Error bounds revisited

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.

Keywords:

• Error bound
• slope
• subdifferential
• subregularity
• semi-infinite programming

AMS CLASSIFICATIONS:

• 49J52
• 49J53
• 49K40
• 90C30
• 90C46

Acknowledgments

We would like to thank the referees for their constructive comments. We are also very grateful to the following colleagues, who provided feedback on the preprint of the paper and made many helpful suggestions: Jean-Noël Corvellec, Joydeep Dutta, Marco López, Ngai Huynh Van, Michel Théra and Jane Ye.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie Grant Agreement No. 823731 CONMECH, and Conicyt REDES program 180032.