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

TL;DR

The paper addresses uniform stability of strong metric subregularity around a reference point $(x_bar, y_bar)$ in Banach spaces under perturbations. It develops perturbation results for sums of single-valued and set-valued mappings, showing that when a perturbation $g$ has modulus $\mu$ with $\kappa\mu < 1$, the sum $g+F$ is strongly metrically subregular with constant $\kappa_prime = \kappa/(1 - \kappa \mu)$ locally, and extends to a uniform version on compact sets via a finite subcover. These uniform results yield common constants $(\kappa, a, b)$ applicable to all points in a compact set and are extended to parametric settings and continuous solution trajectories. The findings provide a solid theoretical foundation for robust algorithms in parametric optimization and control that rely on stable generalized equations under perturbations.

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.

Theorems & Definitions (15)

• Definition 2.1
• Definition 2.2
• Remark 2.1
• Theorem 2.1
• Example 2.1
• Proposition 2.1
• Proof 1
• Theorem 3.1
• Proof 2
• Theorem 3.2