Research article

An accelerated conjugate gradient method for the Z-eigenvalues of symmetric tensors

  • Received: 06 February 2023 Revised: 23 March 2023 Accepted: 30 March 2023 Published: 23 April 2023
  • MSC : 15A18, 15A69, 90C55

  • We transform the Z-eigenvalues of symmetric tensors into unconstrained optimization problems with a shifted parameter. An accelerated conjugate gradient method is proposed for solving these unconstrained optimization problems. If solving problem results in a nonzero critical point, then it is a Z-eigenvector corresponding to the Z-eigenvalue. Otherwise, we solve the shifted problem to find a Z-eigenvalue. In our method, the new conjugate gradient parameter is a modified CD conjugate gradient parameter, and an accelerated parameter is presented by using the quasi-Newton direction. The global convergence of new method is proved. Numerical experiments are listed to illustrate the efficiency of the proposed method.

    Citation: Mingyuan Cao, Yueting Yang, Chaoqian Li, Xiaowei Jiang. An accelerated conjugate gradient method for the Z-eigenvalues of symmetric tensors[J]. AIMS Mathematics, 2023, 8(7): 15008-15023. doi: 10.3934/math.2023766

    Related Papers:

  • We transform the Z-eigenvalues of symmetric tensors into unconstrained optimization problems with a shifted parameter. An accelerated conjugate gradient method is proposed for solving these unconstrained optimization problems. If solving problem results in a nonzero critical point, then it is a Z-eigenvector corresponding to the Z-eigenvalue. Otherwise, we solve the shifted problem to find a Z-eigenvalue. In our method, the new conjugate gradient parameter is a modified CD conjugate gradient parameter, and an accelerated parameter is presented by using the quasi-Newton direction. The global convergence of new method is proved. Numerical experiments are listed to illustrate the efficiency of the proposed method.



