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Approximation methods for complex polynomial optimization

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  • 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
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    References listed on IDEAS

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    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.
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    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.

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