The $\mathcal {U}$-Lagrangian, Fast Track, and Partial Smoothness of a Prox-regular Function

Abstract

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: $\mathcal {U}$, over which a special Lagrangian can be defined and has nice smooth properties and $\mathcal {V}$, the orthogonal complement subspace of $\mathcal {U}$. In this paper we generalize the definition of $\mathcal {VU}$-decomposition and $\mathcal {U}$-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.