Continuity of a scalarization in vector optimization with variable ordering structures and application to convergence of minimal solutions

Abstract

We consider a scalarization function, which was introduced by Eichfelder [Variable ordering structures in vector optimization. Berlin: Springer-Verlag; 2014 (Series in vector optimization)], based on the oriented distance of Hiriart–Urruty with respect to a general variable ordering structure (VOS). We first study the continuity of the composition of a set-valued map with the oriented distance. Then, using the obtained results, we study the continuity of the scalarization function by extending some concepts of continuity for cone-valued maps. As an application, convergence in the sense of Painlevé–Kuratowski of sets of weak minimal solutions is provided, with the vector criterion and a VOS. Illustrative examples are also given.

Keywords:

• Scalarization in optimization
• continuity
• variable ordering structure
• oriented distance
• Painlevé–Kuratowski convergence

AMS CLASSIFICATIONs:

• 06A75
• 49J53
• 90C29
• 49K40

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 author(s).

Additional information

Funding

This work was partially supported by Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación (Spain) under project PID2020-112491GB-I00 / AEI / 10.13039/501100011033. The research of the first three authors is also supported by E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), Spain, under grant 2021-MAT09.