Search Advanced

AIMS Mathematics

2024, Volume 9, Issue 8: 22565-22577. doi: 10.3934/math.20241098
Previous Article Next Article
Research article Special Issues

The development of new efficient iterative methods for the solution of absolute value equations

  • 1.
    School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
  • 2.
    Department of Quantitative analysis, College of Business Administrations, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia
  • Received: 22 May 2024 Revised: 02 July 2024 Accepted: 16 July 2024 Published: 22 July 2024

  • MSC : 49M20, 90C33

  • The use of absolute value equations (AVEs) is widespread across a wide range of fields, including scientific computing, management science, and engineering. Our aim in this study is to introduce two new methods for solving AVEs and to explore their convergence characteristics. Furthermore, numerical experiments will be carried out to demonstrate their feasibility, robustness, and efficacy.

    Citation: Rashid Ali, Fuad A. Awwad, Emad A. A. Ismail. The development of new efficient iterative methods for the solution of absolute value equations[J]. AIMS Mathematics, 2024, 9(8): 22565-22577. doi: 10.3934/math.20241098

    Related Papers:

  • The use of absolute value equations (AVEs) is widespread across a wide range of fields, including scientific computing, management science, and engineering. Our aim in this study is to introduce two new methods for solving AVEs and to explore their convergence characteristics. Furthermore, numerical experiments will be carried out to demonstrate their feasibility, robustness, and efficacy.



