Perturbation Analysis of Error Bounds for Convex Functions on Banach Spaces

TL;DR

This paper investigates the stability of local and global error bounds for convex functions on Banach spaces, providing new estimates using directional derivatives without dual space reliance.

Contribution

It offers novel criteria linking error bound stability to directional derivatives, enhancing understanding of convex function behavior in Banach spaces.

Findings

Stability of local error bounds is tied to the directional derivative's minimum being non-zero.

Global error bound stability relates to the directional derivatives being bounded away from zero.

Provides precise estimates of error bound moduli using directional derivatives.

Abstract

This paper focuses on the stability of both local and global error bounds for a proper lower semicontinuous convex function defined on a Banach space. Without relying on any dual space information, we first provide precise estimates of error bound moduli using directional derivatives. For a given proper lower semicontinuous convex function on a Banach space, we prove that the stability of local error bounds under small perturbations is equivalent to the directional derivative at a reference point having a non-zero minimum over the unit sphere. Additionally, the stability of global error bounds is shown to be equivalent to the infimum of the directional derivatives, at all points on the boundary of the solution set, being bounded away from zero over some neighborhood of the unit sphere.