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Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization

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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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Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Ciudad Universitaria, c/ Juan del Rosal 12, Madrid, 28040, Spain

    L. Huerga, B. Jiménez, V. Novo & A. Vílchez

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  1. L. Huerga
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  2. B. Jiménez
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  3. V. Novo
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  4. A. Vílchez
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Correspondence to V. Novo.

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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

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