A new scalarization approach and applications in set optimization

Abstract

In this paper, we introduce a new nonlinear scalarization function and apply it to investigate optimality conditions and scalar representations for minimal sets and set optimization problems with the set less order relation. First, we propose a new scalarization function for sets belonging to the power set of a normed space and discuss its properties. Next, we apply such properties to formulate sufficient and necessary optimality conditions for strictly minimal and weakly minimal sets. Finally, under suitable adjustments, we provide nonlinear scalarization functions and investigate scalar representations of strictly efficient solution and weakly efficient solution sets of the reference problems.

Keywords:

• Set optimization problem
• set less order relation
• nonlinear scalarization
• optimality condition
• scalar representation

AMS Classifications:

• 49K35
• 49M37
• 65K05
• 90C30

Acknowledgments

The authors are very grateful to the Editors and anonymous Referees for their helpful remarks and suggestions that helped us significantly improve the paper.

Disclosure statement

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

Additional information

Funding

This is a result of the project supported by The Ministry of Education and Training of Viet Nam [grant number B2023-TCT-05].