Abstract
A symmetric tensor is called copositive if it generates a multivariate form taking nonnegative values over the nonnegative orthant. Copositive tensors have found important applications in polynomial optimization, tensor complementarity problems and vacuum stability of a general scalar potential. In this paper, we consider copositivity detection of tensors from both theoretical and computational points of view. After giving several necessary conditions for copositive tensors, we propose several new criteria for copositive tensors based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex and simplicial partitions. It is verified that, as the partition gets finer and finer, the concerned conditions eventually capture all strictly copositive tensors. Based on the obtained theoretical results with the help of simplicial partitions, we propose a numerical method to judge whether a tensor is copositive or not. The preliminary numerical results confirm our theoretical findings.
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Acknowledgements
The authors are very thankful to the anonymous reviewers for their useful comments and constructive advice. This research was done during the first author’s postdoctoral period in Qufu Normal University. Furthermore, the first author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11601261) and Natural Science Foundation of Shandong Province (Grant No. ZR2016AQ12). The second author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11431002), and the third author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 502111, 501212, 501913 and 15302114).
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Communicated by Guoyin Li.
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Chen, H., Huang, ZH. & Qi, L. Copositivity Detection of Tensors: Theory and Algorithm. J Optim Theory Appl 174, 746–761 (2017). https://doi.org/10.1007/s10957-017-1131-2
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DOI: https://doi.org/10.1007/s10957-017-1131-2