Qualification conditions-free characterizations of the $\varepsilon$-subdifferential of convex integral functions

Abstract

We provide formulae for the $\varepsilon$-subdifferential of the integral function $ I_f(x):=\int_T f(t,x) d\mu(t), $ where the integrand $f:T\times X \to [-\infty,+\infty]$ is measurable in $(t,x)$ and convex in $x$. The state variable lies in a locally convex space, possibly non-separable, while $T$ is given a structure of a nonnegative complete $\sigma$-finite measure space $(T,\mathcal{A},\mu)$. The resulting characterizations are given in terms of the $\epsilon$-subdifferential of the data functions involved in the integrand, $f$, without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.