Variational Convexity of Functions and Variational Sufficiency in Optimization

TL;DR

This paper explores the concept of variational convexity in functions, providing new characterizations through second-order subdifferentials and applying these insights to optimization problems and nonlinear programming.

Contribution

It introduces new characterizations of variational convexity and strong convexity via second-order subdifferentials, linking these properties to optimization and stability analysis.

Findings

Variational convexity is equivalent to convexity of the Moreau envelope.

New characterizations of variational properties via second-order subdifferentials.

Applications to nonlinear programming and optimization stability.

Abstract

The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these variational properties of an extended-real-valued function are equivalent to, respectively, the conventional (local) convexity and strong convexity of its Moreau envelope. Then we derive new characterizations of both variational convexity and variational strong convexity of general functions via their second-order subdifferentials (generalized Hessians), which are coderivatives of subgradient mappings. We also study relationships of these notions with local minimizers and tilt-stable local minimizers. The obtained results are used for characterizing related notions of variational and strong variational sufficiency in composite optimization with applications to nonlinear programming.