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

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Taoran Fu

  2. Research Center for Management Science and Data Analytics, School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Bo Jiang

  3. Department of Mathematics, University of Portsmouth, Portsmouth, PO1 3HF, UK

    Zhening Li

Authors
  1. Taoran Fu
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  2. Bo Jiang
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  3. Zhening Li
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Correspondence to Bo Jiang.

Additional information

The research of Bo Jiang was supported in part by National Natural Science Foundation of China (Grant 11401364) and Program for Innovative Research Team of Shanghai University of Finance and Economics.

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Fu, T., Jiang, B. & Li, Z. Approximation algorithms for optimization of real-valued general conjugate complex forms. J Glob Optim 70, 99–130 (2018). https://doi.org/10.1007/s10898-017-0561-6

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  • DOI: https://doi.org/10.1007/s10898-017-0561-6

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