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Approximation algorithms for optimization of real-valued general conjugate complex forms

Author

Listed:
  • Taoran Fu

    (Shanghai Jiao Tong University)

  • Bo Jiang

    (Shanghai University of Finance and Economics)

  • Zhening Li

    (University of Portsmouth)

Abstract

Complex polynomial optimization has recently gained more attention in both theory and practice. In this paper, we study optimization of a real-valued general conjugate complex form over various popular constraint sets including the m-th roots of complex unity, the complex unit circle, and the complex unit sphere. A real-valued general conjugate complex form is a homogenous polynomial function of complex variables as well as their conjugates, and always takes real values. General conjugate form optimization is a wide class of complex polynomial optimization models, which include many homogenous polynomial optimization in the real domain with either discrete or continuous variables, and Hermitian quadratic form optimization as well as its higher degree extensions. All the problems under consideration are NP-hard in general and we focus on polynomial-time approximation algorithms with worst-case performance ratios. These approximation ratios improve previous results when restricting our problems to some special classes of complex polynomial optimization, and improve or equate previous results when restricting our problems to some special classes of polynomial optimization in the real domain. The algorithms are based on tensor relaxation and random sampling. Our novel technical contributions are to establish the first set of probability lower bounds for random sampling over the m-th root of unity, the complex unit circle, and the complex unit sphere, and to propose the first polarization formula linking general conjugate forms and complex multilinear forms. Some preliminary numerical experiments are conducted to show good performance of the proposed algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:70:y:2018:i:1:d:10.1007_s10898-017-0561-6
    DOI: 10.1007/s10898-017-0561-6
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. 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.

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