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Modified Newton-Type Iteration Methods for Generalized Absolute Value Equations

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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.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  4. Schäfur, U.: On : the modulus algorithm for the linear complementarity problem. Oper. Res. Lett. 32, 350–354 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Bai, Z.Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu, S.L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169, 705–712 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mangasarian, O.L.: Linear complementarity as absolute value equation solution. Optim. Lett. 8, 1529–1534 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim Lett. 8, 2191–2202 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  17. Prokopyev, O.A.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44, 363–372 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9, 1469–1474 (2015)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  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  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  25. Haghani, F.K.: On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. 166, 619–625 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, C.X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170, 1055–1059 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Han, D.F.: The majorant method and convergence for solving nondifferentiable equations in Banach space. Appl. Math. Comput. 118, 73–82 (2001)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141–152 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  31. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Maryland (2009)

    MATH  Google Scholar 

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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).

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Authors and Affiliations

  1. School of Science, Nantong University, Nantong, 226019, People’s Republic of China

    An Wang

  2. School of Transportation, Nantong University, Nantong, 226019, People’s Republic of China

    Yang Cao

  3. School of Business, Nantong University, Nantong, 226019, People’s Republic of China

    Jing-Xian Chen

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  1. An Wang
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  2. Yang Cao
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  3. Jing-Xian Chen
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Correspondence to Yang Cao.

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Wang, A., Cao, Y. & Chen, JX. Modified Newton-Type Iteration Methods for Generalized Absolute Value Equations. J Optim Theory Appl 181, 216–230 (2019). https://doi.org/10.1007/s10957-018-1439-6

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  • DOI: https://doi.org/10.1007/s10957-018-1439-6

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