Functions and Sets of Smooth Substructure: Relationships and Examples.

The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., ‘amenable functions’, ‘partly smooth functions’, and ‘g o F decomposable Junctions’. Along with these classes a number of structural properties have been proposed, e.g., ‘identifiable suifaces’. ‘fast tracks’, and ‘primal-dual gradient structures’. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g o F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal. [ABSTRACT FROM AUTHOR]