Topological Properties of the Approximate Subdifferential*

The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be obtained as the Kuratowski]Painlev´e limit of approximate subdifferentials from a one-parametric family of Lipschitzian functions. Sharper characterizations are possible for strongly compact subsets. As a consequence, in any Hilbert space the approximate subdifferential of a suitable Lipschitzian function may be homeomor- phic both in the strong and weak topology.to the Cantor set. Further results relate the approximate subdifferential to specific topological types, to the one-di- mensional case which is extraordinary in some sense., and to the value function of a C1-optimization problem.