Error bounds revisited

TL;DR

This paper introduces a comprehensive framework for understanding error bounds in optimization, applicable to various settings and function structures, by unifying primal and dual conditions across different spaces.

Contribution

It presents a unifying theoretical framework for error bounds that encompasses linear, nonlinear, local, and global cases without requiring specific function structures.

Findings

Unified error bound conditions for diverse settings

Inclusion of primal and dual error bounds in a single framework

Application to metric, normed, Banach, and Asplund spaces

Abstract

We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces.