Metric subregularity of the convex subdifferential in Banach spaces

TL;DR

This paper extends the characterization of metric subregularity of convex subdifferentials from Hilbert to Banach spaces, providing new insights into regularity properties and their implications for optimization algorithms.

Contribution

It generalizes existing metric subregularity characterizations to Banach spaces and explores their implications for convergence of proximal algorithms and solution stability.

Findings

Extended metric subregularity characterizations to Banach spaces.

Linked regularity properties to convergence of proximal point algorithm.

Provided new characterizations for solution map properties in parametric equations.

Abstract

In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where the authors extend to Banach spaces the characterization of the strong regularity, we extend as well the characterizations for the metric subregularity and the strong subregularity given in [2] to Banach spaces. We also notice that at least one implication in these characterizations remains valid for the limiting subdifferential without assuming convexity of the function in Asplund spaces. Additionally, we show some direct implications of the characterizations for the convergence of the proximal point algorithm, and we provide some characterizations of the metric subregularity and calmness properties of solution maps to parametric generalized equations.