Interchange Rules for Integral Functions

TL;DR

This paper develops a general abstract principle for interchanging infimization and integration in topological spaces, leading to improved rules for evaluating various integral function operations.

Contribution

It introduces new conditions to convert an abstract interchange principle into practical rules applicable to integral functions and related operations.

Findings

Unified framework for interchange rules

Enhanced methods for Legendre conjugates and subdifferentials

Broader applicability to integral function analysis

Abstract

We first present an abstract principle for the interchange of infimization and integration over spaces of mappings taking values in topological spaces. New conditions on the underlying space and the integrand are then introduced to convert this principle into concrete scenarios that are shown to capture those of various existing interchange rules. These results are leveraged to improve state-of-the-art interchange rules for evaluating Legendre conjugates, subdifferentials, recessions, Moreau envelopes, and proximity operators of integral functions by bringing the corresponding operations under the integral sign.