On the Dini-Hadamard subdifferential of the difference of two functions

TL;DR

This paper derives a formula for the Dini-Hadamard epsilon-subdifferential of the difference of two functions, providing conditions under which it becomes an equality, and applies these results to optimality conditions in cone-constrained optimization.

Contribution

It introduces a general inclusion formula for the Dini-Hadamard epsilon-subdifferential of function differences and establishes equality conditions under specific geometric assumptions.

Findings

Provides a characterization of the Dini-Hadamard epsilon-subdifferential using sponges.

Establishes a formula for the subdifferential of the difference of two functions.

Applies results to optimality conditions in cone-constrained problems.

Abstract

In this paper we first provide a general formula of inclusion for the Dini-Hadamard epsilon-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard epsilon-subdifferential by means of sponges. The achieved results are employed in the formulation of optimality conditions via the Dini-Hadamard subdifferential for cone-constrained optimization problems having the difference of two functions as objective.