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AIMS Mathematics

2023, Volume 8, Issue 2: 4714-4725. doi: 10.3934/math.2023233
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Relaxed modified Newton-based iteration method for generalized absolute value equations

  • Department of Mathematics, College of Sciences, Northeastern University, Shenyang 110819, China
  • Received: 22 September 2022 Revised: 03 November 2022 Accepted: 18 November 2022 Published: 07 December 2022

  • MSC : 65F10, 90C05, 90C30

  • Many problems in different fields may lead to solutions of absolute value equations, such as linear programming problems, linear complementarity problems, quadratic programming, mixed integer programming, the bimatrix game and so on. In this paper, by introducing a nonnegative real parameter to the modified Newton-based iteration scheme, we present a new relaxed modified Newton-based (RMN) iteration method for solving generalized absolute value equations. The famous Picard iteration method and the modified Newton-type iteration method are the exceptional cases of the RMN iteration method. The convergence property of the new method is discussed. Finally, the validity and feasibility of the RMN iteration method are verified by experimental examples.

    Citation: Xin-Hui Shao, Wan-Chen Zhao. Relaxed modified Newton-based iteration method for generalized absolute value equations[J]. AIMS Mathematics, 2023, 8(2): 4714-4725. doi: 10.3934/math.2023233

    Related Papers:

  • Many problems in different fields may lead to solutions of absolute value equations, such as linear programming problems, linear complementarity problems, quadratic programming, mixed integer programming, the bimatrix game and so on. In this paper, by introducing a nonnegative real parameter to the modified Newton-based iteration scheme, we present a new relaxed modified Newton-based (RMN) iteration method for solving generalized absolute value equations. The famous Picard iteration method and the modified Newton-type iteration method are the exceptional cases of the RMN iteration method. The convergence property of the new method is discussed. Finally, the validity and feasibility of the RMN iteration method are verified by experimental examples.


