Integral functionals on nonseparable Banach spaces with applications

TL;DR

This paper develops new theoretical tools for integral functionals on non-separable Banach spaces, including measurable selection and conjugate formulas, with applications in stochastic programming and calculus of variations.

Contribution

It introduces a novel class of integrands and multifunctions, providing explicit formulas for conjugates and subdifferentials in non-separable Banach spaces.

Findings

Measurable selection results for new integrand classes

Explicit conjugate formulas for integral functionals

Applications to stochastic programming and variational problems

Abstract

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results. Then, we provide an interchange formula between integration and infimum, which enables us to get explicit formulas for the conjugate and Clarke subdifferential of integral functionals. Applications to expected functionals from stochastic programming, optimality conditions for a calculus of variation problem and sweeping processes are given.