IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v199y2023i2d10.1007_s10957-023-02307-z.html
   My bibliography  Save this article

Conjugate Duality in Set Optimization via Nonlinear Scalarization

Author

Listed:
  • Yousuke Araya

    (Akita Prefectural University)

Abstract

Two approaches are applied to the set-valued optimization problem. The following problems have been examined by Corley, Luc and their colleagues: Take the union of all objective values and then search for (weakly, properly, etc.) minimal points in this union with respect to the vector ordering. This approach is called the vector approach to set optimization. The concept shifted when the set relations were popularized by Kuroiwa–Tanaka–Ha at the end of the twentieth century. They introduced six types of set relations on the power set of topological vector space using a convex ordering cone C with nonempty interior. Therefore, this approach is called the set relation approach to set optimization. For a given vector optimization problem, several approaches are applied to construct a dual problem. A difficulty lies in the fact that the minimal point in vector optimization problem is not necessarily a singleton, though it becomes a subset of the image space in general. In this paper, we first present new definitions of set-valued conjugate map based on comparison of sets (the set relation approach) followed by introducing some types of weak duality theorems. We also show convexity and continuity properties of conjugate relations. Lastly, we present some types of strong duality theorems using nonlinear scalarizing technique for set that is generalizations of Gerstewitz’s scalarizing function for the vector-valued case.

Suggested Citation

  • Yousuke Araya, 2023. "Conjugate Duality in Set Optimization via Nonlinear Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 466-498, November.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:2:d:10.1007_s10957-023-02307-z
    DOI: 10.1007/s10957-023-02307-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-023-02307-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-023-02307-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Elisa Caprari & Lorenzo Cerboni Baiardi & Elena Molho, 2022. "Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach," Journal of Global Optimization, Springer, vol. 84(2), pages 295-320, October.
    2. Hidefumi Kawasaki, 1982. "A Duality Theorem in Multiobjective Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 95-110, February.
    3. Takashi Maeda, 2012. "On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 263-279, May.
    4. Gemayqzel Bouza & Ernest Quintana & Christiane Tammer, 2021. "A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 711-743, September.
    5. Hidefumi Kawasaki, 1981. "Conjugate Relations and Weak Subdifferentials of Relations," Mathematics of Operations Research, INFORMS, vol. 6(4), pages 593-607, November.
    6. Truong Quang Bao & Christiane Tammer, 2019. "Scalarization Functionals with Uniform Level Sets in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 310-335, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. C. R. Chen & S. J. Li, 2009. "Different Conjugate Dual Problems in Vector Optimization and Their Relations," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 443-461, March.
    2. Eichfelder, Gabriele & Quintana, Ernest, 2024. "Set-based robust optimization of uncertain multiobjective problems via epigraphical reformulations," European Journal of Operational Research, Elsevier, vol. 313(3), pages 871-882.
    3. A. Y. Azimov, 2008. "Duality for Set-Valued Multiobjective Optimization Problems, Part 1: Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 61-74, April.
    4. B. Jiménez & V. Novo & A. Vílchez, 2020. "Characterization of set relations through extensions of the oriented distance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 89-115, February.
    5. Wenyan Han & Guolin Yu, 2024. "Optimality and error bound for set optimization with application to uncertain multi-objective programming," Journal of Global Optimization, Springer, vol. 88(4), pages 979-998, April.
    6. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 931-946, December.
    7. Chuang-Liang Zhang & Nan-jing Huang, 2021. "Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 894-914, September.
    8. Masamichi Kon, 2020. "A scalarization method for fuzzy set optimization problems," Fuzzy Optimization and Decision Making, Springer, vol. 19(2), pages 135-152, June.
    9. Truong Quang Bao & Christiane Tammer, 2019. "Scalarization Functionals with Uniform Level Sets in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 310-335, July.
    10. Lam Quoc Anh & Vo Thi Mong Thuy & Xiaopeng Zhao, 2025. "Qualitative Properties of Robust Benson Efficient Solutions of Uncertain Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 205(1), pages 1-37, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:199:y:2023:i:2:d:10.1007_s10957-023-02307-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.