The Minimal Robust Core of Abstract Subdifferentials

Abstract

This paper introduces the metric-dependent core subdifferential, a local robust affine-support construction for extended-real functions on metrizable topological vector spaces. A core subgradient is a continuous linear slope for which the affine lower support holds on metric balls with an error negligible relative to the supporting radius, along arbitrarily fine admissible scales. The main minimality results show that these slopes are unavoidable: on complete metrizable spaces they belong to the graph closure of any subdifferential satisfying a local-minimum principle together with a mild stability condition under metric-distance perturbations, and on Banach spaces they belong to the nearby graph closure of any abstract subdifferential satisfying the usual fuzzy minimum principle. For the norm metric, the construction contains the Fréchet subdifferential, coincides with the Fenchel subdifferential on convex functions, is contained in the limiting subdifferential whenever the relevant Fréchet fuzzy calculus is available, and is contained in the Clarke--Rockafellar subdifferential for lower semicontinuous functions on Banach spaces. The paper also records metric-dependence phenomena, strict comparison examples, constrained optimality and variational-inequality conditions, a scale-slope/error-bound characterization, and the relation with Goldstein-type stationarity used in finite-dimensional nonsmooth optimization.