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    [1] J. Rohn, A theorem of the alternatives for the equation Ax+B|x| = b, Linear Multilinear A., 52 (2004), 421–426. https://doi.org/10.1080/0308108042000220686 doi: 10.1080/0308108042000220686
    [2] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43–53. https://doi.org/10.1007/s10589-006-0395-5 doi: 10.1007/s10589-006-0395-5
    [3] J. L. Dong, M. Q. Jiang, A modified modulus method for symmetric positive‐definite linear complementarity problems, Numer. Linear Algebra Appl., 16 (2009), 129–143. https://doi.org/10.1002/nla.609 doi: 10.1002/nla.609
    [4] O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359–367. https://doi.org/10.1016/j.laa.2006.05.004 doi: 10.1016/j.laa.2006.05.004
    [5] L. Abdallah, M. Haddou, T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196–207. https://doi.org/10.1016/j.cam.2017.06.019 doi: 10.1016/j.cam.2017.06.019
    [6] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. https://doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [7] J. Rohn, Systems of linear interval equations, Linear Algebra Appl., 126 (1989), 39–78. https://doi.org/10.1016/0024-3795(89)90004-9 doi: 10.1016/0024-3795(89)90004-9
    [8] L. Caccetta, B. Qu, G. L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45–58. https://doi.org/10.1007/s10589-009-9242-9 doi: 10.1007/s10589-009-9242-9
    [9] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108. https://doi.org/10.1007/s11590-008-0094-5 doi: 10.1007/s11590-008-0094-5
    [10] A. Wang, Y. Cao, J. X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory. Appl., 181 (2019), 216–230. https://doi.org/10.1007/s10957-018-1439-6 doi: 10.1007/s10957-018-1439-6
    [11] N. Zainali, T. Lotfifi, On developing a stable and quadratic convergent method for solving absolute value equation, J. Comput. Appl. Math., 330 (2018), 742–747. https://doi.org/10.1016/j.cam.2017.07.009 doi: 10.1016/j.cam.2017.07.009
    [12] H. Y. Zhou, S. L. Wu, C. X. Li, Newton-based matrix splitting method for generalized absolute value equation, J. Comput. Appl. Math., 394 (2021), 113578. https://doi.org/10.1016/j.cam.2021.113578 doi: 10.1016/j.cam.2021.113578
    [13] Y. Cao, Q. Shi, S. L. Zhu, A relaxed generalized Newton iteration method for generalized absolute value equations, AIMS Math., 6 (2021), 1258–1275. https://doi.org/10.3934/math.2021078 doi: 10.3934/math.2021078
    [14] R. Ali, I. Khan, A. Ali, Two new generalized iteration methods for solving absolute value equations using M-matrix, AIMS Math., 7 (2022), 8176–8187. https://doi.org/10.3934/math.2022455 doi: 10.3934/math.2022455
    [15] A. Khan, J. Iqbal, A. Akgül, R. Alia, Y. Du, A. Hussain, et al., A Newton-type technique for solving absolute value equations, Alexandria Eng. J., 2022. https://doi.org/10.1016/j.aej.2022.08.052 doi: 10.1016/j.aej.2022.08.052
    [16] R. Ali, K. Pan, The solution of the absolute value equations using two generalized accelerated overrelaxation methods, Asian-Eur. J. Math., 15 (2022), 2250154. https://doi.org/10.1142/S1793557122501546 doi: 10.1142/S1793557122501546
    [17] R. Ali, K. Pan, The new iteration methods for solving absolute value equations, Appl. Math., 2021, 1–14. https://doi.org/10.21136/AM.2021.0055-21 doi: 10.21136/AM.2021.0055-21
    [18] R. Ali, K. Pan, Two new fixed point iterative schemes for absolute value equations, Japan J. Indust. Appl. Math., 2022, 1–12. https://doi.org/10.1007/s13160-022-00526-x doi: 10.1007/s13160-022-00526-x
    [19] P. Guo, S. L Wu, C. X Li, On the SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107–113. https://doi.org/10.1016/j.aml.2019.03.033 doi: 10.1016/j.aml.2019.03.033
    [20] X. Dong, X. H. Shao, H. L. Shen, A new SOR-like method for solving absolute value equations, Appl. Numer. Math., 156 (2020), 410–421. https://doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
    [21] Y. F. Ke, C. F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195–202. https://doi.org/10.1016/j.amc.2017.05.035 doi: 10.1016/j.amc.2017.05.035
    [22] C. R. Chen, Y. N. Yang, D. M. Yu, D. R. Han, An inverse-free dynamical system for solving the absolute value equations, Appl. Numer. Math., 168 (2021), 170–181. https://doi.org/10.1016/j.apnum.2021.06.002 doi: 10.1016/j.apnum.2021.06.002
    [23] A. Mansoori, M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 333 (2018), 28–35. https://doi.org/10.1016/j.cam.2017.09.032 doi: 10.1016/j.cam.2017.09.032
    [24] A. Mansoori, M. Eshaghnezhad, S. Effati, An efficient neural network model for solving the absolute value equations, IEEE T. Circuits-Ⅱ, 65 (2017), 391–395. https://doi.org/10.1109/TCSII.2017.2750065 doi: 10.1109/TCSII.2017.2750065
    [25] M. Dehghan, A. Shirilord, Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation, Appl. Numer. Math., 158 (2020), 425–438. https://doi.org/10.1016/j.apnum.2020.08.001 doi: 10.1016/j.apnum.2020.08.001
    [26] D. K. Salkuyeh, The Picard–HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191–2202. https://doi.org/10.1007/s11590-014-0727-9 doi: 10.1007/s11590-014-0727-9
    [27] J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett., 8 (2014), 35–44. https://doi.org/10.1007/s11590-012-0560-y doi: 10.1007/s11590-012-0560-y
    [28] D. F. Han, The majorant method and convergence for solving nondifferentiable equations in Banach space, Appl. Math. Comput., 118 (2001), 73–82. https://doi.org/10.1016/S0096-3003(99)00183-6 doi: 10.1016/S0096-3003(99)00183-6
    [29] G. H. Golub, C. F. Van Loan, Matrix computations, 3 Eds., Johns Hopkins University Press, 2013.
    [30] A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl., 119 (1989), 141–152. https://doi.org/10.1016/0024-3795(89)90074-8 doi: 10.1016/0024-3795(89)90074-8
    [31] S. L. Wu, C. X. Li, Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems, J. Comput. Appl. Math., 302 (2016), 327–339. https://doi.org/10.1016/j.cam.2016.02.011 doi: 10.1016/j.cam.2016.02.011
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