Skip to main content
SpringerLink
Log in

Block-Diagonal and Anti-block-Diagonal Splitting Iteration Method for Absolute Value Equation

  • Conference paper
  • First Online:
Simulation Tools and Techniques (SIMUtools 2020)

Included in the following conference series:

  • 898 Accesses

Abstract

In this paper, the absolute value equation (AVE) is equivalently reformulated as a nonlinear equation in the form of 2 times 2 blocks. A block diagonal inverse block diagonal iteration method based on block-diagonal and anti-block-diagonal splitting (BAS) is proposed. Theoretical analysis shows that BAS is convergent, and numerical experiments show that the method is effective.

This research was supported by National Natural Science Foundation of China (No.11961082).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Rohn, J.: A theorem of the alternatives for the equation \(Ax+B|x|=b\). Linear Multilinear A. 52, 421–426 (2004)

    Article  MathSciNet  Google Scholar 

  2. Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)

    Article  MathSciNet  Google Scholar 

  3. Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)

    Article  MathSciNet  Google Scholar 

  4. Wu, S.-L., Guo, P.: Modulus-based matrix splitting algorithms for the quasi-complementarity problems. Appl. Numer. Math. 132, 127–137 (2018)

    Article  MathSciNet  Google Scholar 

  5. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, San Diego (1992)

    MATH  Google Scholar 

  6. Rohn, J.: An algorithm for solving the absolute value equations. Electron. J. Linear Algebra. 18, 589–599 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optimization Letters 8(1), 35–44 (2012). https://doi.org/10.1007/s11590-012-0560-y

    Article  MathSciNet  MATH  Google Scholar 

  8. Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191–2202 (2014)

    Article  MathSciNet  Google Scholar 

  9. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)

    Article  MathSciNet  Google Scholar 

  10. Hu, S.-L., Huang, Z.-H., Zhang, Q.: A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235, 1490–1501 (2011)

    Article  MathSciNet  Google Scholar 

  11. Nguyena, C.T., Saheyab, B., Chang, Y.-L., Chen, J.-S.: Unified smoothing functions for absolute value equation associated with second-order cone. Appl. Numer. Math. 135, 206–227 (2019)

    Article  MathSciNet  Google Scholar 

  12. Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optimization Letters 9(7), 1469–1474 (2015). https://doi.org/10.1007/s11590-015-0893-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MathSciNet  Google Scholar 

  14. Bai, Z.-Z., Yang, X.: On HSS-based iteration methods for weakly nonlinear systems. Appl. Numer. Math. 59, 2923–2936 (2009)

    Article  MathSciNet  Google Scholar 

  15. Zhu, M.-Z., Zhang, G.-F., Liang, Z.-Z.: The nonlinear HSS-like iteration method for absolute value equations, arXiv.org:1403.7013v2 [math.NA] Jan 2, 2018

  16. Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31, 360–374 (2009)

    Article  MathSciNet  Google Scholar 

  17. Wu, S.-L., Li, C.-X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)

    Article  MathSciNet  Google Scholar 

  18. Wu, S.-L., Huang, T.-Z., Zhao, X.-L.: A modified SSOR iterative method for augmented systems. J. Comput. Appl. Math. 228, 424–433 (2009)

    Article  MathSciNet  Google Scholar 

  19. Wang, A.-X., Wang, H.-J., Deng, Y.-K.: Interval algorithm for absolute value equations. Centr. Eur. J. Math. 9, 1171–1184 (2011)

    Article  MathSciNet  Google Scholar 

  20. Ketabchi, S., Moosaei, H.: An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Comput. Math. Appl. 64, 1882–1885 (2012)

    Article  MathSciNet  Google Scholar 

  21. Zhang, C., Wei, Q.-J.: Global and finite convergence of a generalized newton method for absolute value equations. J. Optim. Theory. Appl. 143, 391–403 (2009)

    Article  MathSciNet  Google Scholar 

  22. Li, C.-X.: A preconditioned AOR iterative method for the absolute value equations. Int. J. Comput. Meth. (2017) https://doi.org/10.1142/S0219876217500165

  23. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    Google Scholar 

  24. Jiang, D.-D., Wang, Y.-Q., Lv, Z.-H., Wang, W.-J., Wang, H.-H.: An energy-efficient networking approach in cloud services for IIoT networks. IEEE J. Sel. Area. Comn. 38(5), 928–941 (2020)

    Article  Google Scholar 

  25. Jiang, D.-D., Wang, W.-J., Shi, L., Song, H.-B.: A compressive sensing-based approach to end-to-end network traffic reconstruction. IEEE T. Netw. Sci. Eng. 7(1), 507–519 (2020)

    Article  MathSciNet  Google Scholar 

  26. Jiang, D.-D., Huo, L.-W., Song, H.-B.: Rethinking behaviors and activities of base stations in mobile cellular networks based on big data analysis. IEEE T. Netw. Sci. Eng. 7(1), 80–90 (2020)

    Article  MathSciNet  Google Scholar 

  27. Jiang D.-D., Wang Y.-Q., Lv Z.-H., Qi S., Singh S.: Big data analysis based network behavior insight of cellular networks for industry 4.0 applications. IEEE T. Ind. Inform. 16(2), 1310–1320 (2020)

    Google Scholar 

  28. Jiang, D.-D., Huo, L.-W., Lv, Z.-H., Song, H.-B., Qin, W.-D.: A joint multi-criteria utility-based network selection approach for vehicle-to-infrastructure networking. IEEE T Intell. Transp. 19(10), 3305–3319 (2018)

    Article  Google Scholar 

  29. Jiang, D.-D., Huo, L.-W., Li, Y.: Fine-granularity inference and estimations to network traffic for SDN. Plos One. 13(5), 1–23 (2018)

    Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous referees for their constructive suggestions and helpful comments, which lead to significant improvement of the original manuscript of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Cui-Xia Li & Shi-Liang Wu

Authors
  1. Cui-Xia Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shi-Liang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Embry-Riddle Aeronautical University, Daytona Beach, FL, USA

    Houbing Song

  2. School of Astronautics and Aeronautic, UESTC, Chengdu, China

    Dingde Jiang

Rights and permissions

Reprints and permissions

Copyright information

© 2021 ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Li, CX., Wu, SL. (2021). Block-Diagonal and Anti-block-Diagonal Splitting Iteration Method for Absolute Value Equation. In: Song, H., Jiang, D. (eds) Simulation Tools and Techniques. SIMUtools 2020. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 369. Springer, Cham. https://doi.org/10.1007/978-3-030-72792-5_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-72792-5_45

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-72791-8

  • Online ISBN: 978-3-030-72792-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation