Error bounds and metric subregularity

Abstract

Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

Keywords:

• error bounds
• slope
• metric regularity
• metric subregularity
• calmness

AMS Subject Classifications:

• 49J52
• 49J53
• 58C06
• 47H04
• 54C60

Acknowledgements

The author is very happy to submit his article to the special issue dedicated to the 40th Anniversary of the journal where his first article in English was published in 1988. The author is grateful to the then editor-in-chief, Professor Karl-Heinz Elster for his support and keeps in his archive a postcard signed by Professor Elster informing the author about the acceptance of that article.

Notes

Dedicated to the 40th Anniversary of the journal; its founder and former editor-in-chief, Professor Karl-Heinz Elster; and Professor Alfred Göpfert, an editorial board member since 1988 in celebration of his 80th birthday.

This work was supported by the Australian Research Council [grant number DP110102011].