Characterizations of the subdifferential of convex integral functions under qualification conditions

Abstract

This work provides formulae for the ε-subdifferential of integral functions in the framework of complete σ-finite measure spaces and locally convex spaces. In this work we present here new formulae for this ε-subdifferential under the presence of continuity-type qualification conditions relying on the data involved in the integrand.We provide new formulae for the subdifferential and the ε-subdifferential of the convex integral function given by the following expressionwhere (T, Σ, µ) is a complete σ-finite measure space, and f : T × X → R is a convex normal integrand defined on a locally convex space X.General formulae have been established in [19] using a finite-dimentional reduction approach, without additional assumptions on the data represented by the integrand f . In this paper, we use natural qualifications condition, involving appropriate continuity assumption on the integrand, to give more explicit characterization of the subdifferential and the ε-subdifferential of the function I f .

Subdifferential Calculus Rules for Possibly Nonconvex Integral Functions

“…Our approach consists in using the concept of robust infima (see Definition 3.1), which we combine with some variational principles, applied in the space X as well as in the functional space of p-integrable functions. Using this we extend and improve the results of [28] and [34] (see also [14,15]).…”

“…To ilustrate our results we compute sequential formulae for series of lower semicontinuous functions using the measure space (N, P(N)). This class of functions has been recently studied in the convex case (see, e.g., [14,15,47]), motivated by some applications to entropy minimization. Moreover, in this case we can apply techniques of separable reduction, and extend the results to an arbitrary Asplund space.…”

Formulae for the Conjugate and the Subdifferential of the Supremum Function

“…Under the additional assumptions of the compactness of T and some continuity property of the function t → f t (w) we can prove that (27) is equivalent to (24). More precisely we get the following result.…”

“…the last means x * i ∈ N dom ft i (x) and n i=1 x * i = 0 with not all x * i equal to zero, this contradicts (27). Therefore, condition (a) of Lemma 5.1 must hold, then…”

“…and t∈T (u * t , α t ) = (x * , α). Consequently using (28) (with ε = 0) we get that (27) implies (24). Then, we can apply Theorem 5.1 and following its notation we use the compactness of T and the upper semicontinuity of t → f t (w) (together with similar arguments as in the first part) to prove that S(x, ε 1 ) is contained in S(x, ε 1 ).…”