Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus

Abstract

We present a subdifferential analysis for a general concept of infinite sum f := Σi∈I fi of arbitrary ollections of convex functions fi, called Lebesgue infinite sum. Since this problem cannot be addressed, at least directly, through classical arguments from the theory of normal convex integrands, we perform a reduction analysis showing that the ε-subdifferential of f reduces to that of countable/finite subsums via appropriate lower limit and closure processes. Then, the usual calculus rules of (countable) integral functions give rise to characterizations of the ε-subdifferential of f, which are written exclusively by means of ε-subdifferentials of the data fi. The resulting characterizations do not assume any qualification or boundedness condition.