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September  2021, 11(3): 353-362. doi: 10.3934/naco.2020030

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

Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China

* Corresponding author: C.-Y. Shi

Received  October 2019 Revised  March 2020 Published  May 2020

Fund Project: 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

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.

Citation: Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030
References:
[1]

Z. Z. Bai, New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems, Comput. Math. Appl., 32 (1996), 41-48.  doi: 10.1016/0898-1221(96)00123-X.  Google Scholar

[2]

Z. Z. Bai, A class of asynchronous parallel nonlinear accelerated over relaxation methods for the nonlinear complementarity problem, J. Comput. Appl. Math., 93 (1998), 35-44.  doi: 10.1016/S0377-0427(98)00280-5.  Google Scholar

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Z. Z. Bai, Asynchronous parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems, Appl. Math. Comput., 92 (1998), 85-100.  doi: 10.1016/S0096-3003(97)10020-0.  Google Scholar

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Z. Z. BaiV. MigallónJ. Penadés and D. B. Szyld, Block and asynchronous two-stage methods for mildly nonlinear systems, Numer. Math., 82 (1999), 1-20.  doi: 10.1007/s002110050409.  Google Scholar

[5]

S. Q. Du and Y. Gao, Merit functions for nonsmooth complementarity problems and related descent algorithm, Applied Mathematics - A Journal of Chinese Universities, 25 (2010), 78-84.  doi: 10.1007/s11766-010-2190-4.  Google Scholar

[6]

W. Deng and W. Yin, On the global and linear convergence of the generalized alternating direction method of multipliers, J. Sci. Comput., 66 (2016), 889-916.  doi: 10.1007/s10915-015-0048-x.  Google Scholar

[7]

C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, in Research Notes in Mathematics, 59, Pitman, Boston, London, 1982.  Google Scholar

[8]

M. C. Ferris and C. Kanzow, Complementarity and related problems: A survey, in Handbook of Applied Optimization(eds. P. M. Pardalos, and M. G. C. Resende), Oxford University Press, New York, (2002), 514–530. doi: 10.1007/978-1-4757-5362-2.  Google Scholar

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M. C. Ferris, O. Mangasarian and J. Pang, Complementarity: Applications, Algorithms and Extensions, Springer, New York, 2011. Google Scholar

[10]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un et la résolution par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Laboria Report 115, 1975.  Google Scholar

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K. H. Hoffmann and J. Zou, Parallel solution of variational inequality problems with nonlinear source terms, IMA J. Numer. Anal., 16 (1996), 31-45.  doi: 10.1093/imanum/16.1.31.  Google Scholar

[12]

N. Huang and C. F. Ma, The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems, Numer. Linear Algebra Appl., 23 (2016), 558-569.  doi: 10.1002/nla.2039.  Google Scholar

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C. E. Lemke and J. T. Howson, Equilibrium points of bimatrix games, SIAM J. Appl. Math., 12 (1964), 413-423.   Google Scholar

[15]

R. Li and J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems, Numer. Algor., 75 (2017), 339-358.  doi: 10.1007/s11075-016-0243-3.  Google Scholar

[16]

W. Li and W. W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z matrices, Linear Algebra Appl., 317 (2000), 227-240.  doi: 10.1016/S0024-3795(00)00140-3.  Google Scholar

[17]

W. Li, A general modulus-based matrix splitting method for linear complementarity problems of H-matrices, Appl. Math. Lett., 26 (2013), 1159-1164.  doi: 10.1016/j.aml.2013.06.015.  Google Scholar

[18]

C. F. Ma and N. Huang, Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems, Appl. Numer. Math., 108 (2016), 116-124.  doi: 10.1016/j.apnum.2016.05.004.  Google Scholar

[19]

G. H. Meyer, Free boundary problems with nonlinear source terms, Numer. Math., 43 (1984), 463-482.  doi: 10.1007/BF01390185.  Google Scholar

[20]

M. A. Noor, Fixed point approach for complementarity problems, J. Comput. Appl. Math., 133 (1988), 437-448.  doi: 10.1016/0022-247X(88)90413-1.  Google Scholar

[21]

F. A. Potra and S. J. Wright, Interior-point methods, J. Comput. Appl. Math., 124 (2000), 255-281.  doi: 10.1016/S0377-0427(00)00433-7.  Google Scholar

[22]

L. QiD. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35.  doi: 10.1007/s101079900127.  Google Scholar

[23]

