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A new smoothing spectral conjugate gradient method for solving tensor complementarity problems

  • * Corresponding author: Shou-Qiang Du

    * Corresponding author: Shou-Qiang Du
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  • In recent years, the tensor complementarity problem has attracted widespread attention and has been extensively studied. The research work of tensor complementarity problem mainly focused on theory, solution methods and applications. In this paper, we study the solution method of tensor complementarity problem. Based on the equivalence relation of the tensor complementarity problem and unconstrained optimization problem, we propose a new smoothing spectral conjugate gradient method with Armijo line search. Under mild conditions, we establish the global convergence of the proposed method. Finally, some numerical results are given to show the effectiveness of the proposed method and verify our theoretical results.

    Mathematics Subject Classification: Primary: 90C33, 65K05; Secondary: 15A69.

    Citation:

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  • Figure 1.  Numerical results of Example 4.1 with different vector $ q $

    Figure 2.  Numerical results of Example 4.2 with different vector $ q $

    Figure 3.  Numerical results of Example 4.3 with different vector $ q $

    Figure 4.  Numerical results of Example 4.4 with different initial points

    Figure 5.  Numerical results of Example 4.5 with different initial points

    Table 1.  The numerical results of Example 4.1

    x0 Sol Val
    $ (0.8333, 0.2037, 0.5444, 0.8749)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 1.9158e-15
    $ (0.9460, 0.0916, 0.9084, 0.5100)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.9449e-14
    $ (0.1445, 0.3705, 0.6224, 0.9976)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.1031e-15
    $ (0.6966, 0.0646, 0.7477, 0.4204)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 8.9036e-14
    $ (0.8113, 0.3796, 0.3191, 0.9861)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 4.1630e-14
     | Show Table
    DownLoad: CSV

    Table 2.  The numerical results of Example 4.2

    q Sol K Val
    $ (1, 2, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 7 2.8222e-13
    $ (1, -2, 3)^T $ $ (0.0000, 1.0000, 0.0000)^T $ 70 3.1532e-13
    $ (3, 3, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 6 2.1459e-14
    $ (-3, -2, -3)^T $ $ (1.3161, 1.0000, 1.0000)^T $ 16 1.0256e-15
    $ (-3, -1, -2)^T $ $ (1.3161, 0.8409, 0.9036)^T $ 19 1.1984e-14
     | Show Table
    DownLoad: CSV

    Table 3.  The numerical results of Example 4.3

    q Sol K Val
    $ (5, 3)^T $ $ (0.0000, 0.0000)^T $ 7 8.9458e-14
    $ (2, -3)^T $ $ (0.0000, 1.7321)^T $ 76 2.6588e-13
    $ (-5, -3)^T $ $ (0.3127, 1.9233)^T $ 30 9.4915e-14
    $ (-5, 3)^T $ $ (1.5513, 0.6847)^T $ 55 2.0370e-14
    $ (0, -5)^T $ $ (0.0000, 2.2361)^T $ 34 7.5750e-14
     | Show Table
    DownLoad: CSV

    Table 4.  The numerical results of Example 4.4

    Alg. x0 Sol K Val
    Algorithm 3.1 $ (0.3804, 0.5678)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
    Algorithm 3.1 $ (0.0759, 0.0540)^T $ $ (0.0000, 0.0000)^T $ 7 2.9600e-13
    Algorithm 3.1 $ (0.9340, 0.1299)^T $ $ (0.0000, 0.0000)^T $ 6 9.5895e-15
    Algorithm 3.1 $ (0.1622, 0.7943)^T $ $ (0.0000, 0.0000)^T $ 6 4.5196e-13
    Algorithm 3.1 $ (0.3112, 0.5285)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
    MIP - $ (0.0000, 0.0000)^T $ 39 -
    MIP - $ (0.0000, 0.0000)^T $ 29 -
     | Show Table
    DownLoad: CSV

    Table 5.  The numerical results of Example 4.5

    x0 K Val
    $ (0.7655, 0.7951, 0.1868, 0.4897, 0.4455, 0.6463)^T $ 10 3.0376e-13
    $ (0.7094, 0.7547, 0.2760, 0.6797, 0.6551, 0.1626)^T $ 12 2.7860e-15
    $ (0.5472, 0.1386, 0.1493, 0.2575, 0.8407, 0.2543)^T $ 10 1.9144e-13
    $ (0.8143, 0.2435, 0.9293, 0.3500, 0.1966, 0.2511)^T $ 10 1.7226e-13
    $ (0.6160, 0.4733, 0.3517, 0.8308, 0.5853, 0.5497)^T $ 9 1.2683e-14
     | Show Table
    DownLoad: CSV
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