Proximal analysis in smooth spaces
The authors discuss different strategies for the definition of subdifferentials for lower semicontinuous functions on a Banach space \(X\). Many notions -- especially the Clarke subdifferential and the approximate \(G\)-subdifferential of Ioffe -- are introduced by a topological way in three steps: (i) definition of \(\partial f\) for Lipschitz functions, (ii) definition of normal cones by the subdifferential of the distance function, (iii) definition of \(\partial f\) for a general function \(f\) via normal cone to the epigraph. It is the aim of the paper to give a sequential description for such subdifferentials, i.e., a description of these subdifferentials by limits of ``classical'' differentiability notions. Here, with classical differentiability the authors mean \(\beta\)-differentiability and \(\beta\)-subdifferentiability with respect to an arbitrary bornology \(\beta\). Clearly, using the bornology of all bounded, compact or finite subsets, we get the differentiability notions in the sense of Fréchet, Hadamard and Gâteaux, respectively. The results are summarized in three main theorems. Assuming a \(\beta\)-smooth norm in \(X\), Theorem 1 gives a representation of the \(G\)-normal cone (Ioffe) and the \(C\)-normal cone (Clarke) of a set as special limits of \(\beta\)-normal cones. An analogous description for the \(G\)-subdifferential and the Clarke subdifferential of a lower semicontinuous function as limit of \(\beta\)-subdifferentials is formulated in Theorem 2. Finally, Theorem 3 concerns to the singular \(G\)-subdifferential and the singular Clarke subdifferential in the Fréchet case, i.e., using the bornology of bounded sets.
approximate \(G\)-subdifferential of Ioffe
scientific article; zbMATH DE number 898037
scientific article