The U -Lagrangian, Fast Track, and Partial Smoothness of a Prox-regular Function

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function and a point, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of VU-decomposition and U-Lagrangian to prox-regular functions and show that the closely related notions fast track and partial smoothness are equivalent under some conditions. Some connections with tilt stability are discussed.