On second-order variational analysis of variational convexity of prox-regular functions

TL;DR

This paper develops second-order conditions for variational convexity of prox-regular functions using generalized derivatives, removing the need for subdifferential continuity assumptions.

Contribution

It introduces modified generalized derivatives compatible with $f$-attentive convergence to characterize variational convexity without subdifferential continuity.

Findings

Provides second-order characterizations of variational convexity.

Formulates formulas for the exact bounds of variational convexity.

Explores relations between variational strong convexity, tilt stability, and metric regularity.

Abstract

Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mapping. Up to now, such characterizations are only known under the assumptions of prox-regularity and subdifferential continuity and in this paper we discard the latter. To this aim we slightly modify the definitions of the generalized derivatives to be compatible with the $f$-attentive convergence appearing in the definition of subgradients. We formulate our results in terms of both coderivatives and subspace containing derivatives. We also give formulas for the exact bound of variational convexity and study relations between variational strong convexity, tilt-stable local minimizers and strong metric regularity of some truncation of the subgradient mapping.