Valadier-like formulas for the supremum function II: The compactly indexed case

TL;DR

This paper extends Valadier's formulas for the subdifferential of the supremum of convex functions by removing continuity assumptions, providing a more general and simplified characterization applicable in broader contexts.

Contribution

It generalizes Valadier's original subdifferential formula for supremum functions by eliminating the continuity requirement, offering a more comprehensive and simplified characterization.

Findings

Removed the continuity assumption from Valadier's formula.

Derived a general subdifferential formula for supremum functions.

Provided a simplified version when the supremum is continuous at some point.

Abstract

We generalize and improve the original characterization given by Valadier [20, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to Valadier formula. Our starting result is the characterization given in [10, Theorem 4], which uses the epsilon-subdiferential at the reference point.