IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v193y2022i1d10.1007_s10957-021-01976-y.html
   My bibliography  Save this article

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

Author

Listed:
  • Christian Günther

    (Martin Luther University Halle-Wittenberg)

  • Bahareh Khazayel

    (Martin Luther University Halle-Wittenberg)

  • Christiane Tammer

    (Martin Luther University Halle-Wittenberg)

Abstract

In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.

Suggested Citation

  • Christian Günther & Bahareh Khazayel & Christiane Tammer, 2022. "Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 408-442, June.
  • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01976-y
    DOI: 10.1007/s10957-021-01976-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01976-y
    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/s10957-021-01976-y?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. Adan, M. & Novo, V., 2003. "Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness," European Journal of Operational Research, Elsevier, vol. 149(3), pages 641-653, September.
    2. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    3. C. Gutiérrez & L. Huerga & V. Novo & C. Tammer, 2016. "Duality related to approximate proper solutions of vector optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 117-139, January.
    4. Ovidiu Bagdasar & Nicolae Popovici, 2018. "Unifying local–global type properties in vector optimization," Journal of Global Optimization, Springer, vol. 72(2), pages 155-179, October.
    5. Franco Giannessi, 2018. "Some Perspectives on Vector Optimization via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 906-912, June.
    6. M. Adán & V. Novo, 2005. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 751-751, March.
    7. Vicente Novo & Constantin Zălinescu, 2021. "On Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 277-290, January.
    8. 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. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2018. "Approximate solutions of vector optimization problems via improvement sets in real linear spaces," Journal of Global Optimization, Springer, vol. 70(4), pages 875-901, April.
    2. M. Chinaie & F. Fakhar & M. Fakhar & H. R. Hajisharifi, 2019. "Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem," Journal of Global Optimization, Springer, vol. 75(1), pages 131-141, September.
    3. Z. A. Zhou & J. W. Peng, 2012. "Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 830-841, September.
    4. Zhi-Ang Zhou & Xin-Min Yang, 2014. "Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 680-693, August.
    5. Elham Kiyani & Majid Soleimani-damaneh, 2014. "Algebraic Interior and Separation on Linear Vector Spaces: Some Comments," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 994-998, June.
    6. Vicente Novo & Constantin Zălinescu, 2021. "On Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 277-290, January.
    7. Ovidiu Bagdasar & Nicolae Popovici, 2018. "Unifying local–global type properties in vector optimization," Journal of Global Optimization, Springer, vol. 72(2), pages 155-179, October.
    8. 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.
    9. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "Robustness Characterizations for Uncertain Optimization Problems via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 459-479, August.
    10. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    11. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2020. "An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces," Mathematics, MDPI, vol. 8(1), pages 1-17, January.
    12. Shokouh Shahbeyk & Majid Soleimani-damaneh & Refail Kasimbeyli, 2018. "Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure," Journal of Global Optimization, Springer, vol. 71(2), pages 383-405, June.
    13. Yang-Dong Xu & Cheng-Ling Zhou & Sheng-Kun Zhu, 2021. "Image Space Analysis for Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 311-343, October.
    14. Davide LA TORRE & Nicolae POPOVICI & Matteo ROCCA, 2008. "Scalar characterization of explicitly quasiconvex set-valued maps," Departmental Working Papers 2008-01, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    15. 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.
    16. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2019. "Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 105-121, October.
    17. Truong Q. Bao & Lidia Huerga & Bienvenido Jiménez & Vicente Novo, 2020. "Necessary Conditions for Nondominated Solutions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 826-842, September.
    18. Shengkun Zhu, 2018. "Constrained Extremum Problems, Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 770-787, June.

    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:193:y:2022:i:1:d:10.1007_s10957-021-01976-y. 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.