Abstract :
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.
