Inexact Newton-type method for solving large-scale absolute value equation A x - | x | = b

Jingyong Tang

Applications of Mathematics (2024)

  • Issue: 1, page 49-66
  • ISSN: 0862-7940

Abstract

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Newton-type methods have been successfully applied to solve the absolute value equation A x - | x | = b (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line search technique, which is well-defined and easy to implement. We prove that the proposed method has global and local superlinear convergence under the condition that the interval matrix [ A - I , A + I ] is regular. This condition is much weaker than those used in some Newton-type methods. Numerical results show that our method has fairly good practical efficiency for solving large-scale AVEs.

How to cite

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Tang, Jingyong. "Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$." Applications of Mathematics (2024): 49-66. <http://eudml.org/doc/299200>.

@article{Tang2024,
abstract = {Newton-type methods have been successfully applied to solve the absolute value equation $Ax-|x| = b$ (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line search technique, which is well-defined and easy to implement. We prove that the proposed method has global and local superlinear convergence under the condition that the interval matrix $[A - I,A + I]$ is regular. This condition is much weaker than those used in some Newton-type methods. Numerical results show that our method has fairly good practical efficiency for solving large-scale AVEs.},
author = {Tang, Jingyong},
journal = {Applications of Mathematics},
keywords = {absolute value equation; inexact Newton method; regularity of interval matrices; superlinear convergence},
language = {eng},
number = {1},
pages = {49-66},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$},
url = {http://eudml.org/doc/299200},
year = {2024},
}

TY - JOUR
AU - Tang, Jingyong
TI - Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 49
EP - 66
AB - Newton-type methods have been successfully applied to solve the absolute value equation $Ax-|x| = b$ (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line search technique, which is well-defined and easy to implement. We prove that the proposed method has global and local superlinear convergence under the condition that the interval matrix $[A - I,A + I]$ is regular. This condition is much weaker than those used in some Newton-type methods. Numerical results show that our method has fairly good practical efficiency for solving large-scale AVEs.
LA - eng
KW - absolute value equation; inexact Newton method; regularity of interval matrices; superlinear convergence
UR - http://eudml.org/doc/299200
ER -

