IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v105y2000i1d10.1023_a1004670229860.html
   My bibliography  Save this article

Comparison of Existence Results for Efficient Points

Author

Listed:
  • Y. Sonntag

    (Université de Provence)

  • C. Zalinescu

    (University Al. I. Cuza)

Abstract

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.

Suggested Citation

  • Y. Sonntag & C. Zalinescu, 2000. "Comparison of Existence Results for Efficient Points," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 161-188, April.
    • Handle: RePEc:spr:joptap:v:105:y:2000:i:1:d:10.1023_a:1004670229860
    DOI: 10.1023/A:1004670229860
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1004670229860
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1004670229860?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. Jonathan M. Borwein, 1983. "On the Existence of Pareto Efficient Points," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 64-73, February.
    2. Dinh The Luc, 1989. "An Existence Theorem in Vector Optimization," Mathematics of Operations Research, INFORMS, vol. 14(4), pages 693-699, November.
    3. Hedy Attouch & Hassan Riahi, 1993. "Stability Results for Ekeland's ε-Variational Principle and Cone Extremal Solutions," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 173-201, February.
    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. Nguyen Quang Huy & Do Sang Kim & Nguyen Van Tuyen, 2017. "Existence Theorems in Vector Optimization with Generalized Order," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 728-745, September.
    2. J. Jahn & A. A. Khan & P. Zeilinger, 2005. "Second-Order Optimality Conditions in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 331-347, May.
    3. A. Engau & M. M. Wiecek, 2007. "Cone Characterizations of Approximate Solutions in Real Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 499-513, September.

    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. Nergiz Kasimbeyli, 2015. "Existence and characterization theorems in nonconvex vector optimization," Journal of Global Optimization, Springer, vol. 62(1), pages 155-165, May.
    2. Nguyen Quang Huy & Do Sang Kim & Nguyen Van Tuyen, 2017. "Existence Theorems in Vector Optimization with Generalized Order," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 728-745, September.
    3. K.F. Ng & X.Y. Zheng, 2002. "Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 29-47, October.
    4. Feng Guo & Liguo Jiao, 2023. "A new scheme for approximating the weakly efficient solution set of vector rational optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 905-930, August.
    5. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability for Properly Quasiconvex Vector Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 492-506, November.
    6. Li-Wen Zhou & Nan-Jing Huang, 2013. "Existence of Solutions for Vector Optimization on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 44-53, April.
    7. Botte, Marco & Schöbel, Anita, 2019. "Dominance for multi-objective robust optimization concepts," European Journal of Operational Research, Elsevier, vol. 273(2), pages 430-440.
    8. Jayswal, Anurag & Singh, Shipra & Kurdi, Alia, 2016. "Multitime multiobjective variational problems and vector variational-like inequalities," European Journal of Operational Research, Elsevier, vol. 254(3), pages 739-745.
    9. Harold Benson, 2012. "An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem," Journal of Global Optimization, Springer, vol. 52(3), pages 553-574, March.
    10. Vincent Martinet & Pedro Gajardo & Michel Lara, 2024. "Bargaining on monotonic social choice environments," Theory and Decision, Springer, vol. 96(2), pages 209-238, March.
    11. L. V. Thuan & D. T. Luc, 2000. "On Sensitivity in Linear Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 615-626, December.
    12. T. X. D. Ha, 2005. "Some Variants of the Ekeland Variational Principle for a Set-Valued Map," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 187-206, January.
    13. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 941-961, December.
    14. Yarui Duan & Liguo Jiao & Pengcheng Wu & Yuying Zhou, 2022. "Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 148-171, October.
    15. X. Y. Zheng & X. M. Yang & K. L. Teo, 2007. "Some Geometrical Aspects of Efficient Points in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 275-288, May.
    16. S. Deng, 1998. "Characterizations of the Nonemptiness and Compactness of Solution Sets in Convex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 123-131, January.
    17. W. Song, 1997. "Generalization of the Arrow–Barankin–Blackwell Theorem in a Dual Space Setting," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 225-230, October.
    18. S. Deng, 1998. "On Efficient Solutions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 201-209, January.
    19. F. Flores-Bazán & C. Vera, 2006. "Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 185-207, August.
    20. Tian, Guoqiang & Zhou, Jianxin, 1992. "Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets," MPRA Paper 41227, University Library of Munich, Germany.

    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:105:y:2000:i:1:d:10.1023_a:1004670229860. 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.