New Characterizations of Nonsmooth Convex Functions via Generalized Derivatives

TL;DR

This paper introduces new characterizations of nonsmooth convex functions using advanced generalized derivatives, providing necessary and sufficient conditions for convexity in nonsmooth optimization.

Contribution

It develops novel second-order generalized derivative characterizations of convexity for nonsmooth functions, advancing theoretical understanding in optimization.

Findings

New second-order derivative-based convexity criteria

Necessary and sufficient conditions for nonsmooth convexity

Enhanced tools for analyzing nonsmooth functions

Abstract

This paper studies the convexity properties of nonsmooth extended-real-valued weakly convex functions, a class of functions that is central to modern optimization and its applications. We establish new characterizations of convexity using second-order generalized derivative tools, including subgradient graphical derivatives, second subderivatives, and second-order subdifferentials. These tools allow us to derive necessary and sufficient conditions for convexity in the nonsmooth framework.