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 \(f:T\times X \rightarrow \overline{\mathbb {R}}\) 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 \(\varepsilon \)-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.