The gradient's limit of a definable family of functions admits a variational stratification

Abstract

It is well-known that the convergence of a family of smooth functions does not imply the convergence of its gradients. In this work, we show that if the family is definable in an o-minimal structure (for instance semialgebraic, subanalytic, or any composition of the previous with exp, log), then the gradient's limit admits a variational stratification and, under mild assumptions, is a conservative set-valued field in the sense introduced by Bolte and Pauwels. Immediate implications of this result on convergence guarantees of smoothing methods are discussed. The result is established in a general form, where the functions in the original family might be non Lipschitz continuous, be vector-valued and the gradients are replaced by their Clarke Jacobians or an arbitrary mapping satisfying a definable variational stratification. In passing, we investigate various stability properties of definable variational stratifications which might be of independent interest.