Formulae for the conjugate and the $\varepsilon$-subdifferential of the supremum function

Pedro Pérez-Aros

Published 2017-12-26Version 1

The aim of this work is to provide formulae for the $\varepsilon$-subdifferential and the conjungate function of the supremun function $f=\sup_{t\in T} f_t$, where $T$ is an index set. The work is principally motivated by the case when the functions $\{f_t\}_{t\in T}$ are lower semicontinuous proper and convex functions. Nevertheless, we explore the case when the family of functions is arbitrary, but they satisfy the relations $f^{\ast \ast}=\sup_{t \in T} f^{\ast \ast}_t$, where $(\cdot)^{\ast \ast}$ denotes the biconjugate of a function. The study focuses its attention on when the space $X$ is finite-dimensional, in this case the formulae can be simplified under certain qualification conditions. However, we show how to extend these finite-dimesional results to arbitrary locally convex spaces without any qualification condition.