Subdifferential formulae for the supremum of an arbitrary family of functions

TL;DR

This paper develops calculus rules for the subdifferentials of the supremum of an arbitrary family of lower semicontinuous functions, extending to both finite and infinite-dimensional settings and including convex functions.

Contribution

It provides new formulas for the Fréchet and limiting subdifferentials of supremum functions, generalizing existing results to broader contexts.

Findings

Fuzzy results for the Fréchet subdifferential of the supremum.

Explicit formulas for the limiting subdifferential in finite and infinite dimensions.

General subdifferential formulas for families of convex functions.

Abstract

This work provides calculus for the Fr\'echet and limiting subdifferential of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study showing fuzzy results about the Fr\'echet subdifferential of the supremum function. Posteriorly, we study in finite- and infinite-dimensional settings the limiting subdifferential of the supremum function. Finally, we apply our results to the study of the convex subdifferential; here we recover general formulae for the subdifferential of an arbitrary family of convex functions.