Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis

TL;DR

This paper advances second-order variational analysis by introducing new primal-space concepts of generalized twice differentiability and quadratic bundles for prox-regular functions, with potential applications in optimization.

Contribution

It develops new techniques to characterize generalized twice differentiability and proves the nonemptiness of quadratic bundles for prox-regular functions, expanding the theoretical framework.

Findings

Quadratic bundles of prox-regular functions are nonempty.

New primal-space characterizations of generalized twice differentiability.

Techniques applicable to broad classes of nonsmooth functions.

Abstract

In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.