Variational and Strong Variational Convexity in Infinite-Dimensional Variational Analysis

TL;DR

This paper extends the concepts of variational and strong variational convexity to infinite-dimensional spaces, providing new characterizations and exploring their relationship with tilt stability in Banach spaces.

Contribution

It develops novel techniques for characterizing variational convexity notions in infinite-dimensional settings, advancing understanding beyond finite-dimensional analysis.

Findings

New characterizations of variational and strong variational convexity in Banach spaces

Established relationships between strong variational convexity and tilt stability

Extended finite-dimensional concepts to infinite-dimensional variational analysis

Abstract

This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by Rockafellar, the importance of them has been already recognized and documented in finite-dimensional variational analysis and optimization. Here we address general infinite-dimensional settings and derive comprehensive characterizations of both variational and strong variational convexity notions by developing novel techniques, which are essentially different from finite-dimensional counterparts. As a consequence of the obtained characterizations, we establish new quantitative and qualitative relationships between strong variational convexity and tilt stability of local minimizers in appropriate frameworks of Banach spaces.