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    [1] L. Lim, Singular values and eigenvalues of tensors: a variational approach, In: Computational Advances in Multi-Sensor Adaptive Processing, 2005, 8912515. https://doi.org/10.1109/CAMAP.2005.1574201
    [2] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302–1324. https://doi.org/10.1016/j.jsc.2005.05.007 doi: 10.1016/j.jsc.2005.05.007
    [3] M. Cao, Q. Huang, Y. Yang, A self-adaptive trust region method for extreme B-eigenvalues of symmetric tensors, Numer. Algor., 81 (2019), 407–420. https://doi.org/10.1007/s11075-018-0554-7 doi: 10.1007/s11075-018-0554-7
    [4] C. Hao, C. Cui, Y. Dai, A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensors, Numer. Linear Algebra Appl., 22 (2015), 283–298. https://doi.org/10.1002/nla.1949 doi: 10.1002/nla.1949
    [5] S. Aji, P. Kumam, A. Awwal, M. M. Yahaya, K. Sitthithakerngkiet, An efficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery, AIMS Math., 6 (2021), 8078–8106. http://dx.doi.org/10.3934/math.2021469 doi: 10.3934/math.2021469
    [6] T. Kolda, J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095–1124. https://doi.org/10.1137/100801482 doi: 10.1137/100801482
    [7] T. Kolda, J. Mayo, An adaptive shifted power methods for computing generalized tensor eigenpairs, SIAM J. Matrix Anal. Appl., 35 (2014), 1563–1581. https://doi.org/10.1137/140951758 doi: 10.1137/140951758
    [8] Y. Chen, M. Cao, Y. Yang, Q. Huang, An adaptive trust-region method for generalized eigenvalues of symmetric tensors, J. Comput. Math., 39 (2021), 533–549.
    [9] G. Li, L. Qi, G. Yu, Semismoothness of the maximum eigenvalue function of a symmetric tensor and its application, Linear Algebra Appl., 438 (2013), 813–833. https://doi.org/10.1016/j.laa.2011.10.043 doi: 10.1016/j.laa.2011.10.043
    [10] M. Cao, Y. Yang, T. Hou, C. Li, A nonmonotone accelerated Levenberg-Marquardt method for the B-eigenvalues of symmetric tensors, Inter. Trans. Oper. Res., 29 (2022), 113–129. https://doi.org/10.1111/itor.12954 doi: 10.1111/itor.12954
    [11] M. Ng, L. Qi, G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090–1099. https://doi.org/10.1137/09074838X doi: 10.1137/09074838X
    [12] J. Bai, W. Hager, H. Zhang, An inexact accelerated stochastic ADMM for separable composite convex optimization, Comput. Optim. Appl., 81 (2022), 479–518. https://doi.org/10.1007/s10589-021-00338-8 doi: 10.1007/s10589-021-00338-8
    [13] J. Bai, D. Han, H. Sun, H. Zhang, Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes, CSIAM Trans. Appl. Math., 3 (2022), 448–479.
    [14] L. Qi, W. Sun, Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2017), 501–526. https://doi.org/10.1007/s11464-007-0031-4 doi: 10.1007/s11464-007-0031-4
    [15] T. Wei, P. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A, 68 (2003), 042307. https://doi.org/10.1103/PhysRevA.68.042307 doi: 10.1103/PhysRevA.68.042307
    [16] L. Qi, G. Yu, E. Wu, Higher order positive semi-definite diffusion tensor imaging, SIAM J. Imaging Sci., 3 (2010), 416–433. https://doi.org/10.1137/090755138 doi: 10.1137/090755138
    [17] L. Qi, F. Wang, Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), 301–316. https://doi.org/10.1007/s10107-007-0193-6 doi: 10.1007/s10107-007-0193-6
    [18] L. Han, An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors, Numer. Algebr. Control Optim., 3 (2013), 583–599.
    [19] C. Hao, C. Cui, Y. Dai, A feasible trust-region method for calculating extreme Z-eigenvalues of symmetric tensors, Pac. J. Optim., 11 (2015), 291–307.
    [20] J. Nocedal, S. J. Wright, Numerical optimization, New York: Springer, 1999.
    [21] Y. Yuan, W. Sun, Optimization theory and methods, Beijing: Science Press, 1997.
    [22] G. Auchmuty, Globally and rapidly convergent algorithm for symmetric eigenproblems, SIAM J. Matrix Anal. Appl., 12 (1991), 690–706. https://doi.org/10.1137/0612053 doi: 10.1137/0612053
    [23] A. Peressini, F. Sullivan, J. Uhl, The mathematics of nonlinear programmming, Berlin: Springer, 1988.
    [24] A. Neculai, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett., 20 (2007), 645–650. https://doi.org/10.1016/j.aml.2006.06.015 doi: 10.1016/j.aml.2006.06.015
    [25] A. Neculai, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Meth. Soft., 22 (2007), 561–571. https://doi.org/10.1080/10556780600822260 doi: 10.1080/10556780600822260
    [26] G. Zoutendijk, Nonlinear programming, computational method, Integer Nonlinear Program., 1970, 37–86.
    [27] C. Cui, Y. Dai, J. Nie, All real eigenvalues of symmetric tensors, SIAM J. Matrix Anal. Appl., 4 (2014), 1582–1601. https://doi.org/10.1137/140962292 doi: 10.1137/140962292
    [28] E. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201–213. https://doi.org/10.1007/s101070100263 doi: 10.1007/s101070100263
    [29] M. Zeng, Q. Ni, Quasi-Newton method for computing Z-eigenvalues of a symmetric tensor, Pac. J. Optim., 2 (2015), 279–290.
    [30] X. Dong, D. Han, Z. Dai, L. Li, J. Zhu, An accelerated three-term conjugate gradient method with sufficient descent condition and conjugacy condition, J. Optim. The. Appl., 179 (2018), 944–961. https://doi.org/10.1007/s10957-018-1377-3 doi: 10.1007/s10957-018-1377-3
    [31] B. Ivanov, G. Milovanovi$\acute{c}$, P. Stanimirovi$\acute{c}$, Accelerated Dai-Liao projection method for solving systems of monotone nonlinear equations with application to image deblurring, J. Glob. Optim., 85 (2023), 377–420. https://doi.org/10.1007/s10898-022-01213-4 doi: 10.1007/s10898-022-01213-4
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