Prox-regular functions in Hilbert spaces
The authors extend results about prox-regular functions known for finite-dimensional spaces to infinite-dimensional Hilbert spaces. After the introduction of prox-regularity of a function \(f\) by means of the limiting subdifferential \(\partial f\) it is pointed out that convex functions, so-called convexly \(C^{1,+}\)-composite functions and primal lower nice functions (in sense of Poliquin) are proxregular. With the help of the Moreau envelopes for \(f\) and the associated proximal mappings, a subdifferential characterization of prox-regularity is given showing that certain localizations of the subdifferential (the so-called \(f\)-attentive \(\varepsilon\)-localization of \(\partial f\)) admit a monotonicity behavior. Moreover, links between these locations and the proximal mappings of \(f\) are pointed out. At the end of the paper, \(C^{1,+}\)-regularity properties of the Moreau envelopes and Lipschitz properties of the proximal mappings of prox-regular functions are presented.
scientific article; zbMATH DE number 2141737
scientific article