Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

TL;DR

This paper explores the relationships between codifferentials, coexhausters, and DC functions, establishing their equivalences and extending optimality conditions to nonhomogeneous approximations and directional derivatives.

Contribution

It demonstrates the equivalence of boundedness and optimality conditions across codifferentials and coexhausters, and links nonhomogeneous approximations with directional derivatives.

Findings

Proves the equivalence of boundedness conditions for codifferentials and coexhausters.

Extends optimality conditions to nonhomogeneous approximations.

Establishes connections between nonhomogeneous approximations and directional derivatives.

Abstract

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.