Z. Sun and J. P. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math., 235 (2011), 1261-1274.  doi: 10.1016/j.cam.2010.08.012.  Google Scholar

[24]

M. Tao and X. Yuan, On Glowinski's open question on the alternating direction method of multipliers, Journal of Optimization Theory and Applications, 179 (2018), 163–196. Google Scholar

[25]

N. H. Xiu and J. Zhang, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math., 152 (2003), 559-585.  doi: 10.1016/S0377-0427(02)00730-6.  Google Scholar

[26]

S. L. XieH. R. Xu and J. P. Zeng, Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Linear Algebra Appl., 494 (2016), 1-10.  doi: 10.1016/j.laa.2016.01.002.  Google Scholar

[27]

Z. Xia and C. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem, Appl. Math. Comput., 271 (2015), 34-42. doi: 10.1016/j.amc.2015.08.108.  Google Scholar

[28]

L. Yong, Nonlinear complementarity problem and solution methods, in Proceedings of the 2010 International Conference on Artificial Intelligence and Computational Intelligence, Part I, Springer-Verlag, (2010), 461–469.  Google Scholar

[29]

H. ZhengS. Vong and L. Liu, The relaxation modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems, Int. J. Comput. Math., 96 (2018), 1648-1667.  doi: 10.1080/00207160.2018.1504928.  Google Scholar

[30]

H. Zheng and S. Vong, The modulus-based nonsmooth Newtons method for solving a class of nonlinear complementarity problems of P-matrices, Calcolo, 55 (2018), 37. doi: 10.1007/s10092-018-0279-y.  Google Scholar

[31]

H. ZhengS. Vong and L. Liu, A direct preconditioned modulus-based iteration method for solving nonlinear complementarity problems of H-matrices, Appl. Math. Comput., 353 (2019), 396-405.  doi: 10.1016/j.amc.2019.02.015.  Google Scholar

[32]

J. J. Zhang, MSSOR-based alternating direction method for symmetric positive-definite linear complementarity problems, Numer. Algor., 68 (2015), 631-644.  doi: 10.1007/s11075-014-9864-6.  Google Scholar

[33]

J. J. ZhangJ. L. Zhang and W. Z. Ye, An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems, Numer. Algor., 78 (2018), 895-910.  doi: 10.1007/s11075-017-0405-y.  Google Scholar

show all references

References:
[1]

Z. Z. Bai, New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems, Comput. Math. Appl., 32 (1996), 41-48.  doi: 10.1016/0898-1221(96)00123-X.  Google Scholar

[2]

Z. Z. Bai, A class of asynchronous parallel nonlinear accelerated over relaxation methods for the nonlinear complementarity problem, J. Comput. Appl. Math., 93 (1998), 35-44.  doi: 10.1016/S0377-0427(98)00280-5.  Google Scholar

[3]

Z. Z. Bai, Asynchronous parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems, Appl. Math. Comput., 92 (1998), 85-100.  doi: 10.1016/S0096-3003(97)10020-0.  Google Scholar

[4]

Z. Z. BaiV. MigallónJ. Penadés and D. B. Szyld, Block and asynchronous two-stage methods for mildly nonlinear systems, Numer. Math., 82 (1999), 1-20.  doi: 10.1007/s002110050409.  Google Scholar

[5]

S. Q. Du and Y. Gao, Merit functions for nonsmooth complementarity problems and related descent algorithm, Applied Mathematics - A Journal of Chinese Universities, 25 (2010), 78-84.  doi: 10.1007/s11766-010-2190-4.  Google Scholar

[6]

W. Deng and W. Yin, On the global and linear convergence of the generalized alternating direction method of multipliers, J. Sci. Comput., 66 (2016), 889-916.  doi: 10.1007/s10915-015-0048-x.  Google Scholar

[7]

C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, in Research Notes in Mathematics, 59, Pitman, Boston, London, 1982.  Google Scholar

[8]

M. C. Ferris and C. Kanzow, Complementarity and related problems: A survey, in Handbook of Applied Optimization(eds. P. M. Pardalos, and M. G. C. Resende), Oxford University Press, New York, (2002), 514–530. doi: 10.1007/978-1-4757-5362-2.  Google Scholar

[9]

M. C. Ferris, O. Mangasarian and J. Pang, Complementarity: Applications, Algorithms and Extensions, Springer, New York, 2011. Google Scholar

[10]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un et la résolution par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Laboria Report 115, 1975.  Google Scholar

[11]

