Advertisement

Calcolo

, 55:37 | Cite as

The modulus-based nonsmooth Newton’s method for solving a class of nonlinear complementarity problems of P-matrices

  • Hua Zheng
  • Seakweng Vong
Article
  • 61 Downloads

Abstract

By applying the Newton’s iteration to the equivalent modulus equations of the nonlinear complementarity problems of P-matrices, a modulus-based nonsmooth Newton’s method is established. The nearly quadratic convergence of the new method is proved under some assumptions. The strategy of choosing the initial iteration vector is given, which leads to a modified method. Numerical examples show that the new methods have higher convergence precision and faster convergence rate than the known modulus-based matrix splitting iteration method.

Keywords

Nonlinear complementarity problem Modulus-based method Nonsmooth Newton’s method P-matrix 

Mathematics Subject Classification

90C33 65F10 

Notes

Acknowledgements

The authors would like to thank the referee for the helpful comments. The work was supported by the National Natural Science Foundation of China (Grant No. 11601340), University of Macau (Grant No. MYRG2017-00098-FST), Macao Science and Technology Development Fund (Grant No. 050/2017/A), the Opening Project of Guangdong Provincial Engineering Technology Research Center for Data Sciences(Grant No. 2016KF11) and Science and Technology Planning Project of Shaoguan(Grant No. SHAOKE [2016]44/15).

References

  1. 1.
    Bai, Z.-Z.: On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J. Matrix Anal. Appl. 21, 67–78 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms 62, 59–77 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  6. 6.
    Berman, A., Plemmons, R.J.: Nonnegative Matrix in the Mathematical Sciences. SIAM Publisher, Philadelphia (1994)CrossRefGoogle Scholar
  7. 7.
    Chen, X.-J.: Smoothing methods for complementarity problems and their applications: a survey. J. Oper. Res. Soc. Jpn. 43, 32–47 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, X.-J., Ye, Y.-Y.: On homotopy-smoothing methods for variational inequalities. SIAM J. Control Optim. 37, 587–616 (1999)MathSciNetGoogle Scholar
  9. 9.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  10. 10.
    Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, SanDiego (1992)zbMATHGoogle Scholar
  11. 11.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)CrossRefGoogle Scholar
  12. 12.
    Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129–143 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Elliott, C.M., Ockendon, I.R.: Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, vol. 59. Pitman, London (1982)zbMATHGoogle Scholar
  14. 14.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II. Springer, New York (2003)zbMATHGoogle Scholar
  15. 15.
    Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hadjidimos, A., Tzoumas, M.: Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem. Linear Algebra Appl. 431, 197–210 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Harker, P.T., Pang, J.-S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hoffmann, K.H., Zou, J.: Parallel solution of variational inequality problems with nonlinear source terms. IMA J. Numer. Anal. 16, 31–45 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Huang, N., Ma, C.-F.: The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems. Numer. Linear Algebra Appl. 23, 558–569 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Harker, P., Xiao, B.-C.: Newton’s Method for the nonlinear complementarity problem: a B-differentiable equation approach. Math. Program. 48, 339–357 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, R., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems. Numer. Algorithms 75, 339–358 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of \(H\)-matrices. Appl. Math. Lett. 26, 1159–1164 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, W., Zheng, H.: A preconditioned modulus-based iteration method for solving linear complementarity problems of \(H\)-matrices. Linear Multilinear Algebra 64, 1390–1403 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, S.-M., Zheng, H., Li, W.: A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. Calcolo 53, 189–199 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Luca, T.D., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ma, C.-F., Huang, N.: Modified modulus-based matrix splitting algorithms for aclass of weakly nondifferentiable nonlinear complementarity problems. Appl. Numer. Math. 108, 116–124 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Meyer, G.H.: Free boundary problems with nonlinear source terms. Numer. Math. 43, 463–482 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988)zbMATHGoogle Scholar
  29. 29.
    Noor, M.A.: Fixed point approach for complementarity problems. J. Comput. Appl. Math. 133, 437–448 (1988)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Qi, L.-Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sun, Z., Zeng, J.-P.: A monotone semismooth Newton type method for a class of complementarity problems. J. Comput. Appl. Math. 235, 1261–1274 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    van Bokhoven, W.M.G.: Piecewise-linear Modelling and Analysis. Proefschrift, Eindhoven (1981)Google Scholar
  33. 33.
    Wen, B.-L., Zheng, H., Li, W., Peng, X.-F.: The relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems of positive definite matrices. Appl. Math. Comput. 321, 349–357 (2018)MathSciNetGoogle Scholar
  34. 34.
    Xia, Z.-C., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem. Appl. Math. Comput. 271, 34–42 (2015)MathSciNetGoogle Scholar
  35. 35.
    Xie, S.-L., Xu, H.-R., Zeng, J.P.: Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Linear Algebra Appl. 494, 1–10 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhang, L.-L.: Two-step modulus based matrix splitting iteration for linear complementarity problems. Numer. Algorithms 57, 83–99 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, L.-L.: Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems. J. Optim. Theory Appl. 160, 189–203 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhang, L.-L.: Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems. J. Comput. Math. 33, 100–112 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhang, L.-L., Ren, Z.-R.: Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl. Math. Lett. 26, 638–642 (2013)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zheng, H.: Improved convergence theorems of modulus-based matrix splitting iteration method for nonlinear complementarity problems of \(H\)-matrices. Calcolo 54, 1481–1490 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Zheng, H., Li, W.: The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J. Comput. Appl. Math. 288, 116–126 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zheng, H., Li, W., Vong, S.: A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems. Numer. Algorithms 74, 137–152 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zheng, H., Liu, L.: A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems of \(H_+\)-matrices. Comput. Appl. Math.  https://doi.org/10.1007/s40314-018-0646-y MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zheng, H., Vong, S.: Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems. Linear Multilinear Algebra  https://doi.org/10.1080/03081087.2018.1470602
  45. 45.
    Zheng, H., Vong, S., Liu, L.: The relaxation modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems. Int. J. Comput. Math.  https://doi.org/10.1080/00207160.2018.1504928
  46. 46.
    Zheng, N., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Algorithms 64, 245–262 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShaoguan UniversityShaoguanPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MacauMacauPeople’s Republic of China

Personalised recommendations