other authors by obtaining sharper optimality conditions. In case \(C=E\) the equivalence between the constraint qualification offered in the paper and the boundedness and nonemptiness of the Kuhn-Tucker multipliers set is established under locally Lipschitzian conditions.