IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v173y2017i1d10.1007_s10957-017-1084-5.html
   My bibliography  Save this article

Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

Author

Listed:
  • Dezhou Kong

    (Shandong Agricultural University)

  • Lishan Liu

    (Qufu Normal University
    Curtin University)

  • Yonghong Wu

    (Curtin University)

Abstract

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

Suggested Citation

  • Dezhou Kong & Lishan Liu & Yonghong Wu, 2017. "Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 117-130, April.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:1:d:10.1007_s10957-017-1084-5
    DOI: 10.1007/s10957-017-1084-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-017-1084-5
    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/s10957-017-1084-5?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. S. Németh & G. Zhang, 2015. "Extended Lorentz cones and mixed complementarity problems," Journal of Global Optimization, Springer, vol. 62(3), pages 443-457, July.
    2. Pedro Gajardo & Alberto Seeger, 2014. "Equilibrium problems involving the Lorentz cone," Journal of Global Optimization, Springer, vol. 58(2), pages 321-340, February.
    3. Hiroki Nishimura & Efe A. Ok, 2012. "Solvability of Variational Inequalities on Hilbert Lattices," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 608-625, November.
    4. Samir Adly & Hadia Rammal, 2015. "A New Method for Solving Second-Order Cone Eigenvalue Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 563-585, May.
    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. Dezhou Kong & Lishan Liu & Yonghong Wu, 2020. "Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 88-104, October.
    2. Dezhou Kong & Lishan Liu & Yonghong Wu, 2017. "Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 341-355, November.
    3. Xiao-Peng Yang, 2019. "Evaluation Model and Approximate Solution to Inconsistent Max-Min Fuzzy Relation Inequalities in P2P File Sharing System," Complexity, Hindawi, vol. 2019, pages 1-11, March.

    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. O. P. Ferreira & S. Z. Németh, 2018. "How to project onto extended second order cones," Journal of Global Optimization, Springer, vol. 70(4), pages 707-718, April.
    2. Dezhou Kong & Lishan Liu & Yonghong Wu, 2017. "Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 341-355, November.
    3. S. Németh & G. Zhang, 2015. "Extended Lorentz cones and mixed complementarity problems," Journal of Global Optimization, Springer, vol. 62(3), pages 443-457, July.
    4. Massimo Marinacci & Luigi Montrucchio, 2017. "Unique Tarski Fixed Points," Working Papers 604, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    5. Yuehu Wang & Baoqing Liu, 2019. "Order-Preservation Properties of Solution Mapping for Parametric Equilibrium Problems and Their Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 881-901, December.
    6. Yingchao Gao & Sándor Zoltán Németh & Roman Sznajder, 2022. "The Monotone Extended Second-Order Cone and Mixed Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 381-407, June.
    7. Roman Sznajder, 2016. "The Lyapunov rank of extended second order cones," Journal of Global Optimization, Springer, vol. 66(3), pages 585-593, November.
    8. Sándor Zoltán Németh & Guohan Zhang, 2016. "Extended Lorentz Cones and Variational Inequalities on Cylinders," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 756-768, March.
    9. Massimo Marinacci & Luigi Montrucchio, 2019. "Unique Tarski Fixed Points," Management Science, INFORMS, vol. 44(4), pages 1174-1191, November.
    10. Niu, Yi-Shuai & Júdice, Joaquim & Le Thi, Hoai An & Pham, Dinh Tao, 2019. "Improved dc programming approaches for solving the quadratic eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 95-113.
    11. Orizon Pereira Ferreira & Yingchao Gao & Sándor Zoltán Németh & Petra Renáta Rigó, 2024. "Gradient projection method on the sphere, complementarity problems and copositivity," Journal of Global Optimization, Springer, vol. 90(1), pages 1-25, September.
    12. Brás, Carmo P. & Fischer, Andreas & Júdice, Joaquim J. & Schönefeld, Klaus & Seifert, Sarah, 2017. "A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 36-48.
    13. Olszewski, Wojciech, 2021. "On sequences of iterations of increasing and continuous mappings on complete lattices," Games and Economic Behavior, Elsevier, vol. 126(C), pages 453-459.
    14. Sándor Zoltán Németh & Lianghai Xiao, 2018. "Linear Complementarity Problems on Extended Second Order Cones," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 269-288, February.
    15. Masao Fukushima & Joaquim Júdice & Welington Oliveira & Valentina Sessa, 2020. "A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem," Computational Optimization and Applications, Springer, vol. 77(3), pages 711-728, December.

    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:173:y:2017:i:1:d:10.1007_s10957-017-1084-5. 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.