A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings

TL;DR

This paper presents a unified framework that bridges convex and nonconvex generalized differentiation theories, enhancing understanding of nonsmooth functions and set-valued mappings.

Contribution

It introduces a novel unified approach that integrates convex and nonconvex generalized differentiation theories using Mordukhovich's concepts.

Findings

Provides a comprehensive theoretical framework for generalized differentiation.

Clarifies the relationship between convex and nonconvex differentiation theories.

Offers new tools for analyzing nonsmooth and set-valued functions.

Abstract

In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the pioneer works by Clarke considering generalized differentiation theory of Lipschitz continuous functions. Although Clarke generalized differentiation theory is applicable for nonconvex functions, convexity still plays a crucial role in Clarke subdifferential calculus. In the mid 1970's, Mordukhovich developed another generalized differentiation theory for nonconvex functions and set-valued mappings in which the "umbilical cord with convexity" no longer exists. The primary goal of this paper is to present a unified approach and shed new light on convex and Clarke generalized differentiation theories using the concepts and techniques from Mordukhovich's developments.