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On solving biquadratic optimization via semidefinite relaxation

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  • Yuning Yang
  • Qingzhi Yang

Abstract

In this paper, we study a class of biquadratic optimization problems. We first relax the original problem to its semidefinite programming (SDP) problem and discuss the approximation ratio between them. Under some conditions, we show that the relaxed problem is tight. Then we consider how to approximately solve the problems in polynomial time. Under several different constraints, we present variational approaches for solving them and give provable estimation for the approximation solutions. Some numerical results are reported at the end of this paper. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Yuning Yang & Qingzhi Yang, 2012. "On solving biquadratic optimization via semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 53(3), pages 845-867, December.
    • Handle: RePEc:spr:coopap:v:53:y:2012:i:3:p:845-867
    DOI: 10.1007/s10589-012-9462-2
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    References listed on IDEAS

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    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. NESTEROV, Yurii, 1997. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Discussion Papers CORE 1997044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    4. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    5. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    6. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
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    Cited by:

    1. 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.
    2. Shigui Li & Linzhang Lu & Xing Qiu & Zhen Chen & Delu Zeng, 2024. "Tighter bound estimation for efficient biquadratic optimization over unit spheres," Journal of Global Optimization, Springer, vol. 90(2), pages 323-353, October.
    3. Haibin Chen & Hongjin He & Yiju Wang & Guanglu Zhou, 2022. "An efficient alternating minimization method for fourth degree polynomial optimization," Journal of Global Optimization, Springer, vol. 82(1), pages 83-103, January.
    4. Pengfei Huang & Qingzhi Yang & Yuning Yang, 2022. "Finding the global optimum of a class of quartic minimization problem," Computational Optimization and Applications, Springer, vol. 81(3), pages 923-954, April.

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