Proximal Smoothness and the Lower{C 2 Property 论文

摘要

Dedicated to R. T. Rockafellar on his 60th Birthday A subset X of a real Hilbert space H is said to be proximally smooth provided that the function dX : H! R (the distance to X) is continuously dieren tiable on an open tube U around X. It is proven that this property is equivalent to dX having a nonempty proximal subgradient at every point of U, and that the (G^ ateaux = Fr echet) derivative is locally Lipschitz on U. The Lipschitz behavior of the derivative is a consequence of the fact that under proximal smoothness, the metric projection onto X is single valued and Lipschitz on U. Alternate characterizations of proximal smoothness are given as well, in terms of properties of the proximal normal cone multifunction on X and on nearby closed neighborhoods of X. In case X is weakly closed, the list of equivalences is extended to include each point of U admitting a unique closest point in X. Further specializations are given in nite dimensions. In that setting, we discuss properties of locally Lipschitz real valued functions whose epigraphs are proximally smooth in a local sense. It is demonstrated that this function class coincides with the lower{C 2 functions studied by