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Spherical optimization with complex variablesfor computing US-eigenpairs

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Abstract

The aim of this paper is to compute unitary symmetric eigenpairs (US-eigenpairs) of high-order symmetric complex tensors, which is closely related to the best complex rank-one approximation of a symmetric complex tensor and quantum entanglement. It is also an optimization problem of real-valued functions with complex variables. We study the spherical optimization problem with complex variables including the first-order and the second-order Taylor polynomials, optimization conditions and convex functions of real-valued functions with complex variables. We propose an algorithm and show that it is guaranteed to approximate a US-eigenpair of a symmetric complex tensor. Moreover, if the number of US-eigenpair is finite, then the algorithm is convergent to a US-eigenpair. Numerical examples are presented to demonstrate the effectiveness of the proposed method in finding US-eigenpairs.

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Acknowledgments

The authors like to thank the comments of the anonymous referees, which greatly improved the presentation of this paper. We also give thanks to the associate editor for pointing out two relative references. This work was supported by the National Natural Science Foundation of China (No. 11571098) and Hunan Provincial Natural Science Foundation of China (No. 14JJ2063).

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, Hunan, China

    Guyan Ni

  2. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, Hunan, China

    Minru Bai

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  1. Guyan Ni
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  2. Minru Bai
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Correspondence to Guyan Ni.

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Ni, G., Bai, M. Spherical optimization with complex variablesfor computing US-eigenpairs. Comput Optim Appl 65, 799–820 (2016). https://doi.org/10.1007/s10589-016-9848-7

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  • DOI: https://doi.org/10.1007/s10589-016-9848-7

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