Qualification Conditions-Free Characterizations of the ε-Subdifferential of Convex Integral Functions

Abstract

We provide formulae for the e-subdifferential of the integral function I f (x) := similar to T f (t, x)d mu(t), where the integrand f : T xX. 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 s-finite measure space (T, A, mu). The resulting characterizations are given in terms of the e-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 finitedimensional setting.