Subdifferential and Conjugate Calculus of Integral Functions with and without Qualification Conditions

Abstract

We characterize the subdifferential and the Fenchel conjugate of convex integral functions by means of respectively the approximate subdifferential and the conjugate of the associated convex normal integrands. The results are stated in Suslin locally convex spaces, and do not require continuity-type qualification conditions on the functions, nor special topological or algebraic structures on the index set. Consequently, when confined to separable Banach spaces, the characterizations of such a subdifferential are obtained using only the exact subdifferential of the given integrand but at nearby points. We also provide some simplifications of our formulas when additional continuity conditions are in force.