Sequential and exact formulae for the subdifferential of nonconvex integral functionals

Abstract

This work concerns the study of the subdifferential of the integral functional $$E_f(x)={\int }_{T}f(t,x)d\mu (t),$$ where $f$ is a (not necessarily convex) normal integrand, $(T,\mathcal{A},\mu )$ is a $\sigma$-finite measure space, while the decision variables vary in a separable Asplund space. First, using techniques of variational analysis we establish sequential approximate formulae for the Fréchet subdifferential of $E_f$. Secondly, we introduce a Lipschitz-like condition, which allows us to give an upper-estimation for the limiting subdifferential of $E_f$ even when this functional is non-Lipschitz.

28 pages