A unifying theory of exactness of linear penalty functions

Abstract

In this article, we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature. We discuss several approaches to the study of both locally and globally exact linear penalty functions, and obtain various necessary and sufficient conditions for the exactness of a linear penalty function. We pay more attention than usual to necessary conditions, which allows us to deeply understand the exact penalty technique.

Keywords:

• Penalty function
• exact penalization
• error bounds
• calmness
• perturbation function
• Palais–Smale condition

AMS Subject Classifications:

• 65K05
• 90C30

Acknowledgements

The author wishes to express sincere gratitude and thanks to the late professor V.F. Demyanov. His advice, encouragement, as well as, inspiring lectures had a great influence on the author’s decision of becoming a professional mathematician. In addition, the idea of writing this article arose from the author’s desire to refine some of the very interesting results of professor V.F. Demyanov on exact penalty functions. Also, the author thanks the anonymous referee for valuable comments that helped to improve the quality of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author was supported by the Russian Foundation for Basic Research [grant number 14-01-31521]; the Saint Petersburg State University [grant number 9.38.205.2014].