Integration formulas via the Fenchel subdifferential of nonconvex functions

Abstract

Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the epsilon-subdifferential and the Fenchel subdifferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction of the lsc convex envelope either in terms of the epsilon-subdifferential of the nominal function or of the subdifferential of its weak lsc envelope.