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

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.