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An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems

  • * Corresponding author: C.-Y. Shi

    * Corresponding author: C.-Y. Shi 

This research is funded by The Science and Technology Development Fund, Macau SAR (File no. 0005/2019/A) and the grant MYRG2018-00047-FST, MYRG2017-00098-FST from University of Macau

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  • Many kinds of practical problems can be formulated as nonlinear complementarity problems. In this paper, an inexact alternating direction method of multipliers for the solution of a kind of nonlinear complementarity problems is proposed. The convergence analysis of the method is given. Numerical results confirm the theoretical analysis, and show that the proposed method can be more efficient and faster than the modulus-based Jacobi, Gauss-Seidel and Successive Overrelaxation method when the dimension of the problem being solved is large.

    Mathematics Subject Classification: 90C33, 65F10, 65F50, 65G40.


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  • Figure 1.  Porous Flow Through a Dam

    Table 1.  Numerical results of Example 1

    m Algorithm 1 MJ MGS MSOR
    $ 2^3 $ IT 57 170 83 81
    CPU 1.98E-03 1.35E-03 6.49E-04 6.82E-04
    RES 9.72E-07 9.86E-07 9.58E-07 8.21E-07
    $ 2^4 $ IT 106 644 272 270
    CPU 1.16E-02 1.62E-02 7.74E-03 7.14E-03
    RES 9.78E-07 9.98E-07 9.25E-07 9.97E-07
    $ 2^5 $ IT 210 2531 1131 865
    CPU 7.13E-02 1.36E-01 6.66E-02 5.05E-02
    RES 9.84E-07 9.98E-07 9.88E-07 9.89E-07
    $ 2^6 $ IT 435 4839 4729
    CPU 0.613 0.981 1.033
    RES 9.80E-07 9.99E-07 9.97E-07
    $ 2^7 $ IT 900
    CPU 5.57
    RES 9.96E-07
    $ 2^8 $ IT 1875
    CPU 75.5
    RES 9.99E-07
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical results of Example 2

    M Algorithm 1 MJ MGS MSOR
    4 IT 82 925 365 308
    CPU 5.80E-03 7.80E-03 3.40E-03 2.90E-03
    RES 8.99E-07 9.97E-07 9.76E-07 9.56E-07
    5 IT 157 3862 1578 1203
    CPU 4.95E-02 1.15E-01 5.34E-02 4.14E-02
    RES 9.95E-07 9.99E-07 9.93E-07 9.94E-07
    6 IT 310 7536 7402
    CPU 0.370 0.976 0.972
    RES 9.99E-07 1.00E-06 9.97E-07
    7 IT 622
    CPU 3.52
    RES 9.99E-07
    8 IT 1255
    CPU 43.2
    RES 1.00E-06
    9 IT 2551
    CPU 584
    RES 9.98E-07
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results of Example 3

    M Algorithm 1 MJ MGS MSOR
    3 IT 337 730 311 307
    CPU 3.90E-02 1.09E-02 5.10E-03 4.90E-03
    RES 9.56E-07 9.91E-07 9.91E-07 9.46E-07
    4 IT 345 3039 1201 1031
    CPU 1.58E-01 1.37E-01 6.55E-02 5.25E-02
    RES 9.95E-07 9.96E-07 9.87E-07 9.97E-07
    5 IT 481 5610 4569
    CPU 0.861 1.13 1.02
    RES 9.89E-07 9.99E-07 9.96E-07
    6 IT 993
    CPU 8.80
    RES 1.00E-06
    7 IT 2069
    CPU 117
    RES 9.97E-07
    8 IT 4340
    CPU 1519
    RES 9.95E-07
     | Show Table
    DownLoad: CSV
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