A note on the uniformity of strong subregularity around the reference point

TL;DR

This paper studies the stability and uniformity of strong metric subregularity in Banach spaces under Lipschitz perturbations, providing theoretical insights for robust variational analysis and parametric optimization.

Contribution

It establishes the preservation of strong subregularity under Lipschitz perturbations and demonstrates its uniformity over compact sets in Banach spaces.

Findings

Strong subregularity is stable under small Lipschitz perturbations.

Uniformity of strong subregularity holds over compact sets.

Application to parametric inclusion problems shows practical relevance.

Abstract

This paper investigates strong metric subregularity around a reference point as introduced by H. Gfrerer and J. V. Outrata. In the setting of Banach spaces, we analyse its stability under Lipschitz continuous perturbations and establish its uniformity over compact sets. Our results ensure that the property is preserved under small Lipschitz perturbations, which is crucial for maintaining robustness in variational analysis. Furthermore, we apply the developed theory to parametric inclusion problems. The analysis demonstrates that the uniformity of strong metric subregularity provides a theoretical foundation for addressing stability issues in parametrized optimization and control applications.