Formulae for the Conjugate and the Subdifferential of the Supremum Function

TL;DR

This paper derives formulas for the subdifferential and conjugate of the supremum function over arbitrary families, with special focus on convex functions in finite-dimensional and locally convex spaces, extending existing results.

Contribution

It provides new explicit formulas for the subdifferential and conjugate of supremum functions, including cases without qualification conditions and in general locally convex spaces.

Findings

Formulas for subdifferential and conjugate of supremum functions in finite-dimensional spaces.

Extension of results to arbitrary locally convex spaces without qualification conditions.

Simplification of formulas under certain conditions for convex functions.

Abstract

The aim of this work is to provide formulae for the subdifferential and the conjungate function of the supremun function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces, in this case the formulae can be simplified under certain qualification conditions. However, we show how to extend these results to arbitrary locally convex spaces without any qualification condition.