On new generalized differentials with respect to a set and their applications

TL;DR

This paper introduces new generalized differentials based on normal cones with respect to a set, providing tools for analyzing set-related properties and optimality conditions in variational analysis.

Contribution

It develops the concepts of limiting coderivative and subdifferential with respect to a set, extending existing theories and deriving conditions for the Aubin property and optimality.

Findings

Characterization of normal cones with respect to a set

Necessary and sufficient conditions for the Aubin property

Optimality criteria for local solutions in set-based problems

Abstract

The notions and certain fundamental characteristics of the proximal and limiting normal cones with respect to a set are first presented in this paper. We present the ideas of the limiting coderivative and subdifferential with respect to a set of multifunctions and singleton mappings, respectively, based on these normal cones. The necessary and sufficient conditions for the Aubin property with respect to a set of multifunctions are then described by using the limiting coderivative with respect to a set. As a result of the limiting subdifferential with respect to a set, we offer the requisite optimality criteria for local solutions to optimization problems. In addition, we also provide examples to demonstrate the outcomes.