IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v55y2013i2p379-398.html
   My bibliography  Save this article

Tightening a copositive relaxation for standard quadratic optimization problems

Author

Listed:
  • Yong Xia
  • Ruey-Lin Sheu
  • Xiaoling Sun
  • Duan Li

Abstract

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of its SDP relaxations known as “strengthened Shor’s relaxation”. To estimate the duality gap, we utilize the duality information of the SDP relaxation to construct a graph G ∗ . The estimation can be then reduced to a two-phase problem of enumerating first all the minimal vertex covers of G ∗ and solving next a family of second-order cone programming problems. When there is a nonzero duality gap, this duality gap estimation can lead to a strictly tighter lower bound than the strengthened Shor’s SDP bound. With the duality gap estimation improving scheme, we develop further a heuristic algorithm for obtaining a good approximate solution for standard QP. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Yong Xia & Ruey-Lin Sheu & Xiaoling Sun & Duan Li, 2013. "Tightening a copositive relaxation for standard quadratic optimization problems," Computational Optimization and Applications, Springer, vol. 55(2), pages 379-398, June.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:2:p:379-398
    DOI: 10.1007/s10589-012-9522-7
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-012-9522-7
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-012-9522-7?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. X. Sun & C. Liu & D. Li & J. Gao, 2012. "On duality gap in binary quadratic programming," Journal of Global Optimization, Springer, vol. 53(2), pages 255-269, June.
    2. Xiaojin Zheng & Xiaoling Sun & Duan Li & Yong Xia, 2010. "Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 864-880, November.
    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. Chunli Liu & Jianjun Gao, 2015. "A polynomial case of convex integer quadratic programming problems with box integer constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 661-674, August.
    2. Liu, Weiwei & Kong, Nan & Wang, Mingzheng & Zhang, Lingling, 2021. "Sustainable multi-commodity capacitated facility location problem with complementarity demand functions," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 145(C).
    3. Gary Kochenberger & Jin-Kao Hao & Fred Glover & Mark Lewis & Zhipeng Lü & Haibo Wang & Yang Wang, 2014. "The unconstrained binary quadratic programming problem: a survey," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 58-81, July.
    4. X. Sun & C. Liu & D. Li & J. Gao, 2012. "On duality gap in binary quadratic programming," Journal of Global Optimization, Springer, vol. 53(2), pages 255-269, June.
    5. Xia, Yong & Sheu, Ruey-Lin & Sun, Xiaoling & Li, Duan, 2012. "Improved estimation of duality gap in binary quadratic programming using a weighted distance measure," European Journal of Operational Research, Elsevier, vol. 218(2), pages 351-357.

    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:coopap:v:55:y:2013:i:2:p:379-398. 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.