Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles

TL;DR

This paper characterizes variational s-convexity and tilt stability for prox-regular functions using quadratic bundles, extending second-order analysis without relying on subdifferential continuity.

Contribution

It introduces new characterizations of variational convexity and tilt stability via quadratic bundles, revising their definitions for primal-dual second-order derivatives.

Findings

Characterizations established without subdifferential continuity

Revised quadratic bundles for effective point-based form

Connection between generalized and classical twice differentiability

Abstract

In this paper, we establish characterizations of variational $s$-convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order derivatives recently introduced by Rockafellar. Deriving such characterizations in the effective pointbased form requires a certain revision of quadratic bundles investigated below. Our device is based on the notion of generalized twice differentiability and its novel characterization via classical twice differentiability of the associated Moreau envelopes combined with various limiting procedures for functions and sets.