Critical and critical tangent cones in optimization problems

In this paper the notion of critical tangent cone CT (x|Q) to Q at x is introduced for the case when Q is a convex subset of a normed space X. If Q is closed with nonempty interior, and x is an element of Q, the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is then characterized by the condition d is an element of CT (x|Q). Furthermore, the support function of V (x, d| Q) is shown to be evaluated in terms of that support function of Q. For the cases when Q is the set of continuous or L(infinity) selections of a certain set-valued map, the corresponding characterization of the cone CT (x|Q) and the formula for the support function of V (x, d|Q) are obtained in terms of more verifiable conditions.