Abstract
In the setting of normed spaces ordered by a convex cone not necessarily solid, we use six set scalarization functions, which are extensions of the oriented distance of Hiriart-Urruty, and we discuss convexity and continuity properties of their composition with two set-valued maps. Furthermore, as an application, we derive a multiplier rule for weak minimal solutions of a convex set optimization problem, with respect to the lower set less preorder of Kuroiwa. Some illustrative examples are also given.
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Acknowledgements
The authors are grateful to the anonymous referees for their useful suggestions and remarks, which have contributed to get a meaningful improvement of the paper. This work, for the first three authors, was partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE), and also by ETSI Industriales, Universidad Nacional de Educación a Distancia (Spain) under Grant 2020-Mat09.
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Huerga, L., Jiménez, B., Novo, V. et al. Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization. Math Meth Oper Res 93, 413–436 (2021). https://doi.org/10.1007/s00186-020-00736-4
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DOI: https://doi.org/10.1007/s00186-020-00736-4
Keywords
- Oriented distance
- Set optimization
- Set order relations
- Scalarization in set optimization
- Convexity
- Continuity
- Lagrange multipliers