SUBDIFFERENTIAL CALCULUS USING EPSILON-SUBDIFFERENTIALS

In applications of convex analysis it is important to be able to calculate the subdifferentials of various combinations of (proper and lower semicontinuous) convex functions, such as the sum of two such functions, or their inf-convolution ("epi-sum"), as well as the pre-composition of a convex function with an affine map or the “marginal" function obtained from a convex function and a linear map. The classical formulas for such calculations all require additional hypotheses, same of which may be difficult to check or are not always satisfied in a given variational problem. In this paper we present formulas for the subdifferentials of such combinations without assuming any additional hypotheses, by utilizing ε(lunate)-subdifferentials. This wider applicability comes at the price of somewhat more complicated formulas. © 1993 Academic Press Limited.