Explicit data-dependent characterizations of the subdifferential of convex pointwise suprema and optimality conditions

TL;DR

This paper provides explicit, data-dependent characterizations of the subdifferential for convex function suprema, leading to precise optimality conditions in convex optimization.

Contribution

It introduces a novel approach that directly relates subdifferentials to data functions, simplifying the derivation of optimality conditions without geometric constructions.

Findings

Explicit subdifferential formulas in terms of data functions

Sharp KKT and Fritz-John optimality conditions derived

Active and non-active functions contribute equally in the characterization

Abstract

We establish explicit data-dependent and symmetric characterizations of the subdifferential of the supremum of convex functions, formulated directly in terms of the underlying data functions. In our approach, both active and non-active functions contribute equally through their subdifferentials, thereby avoiding the need for additional geometric constructions, such as the domain of the supremum, that arise in previous developments. Applications to infinite convex optimization yield sharp Karush-Kuhn-Tucker and Fritz-John optimality conditions, expressed exclusively in terms of the objective and constraint functions and clearly distinguishing the roles of (almost) active and non-active constraints.