Topological Properties of the Approximate Subdiierential Ren E Henrion

The approximate subdiierential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdiierentiability might be related to its non-convexity. 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 obtained as the Kuratowski-Painlev e limit of the approximate subdiierentials of some family of Lipschitzian functions. As a consequence, apart from niteness, there is no restriction on the number of connected components of the subdiierential. In the nite dimensional case, each topological type of a compact set may be realized by an approximate subdiierential of some Lipschitzian function. These are clear diierences for instance to Clarke's subdiierential. The results stated above require the deenition 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 subdiierential is limited by two.