K. H. Hoffmann and J. Zou, Parallel solution of variational inequality problems with nonlinear source terms, IMA J. Numer. Anal., 16 (1996), 31-45.  doi: 10.1093/imanum/16.1.31.  Google Scholar

[12]

N. Huang and C. F. Ma, The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems, Numer. Linear Algebra Appl., 23 (2016), 558-569.  doi: 10.1002/nla.2039.  Google Scholar

[13]

P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220.  doi: 10.1007/BF01582255.  Google Scholar

[14]

C. E. Lemke and J. T. Howson, Equilibrium points of bimatrix games, SIAM J. Appl. Math., 12 (1964), 413-423.   Google Scholar

[15]

R. Li and J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems, Numer. Algor., 75 (2017), 339-358.  doi: 10.1007/s11075-016-0243-3.  Google Scholar

[16]

W. Li and W. W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z matrices, Linear Algebra Appl., 317 (2000), 227-240.  doi: 10.1016/S0024-3795(00)00140-3.  Google Scholar

[17]

W. Li, A general modulus-based matrix splitting method for linear complementarity problems of H-matrices, Appl. Math. Lett., 26 (2013), 1159-1164.  doi: 10.1016/j.aml.2013.06.015.  Google Scholar

[18]

C. F. Ma and N. Huang, Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems, Appl. Numer. Math., 108 (2016), 116-124.  doi: 10.1016/j.apnum.2016.05.004.  Google Scholar

[19]

G. H. Meyer, Free boundary problems with nonlinear source terms, Numer. Math., 43 (1984), 463-482.  doi: 10.1007/BF01390185.  Google Scholar

[20]

M. A. Noor, Fixed point approach for complementarity problems, J. Comput. Appl. Math., 133 (1988), 437-448.  doi: 10.1016/0022-247X(88)90413-1.  Google Scholar

[21]

F. A. Potra and S. J. Wright, Interior-point methods, J. Comput. Appl. Math., 124 (2000), 255-281.  doi: 10.1016/S0377-0427(00)00433-7.  Google Scholar

[22]

L. QiD. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35.  doi: 10.1007/s101079900127.  Google Scholar

[23]

Z. Sun and J. P. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math., 235 (2011), 1261-1274.  doi: 10.1016/j.cam.2010.08.012.  Google Scholar

[24]

M. Tao and X. Yuan, On Glowinski's open question on the alternating direction method of multipliers, Journal of Optimization Theory and Applications, 179 (2018), 163–196. Google Scholar

[25]

N. H. Xiu and J. Zhang, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math., 152 (2003), 559-585.  doi: 10.1016/S0377-0427(02)00730-6.  Google Scholar

[26]

S. L. XieH. R. Xu and J. P. Zeng, Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Linear Algebra Appl., 494 (2016), 1-10.  doi: 10.1016/j.laa.2016.01.002.  Google Scholar

[27]

Z. Xia and C. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem, Appl. Math. Comput., 271 (2015), 34-42. doi: 10.1016/j.amc.2015.08.108.  Google Scholar

[28]

L. Yong, Nonlinear complementarity problem and solution methods, in Proceedings of the 2010 International Conference on Artificial Intelligence and Computational Intelligence, Part I, Springer-Verlag, (2010), 461–469.  Google Scholar

[29]

H. ZhengS. Vong and L. Liu, The relaxation modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems, Int. J. Comput. Math., 96 (2018), 1648-1667.  doi: 10.1080/00207160.2018.1504928.  Google Scholar

[30]

H. Zheng and S. Vong, The modulus-based nonsmooth Newtons method for solving a class of nonlinear complementarity problems of P-matrices, Calcolo, 55 (2018), 37. doi: 10.1007/s10092-018-0279-y.  Google Scholar

[31]

H. ZhengS. Vong and L. Liu, A direct preconditioned modulus-based iteration method for solving nonlinear complementarity problems of H-matrices, Appl. Math. Comput., 353 (2019), 396-405.  doi: 10.1016/j.amc.2019.02.015.  Google Scholar

[32]

J. J. Zhang, MSSOR-based alternating direction method for symmetric positive-definite linear complementarity problems, Numer. Algor., 68 (2015), 631-644.  doi: 10.1007/s11075-014-9864-6.  Google Scholar

[33]

J. J. ZhangJ. L. Zhang and W. Z. Ye, An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems, Numer. Algor., 78 (2018), 895-910.  doi: 10.1007/s11075-017-0405-y.  Google Scholar

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
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
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
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
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
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
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