Are the approximate and the Clarke subgradients generically equal?
Two of the well-known subdifferential notions for a real-valued function \(f : \mathbb{R}^n \to \mathbb{R}\) -- the Clarke subdifferential \(\partial_c f(x)\) and the approximate subdifferential \(\partial_a f(x)\) of Joffe - - are compared in the present paper. Starting with the formerly proved inclusion \[ \partial_a f(x) \subseteq \partial_c f(x) \] it is observed if both sets are generically equal, either in sense of measure theory (i.e., equal on a set of full measure) or in the topological sense (i.e., equal on a dense \(G_\delta\)-set). In the one-dimensional case, positive answers are given showing that both subdifferentials are different at most on a countable set. For higher- dimensional spaces, however, the problem is more complicated. The author constructs a function for which the subdifferentials are different on a set with positive measure. In this example, the set on which the subdifferentials are equal is of positive measure, too, and a dense \(G_\delta\)-set. Hence, the question for the generic equality in topological sense remains unanswered.
scientific article; zbMATH DE number 811192