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A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems

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Abstract

This paper discusses second-order cone tensor eigenvalue complementarity problem. We reformulate second-order cone tensor eigenvalue complementarity problem as two constrained polynomial optimizations. For these two reformulated optimizations, Lasserre-type semidefinite relaxation methods are proposed to compute all second-order cone tensor complementarity eigenpairs. The proposed algorithms terminate when there are finitely many second-order cone complementarity eigenvalues. Numerical examples are reported to show the efficiency of the proposed algorithms.

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References

  1. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)

    Book  Google Scholar 

  2. Chen, Z., Qi, L.: A semismooth Newton method for tensor eigenvalue complementarity problem. Comput. Optim. Appl. 65, 109–126 (2016)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z., Yang, Q., Ye, L.: Generalized eigenvalue complementarity problem for tensors. Pac. J. Optim. 13, 527–545 (2017)

    MathSciNet  Google Scholar 

  4. Cui, C., Dai, Y., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35, 1582–1601 (2014)

    Article  MathSciNet  Google Scholar 

  5. Curto, R., Fialkow, L.: Truncated K-moment problems in several variables. J. Oper. Theory 54, 189–226 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Ding, W., Wei, Y.: Generalized tensor eigenvalue problems. SIAM J. Matrix Anal. Appl. 36, 1073–1099 (2015)

    Article  MathSciNet  Google Scholar 

  7. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)

    MATH  Google Scholar 

  8. Fan, J., Nie, J., Zhao, R.: The maximum tensor complementarity eigenvalues. Optim. Methods Softw. (2018). https://doi.org/10.1080/10556788.2018.1528251

    Article  Google Scholar 

  9. Fan, J., Nie, J., Zhou, A.: Tensor eigenvalue complementarity problems. Math. Program. 170, 507–539 (2018)

    Article  MathSciNet  Google Scholar 

  10. Fukushima, M., Luo, Z., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)

    Article  MathSciNet  Google Scholar 

  11. Helton, J., Nie, J.: A semidefinite approach for truncated K-moment problems. Found. Comput. Math. 12, 851–881 (2012)

    Article  MathSciNet  Google Scholar 

  12. Henrion, D., Lasserre, J., Lofberg, Y.: GloptiPoly 3: moments, optimization and semidfinite programming. Optim. Methods Softw. 24, 761–779 (2009)

    Article  MathSciNet  Google Scholar 

  13. Hou, J., Ling, C., He, H.: A class of second-order cone eigenvalue complementarity problems for higher-order tensors. J. Oper. Res. Soc. China 5, 45–64 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hu, S., Huang, Z.-H., Ling, C., Qi, L.: On determinants and eigenvalue theory of tensors. J. Symb. Comput. 50, 508–531 (2013)

    Article  MathSciNet  Google Scholar 

  15. Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    Article  MathSciNet  Google Scholar 

  16. Lasserre, J.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)

    Book  Google Scholar 

  17. Lasserre, J.: Introduction to Polynomial and Semi-algebraic Optimization. Cambridge University Press, Cambridge (2015)

    Book  Google Scholar 

  18. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. IMA Volumes in Mathematics and Its Applications, vol. 149, pp. 157–270. Springer, Berlin (2009)

    Chapter  Google Scholar 

  19. Laurent, M.: Optimization over polynomials: selected topics. In: Proceedings of the International Congress of Mathematicians, Seoul (2014)

  20. Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63, 143–168 (2016)

    Article  MathSciNet  Google Scholar 

  21. Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. 142, 485–510 (2013)

    Article  MathSciNet  Google Scholar 

  22. Nie, J.: Polynomial optimization with real varieties. SIAM J. Optim. 23, 1634–1646 (2013)

    Article  MathSciNet  Google Scholar 

  23. Nie, J., Yang, Z., Zhang, X.: A complete semidefinite algorithm for detecting copositive matrices and tensors. SIAM J. Optim. 28, 2902–2921 (2018)

    Article  MathSciNet  Google Scholar 

  24. Nie, J., Zhang, X.: Real eigenvalues of nonsymmetric tensors. Comput. Optim. Appl. 70, 1–32 (2018)

    Article  MathSciNet  Google Scholar 

  25. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  26. Sturm, J.: Sedumi 1.02: a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  27. Xu, F., Ling, C.: Some properties on Pareto-eigenvalues of higher-order tensors. Oper. Res. Trans. 19, 34–41 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Yu, G., Song, Y., Xu, Y., Yu, Z.: Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems. Numer. Algorithms 80, 1181–1201 (2019)

    Article  MathSciNet  Google Scholar 

  29. Zhao, X., Fan, J.: A semidefinite method for tensor complementarity problems. Optim. Methods Softw. 34, 758–769 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Xinzhen Zhang was partially supported by the National Natural Science Foundation of China (Grant Nos. 11871369 and 12071343). Guyan Ni was partially supported by the National Natural Science Foundation of China (Grant No. 11871472).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300354, China

    Lulu Cheng & Xinzhen Zhang

  2. Department of Mathematics, National University of Defense Technology, Changsha, Hunan, China

    Guyan Ni

Authors
  1. Lulu Cheng
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  2. Xinzhen Zhang
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  3. Guyan Ni
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Correspondence to Xinzhen Zhang.

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Cheng, L., Zhang, X. & Ni, G. A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems. J Glob Optim 79, 715–732 (2021). https://doi.org/10.1007/s10898-020-00954-4

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