Advertisement

Note on error bounds for linear complementarity problems of Nekrasov matrices

  • Chaoqian Li
  • Shaorong Yang
  • Hui Huang
  • Yaotang Li
  • Yimin WeiEmail author
Original Paper
  • 26 Downloads

Abstract

García-Esnaola and Peña (Numer. Algor. 67, 655–667, 2014) presented an error bound involving a parameter for linear complementarity problems of Nekrasov matrices. This bound is not effective in some cases because it tends to infinity when the involved parameter tends to zero. In this paper, the optimal value of this error bound is determined completely by using the monotonicity of functions of this parameter. Numerical examples are given to verify the corresponding results.

Keywords

Linear complementarity problems Error bounds Nekrasov matrices P-matrices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors would like to thank the associate editor Prof. Xiaojun Chen and anonymous referees for their valuable suggestions, and also thank Mengting Gan for her discussion on 32 cases.

The first author is supported partly by the Applied Basic Research Programs of Science and Technology Department of Yunnan Province [grant number 2018FB001], Outstanding Youth Cultivation Project for Yunnan Province [grant number 2018YDJQ021], Program for Excellent Young Talents in Yunnan University, and CAS ’Light of West China’ Program. His work was finished, while he was a visiting scholar at Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during 2017 and 2018.

The third author is supported partly by National Natural Science Foundations of China [grant number 11461080].

The fourth author is supported partly by National Natural Science Foundations of China [grant number 11861077].

The fifth author is supported partly by National Natural Science Foundations of China [grant number 11771099] and Shanghai Municipal Education Commission.

References

  1. 1.
    Berman, A., Plemmons, R.J.: Nonnegative Matrix in the Mathematical Sciences. SIAM Publisher, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, T.T., Li, W., Wu, X., Vong, S.: Error bounds for linear complementarity problems of M B-matrices. Numer. Algor. 70(2), 341–356 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, X.J., Xiang, S.H.: Computation of error bounds for P-matrix linear complementarity problems. Math. Program., Ser A 106, 513–525 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X.J., Xiang, S.H.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18, 1250–1265 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)zbMATHGoogle Scholar
  6. 6.
    Peña, J.M.: A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J. Matrix Anal. Appl. 22, 1027–1037 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dai, P.F.: Error bounds for linear complementarity problems of D B-matrices. Linear Algebra Appl. 434, 830–840 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dai, P.F., Li, Y.T., Lu, C.J.: Error bounds for linear complementarity problems for SB-matrices. Numer Algor. 61, 121–139 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems for B-matrices. Appl. Math Lett. 22, 1071–1075 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    García-Esnaola, M., Peña, J.M.: A comparison of error bounds for linear complementarity problems of H-matrices. Linear Algebra Appl. 433, 956–964 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems of Nekrasov matrices. Numer Algor. 67, 655–667 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    García-Esnaola, M., Peña, J.M.: B-Nekrasov matrices and error bounds for linear complementarity problems. Numer Algor. 72, 435–445 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    García-Esnaola, M., Peña, J.M.: On the asymptotic optimality of error bounds for some linear complementarity problems. Numer. Algor. 80, 521–532 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, W.: On Nekrasov matrices. Linear Algebra Appl. 281, 87–96 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, C.Q., Li, Y.T.: Note on error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 57, 108–113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, C.Q., Dai, P.F., Li, Y.T.: New error bounds for linear complementarity problems of Nekrasov matrices and B-Nekrasov matrices. Numer. Algor. 74, 997–1009 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988)zbMATHGoogle Scholar
  18. 18.
    Wang, F., Sun, D.: New error bound for linear complementarity problems for B-matrices. Linear and Multilinear Algebra 66(11), 2156–2167 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsYunnan UniversityKunmingPeople’s Republic of China
  2. 2.College of Arts and SciencesYunnan Normal UniversityKunmingPeople’s Republic of China
  3. 3.School of Mathematics Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China

Personalised recommendations