Six set scalarizations based on the oriented distance: properties and application to set optimization

ABSTRACT

In the framework of normed spaces ordered by a convex cone not necessarily solid, we present six set scalarization functions which are extensions of the oriented distance of Hiriart-Urruty. We investigate their relationships and study some of their properties. Moreover, by means of these functions, we characterize the six set relations given by Kuroiwa. Finally, considering these set relations, we define six set optimization problems with the set criterion of solution and we derive several necessary and sufficient minimality conditions by using the six set scalarization functions above-mentioned. Illustrative examples are also given.

Keywords:

• Set optimization
• scalarization in set optimization
• set order relations
• oriented distance

2010 Mathematics Subject Classifications:

• 06A75
• 49J53
• 90C29

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article

Acknowledgments

The authors are grateful to the anonymous referees for their useful suggestions and remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

B. Jiménez http://orcid.org/0000-0003-1513-8590

V. Novo http://orcid.org/0000-0001-9033-3180

Additional information

Funding

This work, for the first and second author, was partially supported by the Ministerio de Economía y Competitividad (Spain) under the project MTM2015-68103-P (MINECO/FEDER). The second author was also supported by ETSI Industriales, Universidad Nacional de Educación a Distancia (Spain) under Grant 2018-Mat13.