IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v59y2014i1p219-248.html
   My bibliography  Save this article

Approximation methods for complex polynomial optimization

Author

Listed:
  • Bo Jiang
  • Zhening Li
  • Shuzhong Zhang

Abstract

Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models where the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) symmetric conjugate form. On the constraint side, the decision variables belong to one of the following three sets: (1) the $$m$$ m -th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogenous polynomial functions and establish key linkages between complex multilinear forms and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented. Numerical results are reported to illustrate the effectiveness of the proposed approximation algorithms. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Bo Jiang & Zhening Li & Shuzhong Zhang, 2014. "Approximation methods for complex polynomial optimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 219-248, October.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:219-248
    DOI: 10.1007/s10589-014-9640-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-014-9640-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-014-9640-5?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. Aharon Ben-Tal & Arkadi Nemirovski & Cornelis Roos, 2003. "Extended Matrix Cube Theorems with Applications to (mu)-Theory in Control," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 497-523, August.
    2. Simai He & Bo Jiang & Zhening Li & Shuzhong Zhang, 2014. "Probability Bounds for Polynomial Functions in Random Variables," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 889-907, August.
    3. Parinya Sanguansat (ed.), 2012. "Principal Component Analysis," Books, IntechOpen, number 1825, January-J.
    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. Taoran Fu & Bo Jiang & Zhening Li, 2018. "Approximation algorithms for optimization of real-valued general conjugate complex forms," Journal of Global Optimization, Springer, vol. 70(1), pages 99-130, January.

    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. Florian Jarre & Felix Lieder & Ya-Feng Liu & Cheng Lu, 2020. "Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting," Journal of Global Optimization, Springer, vol. 76(4), pages 913-932, April.
    2. Bilian Chen & Zhening Li, 2020. "On the tensor spectral p-norm and its dual norm via partitions," Computational Optimization and Applications, Springer, vol. 75(3), pages 609-628, April.
    3. Xianpeng Mao & Yuning Yang, 2022. "Several approximation algorithms for sparse best rank-1 approximation to higher-order tensors," Journal of Global Optimization, Springer, vol. 84(1), pages 229-253, September.
    4. Yuning Yang, 2022. "On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 821-851, September.
    5. Ke Hou & Anthony Man-Cho So, 2014. "Hardness and Approximation Results for L p -Ball Constrained Homogeneous Polynomial Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1084-1108, November.
    6. Aharon Ben-Tal & Arkadi Nemirovski, 2009. "On Safe Tractable Approximations of Chance-Constrained Linear Matrix Inequalities," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 1-25, February.
    7. Roya Karimi & Jianqiang Cheng & Miguel A. Lejeune, 2021. "A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1015-1036, July.
    8. Houessou, Sandrine O. & Dossa, Luc Hippolyte & Diogo, Rodrigue V.C. & Houinato, Marcel & Buerkert, Andreas & Schlecht, Eva, 2019. "Change and continuity in traditional cattle farming systems of West African Coast countries: A case study from Benin," Agricultural Systems, Elsevier, vol. 168(C), pages 112-122.
    9. Taoran Fu & Bo Jiang & Zhening Li, 2018. "Approximation algorithms for optimization of real-valued general conjugate complex forms," Journal of Global Optimization, Springer, vol. 70(1), pages 99-130, January.

    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:59:y:2014:i:1:p:219-248. 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.