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AIMS Mathematics

2021, Volume 6, Issue 2: 1258-1275. doi: 10.3934/math.2021078
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Research article

A relaxed generalized Newton iteration method for generalized absolute value equations

  • School of Transportation and Civil Engineering, Nantong University, Nantong, 226019, P. R. China
  • Received: 02 September 2020 Accepted: 10 November 2020 Published: 12 November 2020

  • MSC : 65F10

  • To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B|x| = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.

    Citation: Yang Cao, Quan Shi, Sen-Lai Zhu. A relaxed generalized Newton iteration method for generalized absolute value equations[J]. AIMS Mathematics, 2021, 6(2): 1258-1275. doi: 10.3934/math.2021078

    Related Papers:

  • To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B|x| = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.


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    [1] J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b, Linear Multilinear Algebra, 52 (2004), 421-426. doi: 10.1080/0308108042000220686
    [2] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43-53.
    [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. doi: 10.1002/nla.609
    [4] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917-933. doi: 10.1002/nla.680
    [5] R. W. Cottle, J. S. Pang, R. E. Stone, The linear complementarity problem, SIAM: Philadelphia, 2009.
    [6] O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359-367.
    [7] U. Schäfur, On the modulus algorithm for the linear complementarity problem, Oper. Res. Lett., 32 (2004), 350-354.
    [8] Z. Z. Bai, L. L. Zhang, Modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 20 (2013), 425-439. doi: 10.1002/nla.1835
    [9] N. Zheng, J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algor., 64 (2013), 245-262. doi: 10.1007/s11075-012-9664-9
    [10] J. Rohn, On unique solvability of the absolute value equation, Optim. Lett., 3 (2009), 603-606. doi: 10.1007/s11590-009-0129-6
    [11] S. L. Wu, C. X. Li, The unique solution of the absolute value equations, Appl. Math. Lett., 76 (2018), 195-200. doi: 10.1016/j.aml.2017.08.012
    [12] S. L. Wu, C. X. Li, A note on unique solvability of the absolute value equation, Optim. Lett., 14 (2020), 1957-1960. doi: 10.1007/s11590-019-01478-x
    [13] 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. doi: 10.1007/s11590-012-0560-y
    [14] S. L. Wu, P. Guo, On the unique solvability of the absolute value equation, J. Optim. Theory Appl., 169 (2016), 705-712. doi: 10.1007/s10957-015-0845-2
    [15] O. L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3-8.
    [16] O. L. Mangasarian, Linear complementarity as absolute value equation solution, Optim. Lett., 8 (2014), 1529-1534. doi: 10.1007/s11590-013-0656-z
    [17] J. Rohn, An algorithm for solving the absolute value equation, Electron. J. Linear Algebra, 18 (2009), 589-599.
    [18] O. A. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363-372. doi: 10.1007/s10589-007-9158-1
    [19] O. L. Mangasarian, A hybrid algorithm for solving the absolute value equation, Optim. Lett., 9 (2015), 1469-1474. doi: 10.1007/s11590-015-0893-4
    [20] C. X. Li, A preconditioned AOR iterative method for the absolute value equations, Int. J. Comput. Meth., 14 (2017), 1750016. doi: 10.1142/S0219876217500165
    [21] D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2016), 2191-2202.
    [22] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101-108. doi: 10.1007/s11590-008-0094-5
    [23] C. Zhang, Q. J. Wei, Global and finite convergence of a generalized Newton method for absolute value equations, J. Optim. Theory Appl., 143 (2009), 391-403. doi: 10.1007/s10957-009-9557-9
    [24] 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. doi: 10.1007/s10957-018-1439-6
    [25] S. L. Hu, Z. H. Huang, Q. Zhang, A generalized Newton method for absolute value equations associated with second order cones, J. Comput. Appl. Math., 235 (2011), 1490-1501. doi: 10.1016/j.cam.2010.08.036
    [26] L. Caccetta, B. Qu, G. L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45-58. doi: 10.1007/s10589-009-9242-9
    [27] Y. Y. Lian, C. X. Li, S. L. Wu, Weaker convergent results of the generalized Newton method for the generalized absolute value equations, J. Comput. Appl. Math., 338 (2018), 221-226.
    [28] N. Zainali, T. Lotfi, On developing a stable and quadratic convergent method for solving absolute value equation, J. Comput. Appl. Math., 330 (2018), 742-747. doi: 10.1016/j.cam.2017.07.009
    [29] F. K. Haghani, On generalized Traub's method for absolute value equations, J. Optim. Theory Appl., 166 (2015), 619-625. doi: 10.1007/s10957-015-0712-1
    [30] C. X. Li, A modified generalized Newton method for absolute value equations, J. Optim. Theory Appl., 170 (2016), 1055-1059. doi: 10.1007/s10957-016-0956-4
    [31] J. Y. Bello Cruz, O. P. Ferreira, L. F. Prudente, On the global convergence of the inexact semismooth Newton method for absolute value equation, Comput. Optim. Appl., 65 (2016), 93-108. doi: 10.1007/s10589-016-9837-x
    [32] J. M. Feng, S. Y. Liu, A new two-step iterative method for solving absolute value equations, J. Inequal. Appl., 2019 (2019), 39. doi: 10.1186/s13660-019-1969-y
    [33] G. H. Golub, C. F. Van Loan, Matrix computations, 3 Eds., Maryland: The Johns Hopkins University Press, 2009.
    [34] G. Ramadurai, S. V. Ukkusuri, J. Y. Zhao, J. S. Pang, Linear complementarity formulation for single bottleneck model with heterogeneous commuters, Transport. Res. Part B, 44 (2010), 193-214. doi: 10.1016/j.trb.2009.07.005
    [35] R. Li, J. F. Yin, On the convergence of modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems with H+-matrices, J. Comput. Appl. Math., 342 (2018), 202-209. doi: 10.1016/j.cam.2017.12.029
    [36] A. Wang, Y. Cao, Q. Shi, Convergence analysis of modulus-based matrix splitting iterative methods for implicit complementarity problems, J. Inequal. Appl., 2018 (2018), 2. doi: 10.1186/s13660-017-1593-7
    [37] Y. Cao, A. Wang, Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems, Numer. Algor., 82 (2019), 1377-1394. doi: 10.1007/s11075-019-00660-7
    [38] S. L. Wu, P. Guo, Modulus-based matrix splitting algorithms for quasi-complementarity problems, Appl. Numer. Math., 132 (2018), 127-137. doi: 10.1016/j.apnum.2018.05.017
    [39] Q. Shi, Q. Q. Shen, T. P. Tang, A class of two-step modulus-basedmatrix splitting iteration methods for quasi-complementarity problems, Comput. Appl. Math., 39 (2020), 11. doi: 10.1007/s40314-019-0984-4
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