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Tensor eigenvalue complementarity problems

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Abstract

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.

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Acknowledgements

The authors would like to thank two anonymous referees for their useful comments on improving the paper. Jinyan Fan was partially supported by the NSFC Grants 11171217 and 11571234. Jiawang Nie was partially supported by the NSF Grants DMS-1417985 and DMS-1619973. Anwa Zhou was partially supported by the National Postdoctoral Program for Innovative Talents Grant BX201600097 and Project Funded by China Postdoctoral Science Foundation Grant 2016M601562.

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Authors and Affiliations

  1. School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Jinyan Fan

  2. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Anwa Zhou

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  1. Jinyan Fan
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  2. Jiawang Nie
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  3. Anwa Zhou
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Correspondence to Jiawang Nie.

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Fan, J., Nie, J. & Zhou, A. Tensor eigenvalue complementarity problems. Math. Program. 170, 507–539 (2018). https://doi.org/10.1007/s10107-017-1167-y

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