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Weakening convergence conditions of a potential reduction method for tensor complementarity problems

  • * Corresponding author: Yong Wang

    * Corresponding author: Yong Wang

The second author's work was supported by the National Natural Science Foundation of China (grant number 11871051)

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  • Recently, under the condition that the included tensor in the tensor complementarity problem is a diagonalizable and positive definite tensor, the convergence of a potential reduction method for tensor complementarity problems is verified in [a potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2019, 15(2): 429–443]. In this paper, we improve the convergence of this method in the sense that the condition we used is strictly weaker than the one used in the above reference. Preliminary numerical results indicate the effectiveness of the potential reduction method under the new condition.

    Mathematics Subject Classification: Primary: 15A69, 68W40, 90C33.

    Citation:

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  • Figure 1.  The relationships among three classes of tensors

    Table 1.  Numerical Results for Example 4.1

    $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
    (1.2, 0.4, 1.6280, 0.1640) 40 0.3584 (0.4642, 0.0000, 0.0000, 0.1000)
    (2.4, 0.8, 13.7240, 0.6120) 46 0.3714 (0.4642, 0.0000, 0.0000, 0.1000)
    (3.6, 1.2, 46.5560, 1.8280) 49 0.3776 (0.4642, 0.0000, 0.0000, 0.1000)
    (4.8, 1.6,110.4920, 4.1960) 52 0.4612 (0.4642, 0.0000, 0.0000, 0.1000)
    (6.0, 2.0,215.9000, 8.1000) 54 0.3880 (0.4642, 0.0000, 0.0000, 0.1000)
    (7.2, 2.4,373.1480, 13.9240) 56 0.4293 (0.4642, 0.0000, 0.0000, 0.1000)
     | Show Table
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    Table 2.  Numerical Results for Example 4.2

    $ \beta_{0} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
    0.9 (0.9, 0.3, 0.6480, 0.0270) 220 1.7153 (0.0127, 0.0082, 0.0000, 0.0000)
    0.8 (0.9, 0.3, 0.6480, 0.0270) 108 0.7526 (0.0125, 0.0081, 0.0000, 0.0000)
    0.7 (0.9, 0.3, 0.6480, 0.0270) 70 0.5857 (0.0128, 0.0083, 0.0000, 0.0000)
    0.6 (0.9, 0.3, 0.6480, 0.0270) 52 0.3878 (0.0120, 0.0078, 0.0000, 0.0000)
    0.5 (0.9, 0.3, 0.6480, 0.0270) 40 0.2651 (0.0126, 0.0082, 0.0000, 0.0000)
    0.4 (0.9, 0.3, 0.6480, 0.0270) 33 0.1837 (0.0116, 0.0075, 0.0000, 0.0000)
    0.3 (0.9, 0.3, 0.6480, 0.0270) 27 0.2445 (0.0127, 0.0082, 0.0000, 0.0000)
    0.2 (0.9, 0.3, 0.6480, 0.0270) 23 0.1956 (0.0115, 0.0075, 0.0000, 0.0000)
    0.1 (0.9, 0.3, 0.6480, 0.0270) 19 0.1178 (0.0136, 0.0088, 0.0000, 0.0000)
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical Results for Example 4.3

    $ {q}^{\top} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
    $ (-1, 1) $ (2, 1, 28, 2) 47 0.3207 (1, 0, 0, 1)
    $ (-1, 1) $ (4.2, 2.1, 1183.3893, 41.8410) 67 0.4496 (1, 0, 0, 1)
    $ (-6, -2) $ (2.8, 1.4,149.9690, 3.3782) 52 0.3200 (1.6438, 1.1487, 0, 0)
    $ (-6, -2) $ (8, 4, 29690, 1022) 129 1.3179 (1.6438, 1.1487, 0, 0)
    $ (36, -19) $ (4, 2,964, 13) 66 0.4528 (1.8384, 1.8020, 0, 0)
    $ (36, -19) $ (24, 12, 7216164, 248813) 760 7.4286 (1.8384, 1.8020, 0, 0)
     | Show Table
    DownLoad: CSV
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