Abstract convex approximations of nonsmooth functions

TL;DR

This paper introduces a unified framework using abstract convexity to analyze nonsmooth functions, generalizing classical concepts and providing new tools for nonsmooth optimization and optimality conditions.

Contribution

It develops the theory of abstract codifferentiability and convex approximations, unifying and extending classical nonsmooth analysis concepts with applications to optimization.

Findings

Many classical nonsmooth concepts are special cases of abstract notions.

Abstract convex approximations facilitate the derivation of optimality conditions.

The framework simplifies analysis of nonsmooth, nonconvex optimization problems.

Abstract

In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.