Calculus for directional limiting normal cones and subdifferentials

TL;DR

This paper develops a comprehensive calculus for directional limiting normal cones and subdifferentials in finite dimensions, enabling advanced variational analysis with weak qualification conditions.

Contribution

It introduces a unified calculus for directional limiting notions that extends standard differential calculus with minimal qualification assumptions.

Findings

Enhanced tools for stability analysis in variational problems

New calculus rules applicable under weak qualification conditions

Illustrative applications demonstrating practical utility

Abstract

The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.