On e-convex functions and e-subdifferentials in locally convex spaces

Abstract

We introduce and study the notion of e-subdifferential for an e-convex function in locally convex spaces. Some relationships between the e-subdifferential and the Clarke–Rockafellar subdifferential are established. Besides, the e-convex functions are characterized. Moreover, some results regarding the locally Lipschitz properties of e-convex functions are obtained.

Keywords:

• Generalized convex functions
• Lipschitz property
• generalized subdifferentials

2010 Mathematics Subject Classifications:

• 47H05
• 49J53
• 46B10
• 26A27
• 26A48

Acknowledgements

The author expresses his gratitude to Professors Constantin Zalinescu and Nicolas Hadjisavvas for their many insightful comments, suggestions, and discussions, as well as Professor Christiane Tammer, Editor in Chief of Optimization. Also, I would like to thank the two referees for their valuable comments on the manuscript and their suggestions and corrections for improving the document.

Disclosure statement

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