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

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.