Variational Analysis Based on Proximal Subdifferential on Smooth Banach Spaces

This paper first shows that for any p ∈ (1, 2) there exists a continuously differentiable function f on l^p (and L^p) such that the proximal subdifferential of f is empty everywhere, and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces l^p and L^P with p ∈ (1, 2). On the other hand, this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2, which conclude all Hilbert spaces and all the classical spaces l^p and L^p with p ∈ (2, +∞). In particular, in such a smooth space, we provide the proximal subdifferential rules for sum functions, product functions, composite functions and supremum functions, which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces. Some of our main results are new even in the Hilbert space case. As applications, we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.