Stability in generalized differentiability

From a novel general principle of upper set convergence for locally bounded set-valued maps, we derive stability results for various objects in generalized differentiability. First, we obtain a stability result for the Clarke generalized gradient under assumptions of epiconver-gence and equi-F-differentiability for a sequence of locally Lipschitz real-valued functions. We relax in this way the conditions of equi-lower semidifferentiability of Zolezzi [6] and equi-F-subdifferentiability of Penot [5], respectively. Then, we turn to vector-valued functions and prove analogous stability results for two different concepts of a generalized Hessian of C1,1 functions. The first concept was used by Hirriart-Urruty et al. [2], and the second was intro-duced by Páles and Zeidan [4]. In another approach for C1,1 functions, we present a weak ∗ stability result for the second-order Clarke subdifferential. It presupposes for the approximating sequence fn of C1,1 functions that 〈f ′n(x), h 〉 are locally convergent to 〈f ′(x), h 〉 with respect to the Lipschitz seminorm. For such functions we extend the main result of Zolezzi [7] to the second-order subdifferential of Clarke’s type. We apply the latter result to the continuous behaviour of the Lagrange multipliers i