References

top
  1. Arias, C. A., Martínez, H. J., Pérez, R., 10.1016/j.apnum.2019.11.002, Appl. Numer. Math. 150 (2020), 559-575. (2020) Zbl1434.65073MR4046472DOI10.1016/j.apnum.2019.11.002
  2. Caccetta, L., Qu, B., Zhou, G., 10.1007/s10589-009-9242-9, Comput. Optim. Appl. 48 (2011), 45-58. (2011) Zbl1230.90195MR2762957DOI10.1007/s10589-009-9242-9
  3. Chen, C., Yu, D., Han, D., 10.1093/imanum/drab105, IMA J. Numer. Anal. 43 (2023), 1036-1060. (2023) Zbl07673881MR4568439DOI10.1093/imanum/drab105
  4. Haghani, F. K., 10.1007/s10957-015-0712-1, J. Optim. Theory Appl. 166 (2015), 619-625. (2015) Zbl1391.65106MR3371392DOI10.1007/s10957-015-0712-1
  5. Iqbal, J., Iqbal, A., Arif, M., 10.1016/j.cam.2014.11.062, J. Comput. Appl. Math. 282 (2015), 134-138. (2015) Zbl1309.65057MR3313095DOI10.1016/j.cam.2014.11.062
  6. Jiang, X., Zhang, Y., 10.3934/jimo.2013.9.789, J. Ind. Manag. Optim. 9 (2013), 789-798. (2013) Zbl1281.90023MR3119087DOI10.3934/jimo.2013.9.789
  7. Kumar, S., Deepmala, 10.1007/s40009-022-01193-9, Natl. Acad. Sci. Lett. 46 (2023), 129-131. (2023) MR4565846DOI10.1007/s40009-022-01193-9
  8. Kumar, S., Deepmala, 10.2298/YJOR220515036K, (to appear) in Yugosl. J. Oper. Res. MR4633526DOI10.2298/YJOR220515036K
  9. Li, D., Fukushima, M., 10.1080/10556780008805782, Optim. Methods Softw. 13 (2000), 181-201. (2000) Zbl0960.65076MR1785195DOI10.1080/10556780008805782
  10. Mangasarian, O. L., 10.1007/s11590-008-0094-5, Optim. Lett. 3 (2009), 101-108. (2009) Zbl1154.90599MR2453508DOI10.1007/s11590-008-0094-5
  11. Mangasarian, O. L., 10.1007/s11590-011-0347-6, Optim. Lett. 6 (2012), 1527-1533. (2012) Zbl1259.90065MR2980561DOI10.1007/s11590-011-0347-6
  12. Mangasarian, O. L., 10.1007/s11590-015-0893-4, Optim. Lett. 9 (2015), 1469-1474. (2015) Zbl1332.90215MR3396552DOI10.1007/s11590-015-0893-4
  13. Mangasarian, O. L., Meyer, R. R., 10.1016/j.laa.2006.05.004, Linear Algebra Appl. 419 (2006), 359-367. (2006) Zbl1172.15302MR2277975DOI10.1016/j.laa.2006.05.004
  14. Ni, T., Gu, W.-Z., Zhai, J., 10.1016/j.neucom.2013.09.047, Neurocomputing 129 (2014), 127-135. (2014) DOI10.1016/j.neucom.2013.09.047
  15. Noor, M. A., Iqbal, J., Noor, K. I., Al-Said, E., 10.1007/s11590-011-0332-0, Optim. Lett. 6 (2012), 1027-1033. (2012) Zbl1254.90149MR2925637DOI10.1007/s11590-011-0332-0
  16. Qi, L., Sun, D., Zhou, G., 10.1007/s101079900127, Math. Program. 87 (2000), 1-35. (2000) Zbl0989.90124MR1734657DOI10.1007/s101079900127
  17. Qi, L., Sun, J., 10.1007/BF01581275, Math. Program. 58 (1993), 353-367. (1993) Zbl0780.90090MR1216791DOI10.1007/BF01581275
  18. Rex, G., Rohn, J., 10.1137/S0895479896310743, SIAM J. Matrix Anal. Appl. 20 (1998), 437-445. (1998) Zbl0924.15003MR1651396DOI10.1137/S0895479896310743
  19. Rohn, J., 10.1016/0024-3795(89)90004-9, Linear Algebra Appl. 126 (1989), 39-78. (1989) Zbl0712.65029MR1040771DOI10.1016/0024-3795(89)90004-9
  20. Rohn, J., 10.1007/s11590-011-0305-3, Optim. Lett. 6 (2012), 851-856. (2012) Zbl1254.90260MR2925621DOI10.1007/s11590-011-0305-3
  21. Saheya, B., Yu, C.-H., Chen, J.-S., 10.1007/s12190-016-1065-0, J. Appl. Math. Comput. 56 (2018), 131-149. (2018) Zbl1390.26020MR3770379DOI10.1007/s12190-016-1065-0
  22. Tang, J., Zhou, J., 10.1016/j.orl.2019.03.014, Oper. Res. Lett. 47 (2019), 229-234. (2019) Zbl1476.65077MR3937799DOI10.1016/j.orl.2019.03.014
  23. Tang, J., Zhou, J., Zhang, H., 10.1007/s10957-022-02152-6, J. Optim. Theory Appl. 196 (2023), 641-665. (2023) Zbl07675403MR4548588DOI10.1007/s10957-022-02152-6
  24. Wang, A., Wang, H., Deng, Y., 10.2478/s11533-011-0067-2, Cent. Eur. J. Math. 9 (2011), 1171-1184. (2011) Zbl1236.65047MR2824456DOI10.2478/s11533-011-0067-2
  25. Wu, S.-L., 10.1016/j.aml.2021.107029, Appl. Math. Lett. 116 (2021), Article ID 107029, 6 pages. (2021) Zbl1469.15019MR4205122DOI10.1016/j.aml.2021.107029
  26. Yu, D., Chen, C., Han, D., 10.48550/arXiv.2103.10129, Available at https://arxiv.org/abs/2103.10129 (2022), 15 pages. (2022) DOI10.48550/arXiv.2103.10129
  27. Zhang, C., Wei, Q. J., 10.1007/s10957-009-9557-9, J. Optim. Theory Appl. 143 (2009), 391-403. (2009) Zbl1175.90418MR2545959DOI10.1007/s10957-009-9557-9
  28. Zhang, H., Hager, W. W., 10.1137/S1052623403428208, SIAM. J. Optim. 14 (2004), 1043-1056. (2004) Zbl1073.90024MR2112963DOI10.1137/S1052623403428208
  29. Zhang, J.-J., 10.1016/j.amc.2015.05.018, Appl. Math. Comput. 265 (2015), 266-274. (2015) Zbl1410.90224MR3373480DOI10.1016/j.amc.2015.05.018
  30. Zhou, H., Wu, S., 10.3934/math.2021517, AIMS Math. 6 (2021), 8912-8919. (2021) Zbl1485.90065MR4281108DOI10.3934/math.2021517

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