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Functions and sets of smooth substructure: relationships and examples. (English) Zbl 1103.90100
Summary: 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 “$g\circ F$ decomposable functions ”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “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 $g \circ F$ decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.

##### MSC:
 90C52 Methods of reduced gradient type 90C46 Optimality conditions, duality
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##### References:
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