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About Error Bounds in Metrizable Topological Vector Spaces

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Abstract

This paper aims to present some sufficient criteria under which a given function between two spaces that are either topological vector spaces whose topologies are generated by metrics or metrizable subsets of some topological vector spaces, satisfies the error bound property. Then, we discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows to calculate the coefficient of the error bound. The applications of this presentation are illustrated by some examples.

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Acknowledgements

The authors would like to thank Professor Alexander Kruger for useful discussions on the subject, as well as the anonymous referees for their valuable comments that have led to improvements in the presentation.

Funding

This research benefited from the support of the FMJH Program PGMO and from the support of EDF.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, Iran

    Malek Abbasi

  2. XLIM UMR-CNRS 7252, Université de Limoges, France and Centre for Informatics and Applied Optimisation, Federation University Vic, 123, Avenue Albert Thomas, 87060 LIMOGES CEDEX, Ballarat, Australia

    Michel Théra

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  1. Malek Abbasi
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  2. Michel Théra
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Correspondence to Malek Abbasi.

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Dedicated to Miguel Ángel Goberna in recognition of his significant contribution to the development of optimization and semi-infinite programming

Research of the second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

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Abbasi, M., Théra, M. About Error Bounds in Metrizable Topological Vector Spaces. Set-Valued Var. Anal 30, 1291–1311 (2022). https://doi.org/10.1007/s11228-022-00643-2

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