On the sum rules and maximality of generalized subdifferentials

Abstract

Some properties of σ-convex functions are studied. Then some relations between the notions of σ-subdifferentials and σ-convex functions are established. Also, we present some results regarding the sum formula for the σ-subdifferential. Moreover, we obtain some particular relationships between the σ-subdifferential and the $(\sigma ,y)$-conjugate. Finally, by imposing some assumptions on f and σ, the maximal $2\sigma$-monotonicity of σ-subdifferential is studied. Indeed, the maximality of the usual Fenchel subdifferentials of lower semicontinuous convex functions follows from our result.

Keywords:

• Convex functions
• generalized convex functions
• conjugates
• generalized subdifferentials

Mathematics Subject Classifications:

• 49J52
• 90C26

Acknowledgments

The authors express their gratitude to Professor Nicolas Hadjisavvas for his many insightful comments and suggestions. The authors would like to thank the anonymous Referees for their valuable comments on the manuscript for improving the document. Besides, we want to thank and appreciate the anonymous Associate Editor for his∖her comments, as well as Professor Christiane Tammer, Editor in chief of Optimization.

Disclosure statement

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

Additional information

Funding

This work is based upon research funded by Iran National Science Foundation (INSF) [project number 4004526].