On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions

TL;DR

This paper introduces a generalized subdifferential concept using conjugation, linking it to directional derivatives and optimality conditions in DC (difference of convex) optimization problems.

Contribution

It develops a new generalized subdifferential framework and applies it to derive necessary optimality conditions for DC problems.

Findings

Established properties of the generalized subdifferential.

Connected subdifferential with conjugate functions and directional derivatives.

Derived necessary conditions for {}-optimality and global optimality in DC problems.

Abstract

This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the {\epsilon}-directional derivative. In addition, we also present necessary conditions for {\epsilon}-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions will be written via this generalized notion of subdifferential studied in the first sections of the paper.