IDEAS home Printed from https://ideas.repec.org/a/wsi/apjorx/v26y2009i02ns0217595909002171.html
   My bibliography  Save this article

Vector-Valued Implicit Lagrangian For Symmetric Cone Complementarity Problems

Author

Listed:
  • LINGCHEN KONG

    (Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China;
    Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • LEVENT TUNÇEL

    (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • NAIHUA XIU

    (Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China)

Abstract

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

Suggested Citation

  • Lingchen Kong & Levent Tunçel & Naihua Xiu, 2009. "Vector-Valued Implicit Lagrangian For Symmetric Cone Complementarity Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 26(02), pages 199-233.
    • Handle: RePEc:wsi:apjorx:v:26:y:2009:i:02:n:s0217595909002171
    DOI: 10.1142/S0217595909002171
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0217595909002171
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0217595909002171?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. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, 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. Nan Lu & Zheng-Hai Huang, 2014. "A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 446-464, May.
    2. Xiangjing Liu & Sanyang Liu, 2020. "A new nonmonotone smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$ P 0 -property," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 229-247, October.

    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. M. V. Dolgopolik, 2018. "Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property," Journal of Global Optimization, Springer, vol. 71(2), pages 237-296, June.
    2. Jingyong Tang & Hongchao Zhang, 2021. "A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 679-715, June.
    3. Y. D. Chen & Y. Gao & Y.-J. Liu, 2010. "An Inexact SQP Newton Method for Convex SC1 Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 33-49, July.
    4. S. H. Pan & J.-S. Chen, 2009. "Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 167-191, April.
    5. Xiangjing Liu & Sanyang Liu, 2020. "A new nonmonotone smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$ P 0 -property," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 229-247, October.
    6. Nan Lu & Zheng-Hai Huang, 2014. "A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 446-464, May.
    7. Yingnan Wang & Naihua Xiu, 2011. "Strong Semismoothness of Projection onto Slices of Second-Order Cone," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 599-614, September.
    8. Yue Lu & Ching-Yu Yang & Jein-Shan Chen & Hou-Duo Qi, 2020. "The decompositions with respect to two core non-symmetric cones," Journal of Global Optimization, Springer, vol. 76(1), pages 155-188, January.
    9. Jingyong Tang & Jinchuan Zhou & Hongchao Zhang, 2023. "An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 641-665, February.
    10. Lingchen Kong & Levent Tunçel & Naihua Xiu, 2011. "Equivalent Conditions for Jacobian Nonsingularity in Linear Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 364-389, February.
    11. Enzo Busseti & Walaa M. Moursi & Stephen Boyd, 2019. "Solution refinement at regular points of conic problems," Computational Optimization and Applications, Springer, vol. 74(3), pages 627-643, December.
    12. Nooshin Movahedian, 2014. "Nonsmooth Calculus of Semismooth Functions and Maps," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 415-438, February.
    13. Sangho Kum & Sangwoon Yun, 2017. "Incremental Gradient Method for Karcher Mean on Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 141-155, January.
    14. Xin-He Miao & Yu-Lin Chang & Jein-Shan Chen, 2017. "On merit functions for p-order cone complementarity problem," Computational Optimization and Applications, Springer, vol. 67(1), pages 155-173, May.
    15. S. H. Kum & Y. D. Lim, 2009. "Coercivity and Strong Semismoothness of the Penalized Fischer-Burmeister Function for the Symmetric Cone Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 377-383, August.
    16. Xin-He Miao & Shengjuan Guo & Nuo Qi & Jein-Shan Chen, 2016. "Constructions of complementarity functions and merit functions for circular cone complementarity problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 495-522, March.

    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:wsi:apjorx:v:26:y:2009:i:02:n:s0217595909002171. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/apjor/apjor.shtml .

    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.