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Scalarization in set optimization with solid and nonsolid ordering cones

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Abstract

This paper focuses on characterizations via scalarization of several kinds of minimal solutions of set-valued optimization problems, where the objective values are compared through relations between sets (set optimization). For this aim we follow an axiomatic approach based on general order representation and order preservation properties, which works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then, following this approach, we study a recent Gerstewitz scalarization mapping for set-valued optimization problems with \(K\)-proper sets and a solid ordering cone \(K\). In particular we show a dual minimax reformulation of this scalarization. Moreover, in the setting of normed spaces ordered by non necessarily solid ordering cones, we introduce a new scalarization functional based on the so-called oriented distance. Using these scalarization mappings, we obtain necessary and sufficient optimality conditions in set optimization. Finally, whenever the ordering cone is solid, by considering suitable generalized Chebyshev norms with appropriate parameters, we show that the three scalarizations studied in the present work are coincident.

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Acknowledgments

The authors are grateful to Professor Vicente Novo and to the anonymous referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011 , Valladolid, Spain

    C. Gutiérrez

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

    B. Jiménez

  3. Dipartimento di Discipline Matematiche, Finanza Matematica ed Econometria, Università Cattolica del Sacro Cuore, via Necchi 9, 20123 , Milan, Italy

    E. Miglierina

  4. Dipartimento di Scienze Economiche e Aziendali, Università degli Studi di Pavia, via S. Felice 5, 27100 , Pavia, Italy

    E. Molho

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  1. C. Gutiérrez
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  2. B. Jiménez
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  3. E. Miglierina
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  4. E. Molho
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Corresponding author

Correspondence to C. Gutiérrez.

Additional information

This research was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The fourth author was also partially supported by MIUR PRIN MISURA Project, 2013–2015, Italy.

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Gutiérrez, C., Jiménez, B., Miglierina, E. et al. Scalarization in set optimization with solid and nonsolid ordering cones. J Glob Optim 61, 525–552 (2015). https://doi.org/10.1007/s10898-014-0179-x

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  • DOI: https://doi.org/10.1007/s10898-014-0179-x

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