A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by l-type and u-type preorder relations

Abstract

In this paper, we study different classes of generalized convex/quasiconvex set-valued maps, defined by means of the l-type and u-type preorder relations, currently used in set-valued optimization. In particular, we identify those classes of set-valued maps for which it is possible to extend the classical characterization of convex real-valued functions by quasiconvexity of their affine perturbations.

Keywords:

• Set-valued optimization
• generalized convexity
• affine perturbation

Acknowledgements

The authors thank the anonymous referees for their constructive comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Daishi Kuroiwa’s research was supported by a grant of JSPS KAKENHI [grant number 16K05274]. Nicolae Popovici’s research was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI [project number PN-III-P4-ID-PCE-2016-0190], within PNCDI III.