Variational Convexity of Functions in Banach Spaces

TL;DR

This paper explores the concept of variational convexity for functions in Banach spaces, extending finite-dimensional theories to infinite-dimensional settings using advanced geometric and variational analysis tools.

Contribution

It introduces the study of variational convexity in Banach spaces, a novel extension from finite-dimensional cases, employing advanced functional analysis techniques.

Findings

First characterization of variational convexity in Banach spaces

Development of new analytical tools for infinite-dimensional variational analysis

Potential applications to infinite-dimensional optimization problems

Abstract

This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and applied to continuous optimization problems in finite-dimensional spaces. Variational convexity in infinite-dimensional spaces, which is studied here for the first time, is significantly more involved and requires the usage of powerful tools of geometric functional analysis together with variational analysis and generalized differentiation in Banach spaces.