Topological Properties of the Approximate Subdierential

The approximate subdierential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization Potential advantages over other concepts of subdierentiability might be related to its nonconvexity This motivates to study some topological properties more in detail As the main result it is shown that in a Hilbert space setting each weakly compact set may be ob tained as the KuratowskiPainleve limit of the approximate subdierentials of some family of Lipschitzian functions As a consequence apart from niteness there is no restriction on the number of connected components of the subdierential In the nite dimensional case each topological type of a compact set may be realized by an approximate subdierential of some Lipschitzian function These are clear dierences for instance to Clarkes subdierential The results stated above require the denition of Lipschitzian functions on a space which is enlarged by one extra dimension Otherwise they would not hold true any longer since one can show that for a real function the number of connected components of the approximate subdierential is limited by two AMS subject classications C