    加载中


    [1] J. Rohn, A theorem of the alternatives for the equation $Ax + B|x| = b $, Linear Multilinear A., 52 (2004), 421–426. http://dx.doi.org/10.1080/0308108042000220686 doi: 10.1080/0308108042000220686
    [2] O. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43–53. http://dx.doi.org/10.1007/s10589-006-0395-5 doi: 10.1007/s10589-006-0395-5
    [3] O. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108. http://dx.doi.org/10.1007/s11590-008-0094-5 doi: 10.1007/s11590-008-0094-5
    [4] O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363–372. http://dx.doi.org/10.1007/s10589-007-9158-1 doi: 10.1007/s10589-007-9158-1
    [5] R. Ali, Z. Zhang, F. Awwad, The study of new fixed-point iteration schemes for solving absolute value equations, Heliyon, in press. http://dx.doi.org/10.1016/j.heliyon.2024.e34505
    [6] S. Hu, Z. Huang, A note on absolute value equations, Optim. Lett., 4 (2010), 417–424. http://dx.doi.org/10.1007/s11590-009-0169-y doi: 10.1007/s11590-009-0169-y
    [7] O. Mangasarian, R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359–367. http://dx.doi.org/10.1016/j.laa.2006.05.004 doi: 10.1016/j.laa.2006.05.004
    [8] O. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3–8. http://dx.doi.org/10.1007/s11590-006-0005-6 doi: 10.1007/s11590-006-0005-6
    [9] R. Ali, A. Akgül, A new matrix splitting generalized iteration method for linear complementarity problems, Appl. Math. Comput., 464 (2024), 128378. http://dx.doi.org/10.1016/j.amc.2023.128378 doi: 10.1016/j.amc.2023.128378
    [10] N. Yilmaz, A. Sahiner, Smoothing techniques in solving non-Lipschitz absolute value equations, Int. J. Comput. Math., 100 (2023), 867–879. http://dx.doi.org/10.1080/00207160.2022.2163388 doi: 10.1080/00207160.2022.2163388
    [11] R. Ali, A. Ali, The matrix splitting fixed point iterative algorithms for solving absolute value equations, Asian-Eur. J. Math., 16 (2023), 2350106. http://dx.doi.org/10.1142/S1793557123501061 doi: 10.1142/S1793557123501061
    [12] H. Zhou, S. Wu, C. Li, Newton-based matrix splitting method for generalized absolute value equation, J. Comput. Appl. Math., 394 (2021), 113578. http://dx.doi.org/10.1016/j.cam.2021.113578 doi: 10.1016/j.cam.2021.113578
    [13] D. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2016), 2191–2202. http://dx.doi.org/10.1007/s11590-014-0727-9 doi: 10.1007/s11590-014-0727-9
    [14] A. Khan, J. Iqbal, A. Akgul, R. Ali, Y. Du, A. Hussain, et al., A Newton-type technique for solving absolute value equations, Alex. Eng. J., 64 (2023), 291–296. http://dx.doi.org/10.1016/j.aej.2022.08.052 doi: 10.1016/j.aej.2022.08.052
    [15] M. Noor, J. Iqbal, S. Khattri, E. Al-Said, A new iterative method for solving absolute value equations, Int. J. Phys. Sci., 6 (2011), 1793–1797. http://dx.doi.org/10.5897/IJPS11.244 doi: 10.5897/IJPS11.244
    [16] Y. Ke, C. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195–202. http://dx.doi.org/10.1016/j.amc.2017.05.035 doi: 10.1016/j.amc.2017.05.035
    [17] X. Dong, X. Shao, H. Shen, A new SOR-like method for solving absolute value equations, Appl. Numer. Math., 156 (2020), 410–421. http://dx.doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
    [18] J. Tang, Inexact Newton-type method for solving large-scale absolute value equation $ Ax-|x| = b $, Appl. Math., 69 (2024), 49–66. http://dx.doi.org/10.21136/AM.2023.0171-22 doi: 10.21136/AM.2023.0171-22
    [19] 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. http://dx.doi.org/10.1007/s11590-012-0560-y doi: 10.1007/s11590-012-0560-y
    [20] J. Tang, J. Zhou, A quadratically convergent descent method for the absolute value equation $Ax+B|x| = b$, Oper. Res. Lett., 47 (2019), 229–234. http://dx.doi.org/10.1016/j.orl.2019.03.014 doi: 10.1016/j.orl.2019.03.014
    [21] L. Caccetta, B. Qu, G. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45–58. http://dx.doi.org/10.1007/s10589-009-9242-9 doi: 10.1007/s10589-009-9242-9
    [22] C. Zhang, Q. Wei, Global and finite convergence of a generalized Newton method for absolute value equations, J. Optim. Theory Appl., 143 (2009), 391–403. http://dx.doi.org/10.1007/s10957-009-9557-9 doi: 10.1007/s10957-009-9557-9
    [23] A. Wang, Y. Cao, J. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 181 (2019), 216–230. http://dx.doi.org/10.1007/s10957-018-1439-6 doi: 10.1007/s10957-018-1439-6
    [24] Y. Lian, C. Li, S. Wu, Weaker convergent results of the generalized Newton method for the generalized absolute value equations, J. Comput. Appl. Math., 338 (2018), 221–226. http://dx.doi.org/10.1016/j.cam.2017.12.027 doi: 10.1016/j.cam.2017.12.027
    [25] Y. Cao, Q. Shi, S. Zhu, A relaxed generalized Newton iteration method for generalized absolute value equations, AIMS Mathematics, 6 (2020), 1258–1275. http://dx.doi.org/10.3934/math.2021078 doi: 10.3934/math.2021078
    [26] C. Zhou, Y. Cao, Q. Shen, Q. Shi, A modified Newton-based matrix splitting iteration method for generalized absolute value equations, J. Comput. Appl. Math., 442 (2024), 115747. http://dx.doi.org/10.1016/j.cam.2023.115747 doi: 10.1016/j.cam.2023.115747
    [27] X. Lv, S. Miao, An inexact fixed point iteration method for solving absolute value equation, Japan J. Indust. Appl. Math., 41 (2024), 1137–1148. http://dx.doi.org/10.1007/s13160-023-00641-3 doi: 10.1007/s13160-023-00641-3
    [28] X. Lv, S. Miao, A new inexact fixed point iteration method for solving tensor absolute value equation, Appl. Math. Lett., 154 (2024), 109109. http://dx.doi.org/10.1016/j.aml.2024.109109 doi: 10.1016/j.aml.2024.109109
    [29] J. Zhang, S. Miao, A general fast shift-splitting iteration method for nonsymmetric saddle point problems, Comp. Appl. Math., 40 (2021), 229. http://dx.doi.org/10.1007/s40314-021-01618-z doi: 10.1007/s40314-021-01618-z
    [30] S. Wu, C. Li, A special shift splitting iteration method for absolute value equation, AIMS Mathematics, 5 (2020), 5171–5183. http://dx.doi.org/10.3934/math.2020332 doi: 10.3934/math.2020332
    [31] V. Edalatpour, D. Hezari, D. Salkuyeh, A generalization of the Gauss-Seidel iteration method for solving absolute value equations, Appl. Math. Comput., 293 (2017), 156–167. http://dx.doi.org/10.1016/j.amc.2016.08.020 doi: 10.1016/j.amc.2016.08.020
    [32] R. Varga, Matrix iterative analysis, Heidelberg: Springer, 1999. http://dx.doi.org/10.1007/978-3-642-05156-2
    [33] A. Fakharzadeh, N. Shams, An efficient algorithm for solving absolute value equations, J. Math. Ext., 15 (2021), 1–23. http://dx.doi.org/10.30495/JME.2021.1393 doi: 10.30495/JME.2021.1393
    [34] C. Li, A preconditioned AOR iterative method for the absolute value equations, Int. J. Comput. Meth., 14 (2017), 1750016. http://dx.doi.org/10.1142/S0219876217500165 doi: 10.1142/S0219876217500165
    [35] R. Ali, K. Pan, The solution of a type of absolute value equations using two new matrix splitting iterative techniques, Port. Math., 79 (2022), 241–252. http://dx.doi.org/10.4171/PM/2089 doi: 10.4171/PM/2089
    [36] Z. Yu, L. Li, Y. Yuan, A modified multivariate spectral gradient algorithm for solving absolute value equations, Appl. Math. Lett., 121 (2021), 107461. http://dx.doi.org/10.1016/j.aml.2021.107461 doi: 10.1016/j.aml.2021.107461
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(494) PDF downloads(42) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog