Abstract
In the framework of normed spaces ordered by a convex cone not necessarily solid, we consider two set scalarization functions of type sup-inf, which are extensions of the oriented distance of Hiriart-Urruty. We investigate some of their properties and, moreover, we use these functions to characterize the lower and upper set less preorders of Kuroiwa and the strict lower and strict upper set relations. Finally, we apply the obtained results to characterize several concepts of minimal solution to a set optimization problem defined by a set-valued map. Minimal and weak minimal solutions with respect to the lower and upper set less relations are between the concepts considered. Illustrative examples are also given.
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The authors are grateful to the anonymous referees for their useful suggestions and remarks.
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This work, for the first and second author, was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2015-68103-P (MINECO/FEDER). Second author was also supported by ETSI Industriales, Universidad Nacional de Educación a Distancia (Spain) under Grant 2018-Mat13.
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Jiménez, B., Novo, V. & Vílchez, A. Characterization of set relations through extensions of the oriented distance. Math Meth Oper Res 91, 89–115 (2020). https://doi.org/10.1007/s00186-019-00661-1
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DOI: https://doi.org/10.1007/s00186-019-00661-1