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

Author

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  • L. Huerga

    (E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Ciudad Universitaria)

  • B. Jiménez

    (E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Ciudad Universitaria)

  • V. Novo

    (E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Ciudad Universitaria)

  • A. Vílchez

    (E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Ciudad Universitaria)

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.

Suggested Citation

  • L. Huerga & B. Jiménez & V. Novo & A. Vílchez, 2021. "Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 413-436, April.
    • Handle: RePEc:spr:mathme:v:93:y:2021:i:2:d:10.1007_s00186-020-00736-4
    DOI: 10.1007/s00186-020-00736-4
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    References listed on IDEAS

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    1. Yu Han & Nan-jing Huang, 2018. "Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 679-695, June.
    2. Giovanni Crespi & Ivan Ginchev & Matteo Rocca, 2006. "First-order optimality conditions in set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 87-106, February.
    3. Johannes Jahn & Truong Xuan Duc Ha, 2011. "New Order Relations in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 209-236, February.
    4. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
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    Cited by:

    1. L. Huerga & B. Jiménez & V. Novo, 2022. "New Notions of Proper Efficiency in Set Optimization with the Set Criterion," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 878-902, December.

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