IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v151y2011i2d10.1007_s10957-011-9928-x.html
   My bibliography  Save this article

Optimal Elements in Vector Optimization with a Variable Ordering Structure

Author

Listed:
  • Gabriele Eichfelder

    (University of Erlangen-Nuremberg)

Abstract

Optimality concepts for vector optimization problems with a variable ordering structure are examined. These considerations are motivated by an application in medical image registration, where the preferences vary depending on the element in the image space. The idea of variable ordering structures was first introduced by Yu (J. Opt. Theory Appl. 14:319–377, [1974]) in terms of domination structures. Variable ordering structures mean that there is a set-valued map with cone values that associates to each element an ordering. A candidate element is called a nondominated element iff it is not dominated by other reference elements w.r.t. their corresponding ordering. In addition to nondominated elements, another notion of optimal elements, called minimal elements, has also been discussed. For that notion, only the ordering of the candidate element itself is considered. This paper shows that these two different optimality concepts are connected by duality properties. Characterizations and existence results of the above two solution concepts are also given.

Suggested Citation

  • Gabriele Eichfelder, 2011. "Optimal Elements in Vector Optimization with a Variable Ordering Structure," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 217-240, November.
    • Handle: RePEc:spr:joptap:v:151:y:2011:i:2:d:10.1007_s10957-011-9928-x
    DOI: 10.1007/s10957-011-9928-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-011-9928-x
    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/s10957-011-9928-x?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. G. Y. Chen & X. Q. Yang, 2002. "Characterizations of Variable Domination Structures via Nonlinear Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 97-110, January.
    2. Karasakal, Esra Koktener & Michalowski, Wojtek, 2003. "Incorporating wealth information into a multiple criteria decision making model," European Journal of Operational Research, Elsevier, vol. 150(1), pages 204-219, October.
    3. Huang, N.J. & Yang, X.Q. & Chan, W.K., 2007. "Vector complementarity problems with a variable ordering relation," European Journal of Operational Research, Elsevier, vol. 176(1), pages 15-26, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Glaydston Carvalho Bento & Gemayqzel Bouza Allende & Yuri Rafael Leite Pereira, 2018. "A Newton-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 201-221, April.
    2. Christian Sommer, 2014. "Geometrical and Topological Properties of a Parameterized Binary Relation in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 815-840, December.
    3. 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.
    4. Behnam Soleimani, 2014. "Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 605-632, August.
    5. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 947-963, December.
    6. 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.
    7. J. Y. Bello Cruz & G. Bouza Allende, 2014. "A Steepest Descent-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 371-391, August.
    8. T. Q. Bao & B. S. Mordukhovich & A. Soubeyran, 2015. "Variational Analysis in Psychological Modeling," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 290-315, January.
    9. Bettina Zargini, 2022. "Multiobjective Location Problems with Variable Domination Structures and an Application to Select a New Hub Airport," Logistics, MDPI, vol. 6(2), pages 1-13, March.
    10. Elena-Andreea Florea, 2018. "Vector Optimization Problems with Generalized Functional Constraints in Variable Ordering Structure Setting," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 94-118, July.
    11. Truong Q. Bao & Boris S. Mordukhovich, 2014. "Necessary Nondomination Conditions in Set and Vector Optimization with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 350-370, August.
    12. 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.

    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. Behnam Soleimani, 2014. "Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 605-632, August.
    2. Elena-Andreea Florea, 2018. "Vector Optimization Problems with Generalized Functional Constraints in Variable Ordering Structure Setting," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 94-118, July.
    3. Jia-Wei Chen & Zhongping Wan & Yeol Cho, 2013. "Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 33-64, February.
    4. Jinxia Cen & Tahar Haddad & Van Thien Nguyen & Shengda Zeng, 2022. "Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems," Journal of Global Optimization, Springer, vol. 84(3), pages 783-805, November.
    5. T. Q. Bao & B. S. Mordukhovich & A. Soubeyran, 2015. "Variational Analysis in Psychological Modeling," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 290-315, January.
    6. Engau, Alexander, 2009. "Tradeoff-based decomposition and decision-making in multiobjective programming," European Journal of Operational Research, Elsevier, vol. 199(3), pages 883-891, December.
    7. G Özerol & E Karasakal, 2008. "Interactive outranking approaches for multicriteria decision-making problems with imprecise information," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(9), pages 1253-1268, September.
    8. N. J. Huang & J. Li & J. C. Yao, 2007. "Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 201-212, May.
    9. Gang Xiao & Hong Xiao & Sanyang Liu, 2011. "Scalarization and pointwise well-posedness in vector optimization problems," Journal of Global Optimization, Springer, vol. 49(4), pages 561-574, April.
    10. Gabriele Eichfelder, 2014. "Numerical Procedures in Multiobjective Optimization with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 489-514, August.
    11. 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.
    12. T. C. E. Cheng & Y. N. Wu, 2006. "A Multiproduct, Multicriterion Supply-Demand Network Equilibrium Model," Operations Research, INFORMS, vol. 54(3), pages 544-554, June.
    13. Suhel Khan, 2011. "Generalized vector implicit quasi complementarity problems," Journal of Global Optimization, Springer, vol. 49(4), pages 695-705, April.
    14. 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.
    15. S. Li & M. Li, 2009. "Levitin–Polyak well-posedness of vector equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 125-140, March.
    16. Suhel Ahmad Khan & Naeem Ahmad, 2013. "Existence Results for Vector Mixed Quasi-Complementarity Problems," Journal of Mathematics, Hindawi, vol. 2013, pages 1-6, February.
    17. Miettinen, Kaisa & Eskelinen, Petri & Ruiz, Francisco & Luque, Mariano, 2010. "NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point," European Journal of Operational Research, Elsevier, vol. 206(2), pages 426-434, October.
    18. F. Giannessi & G. Mastroeni & X. Yang, 2012. "Survey on Vector Complementarity Problems," Journal of Global Optimization, Springer, vol. 53(1), pages 53-67, May.

    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:151:y:2011:i:2:d:10.1007_s10957-011-9928-x. 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.