Tangent cone and contingent cone to the intersection of two closed sets

For a nonempty closed set CC in a real normed vector space XX and an inequality solution set, we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones. We not only express the contingent cone to a solution set of inequalities and equalities by the directional (or Fréchet) derivatives of the active inequality constraint functions and the Fréchet derivatives of the equality constraint functions but also the tangent cone by the Clarke (or lower Dini, or upper Dini) derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions. By using a simple property of the function dC−dCcdC−dCc, we characterize these cones by the hypertangent and hypercontingent vectors to the set CC. Furthermore, these results allow us to present new constraint qualifications for the Karush–Kuhn–Tucker conditions.