Advertisement

Modified Newton-Type Iteration Methods for Generalized Absolute Value Equations

  • An Wang
  • Yang CaoEmail author
  • Jing-Xian Chen
Article

Abstract

In this paper, by separating the differential and the non-differential parts of the generalized absolute value equations, a class of modified Newton-type iteration methods are proposed. The modified Newton-type iteration method involves the well-known Picard iteration method as the special case. Convergence properties of the new iteration schemes are analyzed in detail. In particular, some specific sufficient conditions are presented for two special coefficient matrices. Finally, two numerical examples are given to illustrate the effectiveness of the proposed modified Newton-type iteration methods.

Keywords

Generalized absolute value equations Newton method Convergence Differential function 

Mathematics Subject Classification

65F10 90C05 90C30 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11771225, 71401082, 71771127) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_1905).

References

  1. 1.
    Rohn, J.: A theorem of the alternatives for the equation \(Ax+B|x|=b\). Linear Multilinear Algebra 52, 421–426 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Schäfur, U.: On : the modulus algorithm for the linear complementarity problem. Oper. Res. Lett. 32, 350–354 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bai, Z.Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bai, Z.Z., Zhang, L.L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Wu, S.L., Li, C.X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8, 35–44 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wu, S.L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169, 705–712 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Mangasarian, O.L.: Linear complementarity as absolute value equation solution. Optim. Lett. 8, 1529–1534 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Rohn, J.: An algorithm for solving the absolute value equation. Electron. J. Linear Algebra 18, 589–599 (2009)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim Lett. 8, 2191–2202 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Prokopyev, O.A.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44, 363–372 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9, 1469–1474 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Caccetta, L., Qu, B., Zhou, G.L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lian, Y.Y., Li, C.X., Wu, S.L.: Weaker convergent results of the generalized Newton method for the generalized absolute value equations. J. Comput. Appl. Math. 338, 221–226 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Zainali, N., Lotfi, T.: On developing a stable and quadratic convergent method for solving absolute value equation. J. Comput. Appl. Math. 330, 742–747 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Haghani, F.K.: On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. 166, 619–625 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Li, C.X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170, 1055–1059 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Bello Cruz, J.Y., Ferreira, O.P., Prudente, L.F.: On the global convergence of the inexact semi-smooth Newton method for absolute value equation. Comput. Optim. Appl. 65, 93–108 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Han, D.F.: The majorant method and convergence for solving nondifferentiable equations in Banach space. Appl. Math. Comput. 118, 73–82 (2001)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Bai, Z.Z., Yang, X.: On HSS-based iteration methods for weakly nonlinear systems. Appl. Numer. Math. 59, 2923–2936 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141–152 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Maryland (2009)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongPeople’s Republic of China
  2. 2.School of TransportationNantong UniversityNantongPeople’s Republic of China
  3. 3.School of BusinessNantong UniversityNantongPeople’s Republic of China

Personalised recommendations