IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v97y2023i3d10.1007_s00186-023-00818-z.html
   My bibliography  Save this article

Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization

Author

Listed:
  • Fernando García-Castaño

    (University of Alicante)

  • Miguel Ángel Melguizo-Padial

    (University of Alicante)

  • G. Parzanese

    (University of Alicante)

Abstract

We show that under a separation property, a $${{{\mathcal {Q}}}}$$ Q -minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.

Suggested Citation

  • Fernando García-Castaño & Miguel Ángel Melguizo-Padial & G. Parzanese, 2023. "Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(3), pages 367-382, June.
    • Handle: RePEc:spr:mathme:v:97:y:2023:i:3:d:10.1007_s00186-023-00818-z
    DOI: 10.1007/s00186-023-00818-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-023-00818-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/s00186-023-00818-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. X. Y. Zheng, 1997. "Proper Efficiency in Locally Convex Topological Vector Spaces," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 469-486, August.
    2. X. Y. Zheng, 2000. "Scalarization of Henig Proper Efficient Points in a Normed Space," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 233-247, April.
    3. E. K. Makarov & N. N. Rachkovski, 1999. "Unified Representation of Proper Efficiency by Means of Dilating Cones," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 141-165, April.
    4. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, February.
    5. Gabriele Eichfelder & Refail Kasimbeyli, 2014. "Properly optimal elements in vector optimization with variable ordering structures," Journal of Global Optimization, Springer, vol. 60(4), pages 689-712, December.
    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. J. H. Qiu & Y. Hao, 2010. "Scalarization of Henig Properly Efficient Points in Locally Convex Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 71-92, October.
    2. Yichen Lu & Chao Yang & Jun Yang, 2022. "A multi-objective humanitarian pickup and delivery vehicle routing problem with drones," Annals of Operations Research, Springer, vol. 319(1), pages 291-353, December.
    3. Wu, Weitiao & Lin, Yue & Liu, Ronghui & Jin, Wenzhou, 2022. "The multi-depot electric vehicle scheduling problem with power grid characteristics," Transportation Research Part B: Methodological, Elsevier, vol. 155(C), pages 322-347.
    4. Bogdana Stanojević & Milan Stanojević & Sorin Nădăban, 2021. "Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers," Mathematics, MDPI, vol. 9(11), pages 1-16, June.
    5. Stelios Rozakis & Athanasios Kampas, 2022. "An interactive multi-criteria approach to admit new members in international environmental agreements," Operational Research, Springer, vol. 22(4), pages 3461-3487, September.
    6. Manuel V. C. Vieira & Margarida Carvalho, 2023. "Lexicographic optimization for the multi-container loading problem with open dimensions for a shoe manufacturer," 4OR, Springer, vol. 21(3), pages 491-512, September.
    7. Petr Iakovlevitch Ekel & Matheus Pereira Libório & Laura Cozzi Ribeiro & Mateus Alberto Dorna de Oliveira Ferreira & Joel Gomes Pereira Junior, 2024. "Multi-Criteria Decision under Uncertainty as Applied to Resource Allocation and Its Computing Implementation," Mathematics, MDPI, vol. 12(6), pages 1-20, March.
    8. E. K. Makarov & N. N. Rachkovski, 1999. "Unified Representation of Proper Efficiency by Means of Dilating Cones," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 141-165, April.
    9. Yeudiel Lara Moreno & Carlos Ignacio Hernández Castellanos, 2024. "A Hierarchical Approach to a Tri-Objective Portfolio Optimization Problem Considering an ESG Index," Mathematics, MDPI, vol. 12(19), pages 1-16, October.
    10. Bennet Gebken & Sebastian Peitz, 2021. "Inverse multiobjective optimization: Inferring decision criteria from data," Journal of Global Optimization, Springer, vol. 80(1), pages 3-29, May.
    11. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, November.
    12. H. T. H. Diem & P. Q. Khanh & L. T. Tung, 2014. "On Higher-Order Sensitivity Analysis in Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 463-488, August.
    13. Abdelaziz, Fouad Ben & Maddah, Bacel & Flamand, Tülay & Azar, Jimmy, 2024. "Store-Wide space planning balancing impulse and convenience," European Journal of Operational Research, Elsevier, vol. 312(1), pages 211-226.
    14. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    15. Doğan, Ilgın & Lokman, Banu & Köksalan, Murat, 2022. "Representing the nondominated set in multi-objective mixed-integer programs," European Journal of Operational Research, Elsevier, vol. 296(3), pages 804-818.
    16. Brouer, Berit D. & Dirksen, Jakob & Pisinger, David & Plum, Christian E.M. & Vaaben, Bo, 2013. "The Vessel Schedule Recovery Problem (VSRP) – A MIP model for handling disruptions in liner shipping," European Journal of Operational Research, Elsevier, vol. 224(2), pages 362-374.
    17. Lourdes Uribe & Johan M Bogoya & Andrés Vargas & Adriana Lara & Günter Rudolph & Oliver Schütze, 2020. "A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems," Mathematics, MDPI, vol. 8(10), pages 1-29, October.
    18. Mesquita-Cunha, Mariana & Figueira, José Rui & Barbosa-Póvoa, Ana Paula, 2023. "New ϵ−constraint methods for multi-objective integer linear programming: A Pareto front representation approach," European Journal of Operational Research, Elsevier, vol. 306(1), pages 286-307.
    19. Steuer, Ralph E. & Utz, Sebastian, 2023. "Non-contour efficient fronts for identifying most preferred portfolios in sustainability investing," European Journal of Operational Research, Elsevier, vol. 306(2), pages 742-753.
    20. Raeesi, Ramin & Zografos, Konstantinos G., 2022. "Coordinated routing of electric commercial vehicles with intra-route recharging and en-route battery swapping," European Journal of Operational Research, Elsevier, vol. 301(1), pages 82-109.

    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:mathme:v:97:y:2023:i:3:d:10.1007_s00186-023-00818-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.