Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus

TL;DR

This paper introduces new characterizations of the normal cone to the domain of supremum functions and derives novel subdifferential formulas that involve only data functions, simplifying the analysis of convex optimization problems.

Contribution

It provides new characterizations of the normal cone and subdifferential formulas that avoid the normal cone to the domain, advancing subdifferential calculus for supremum functions.

Findings

New formulas for subdifferentials involving active and nonactive functions

Normal cone characterization that excludes the domain's normal cone

New optimality conditions for convex optimization

Abstract

The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper.