Archive
2021, Volume 9
2020, Volume 8
2019, Volume 7
2018, Volume 6
2017, Volume 5
2016, Volume 4
2015, Volume 3
2014, Volume 2
2013, Volume 1
Volume 9 , Issue 4 , August 2021 , Pages: 104 - 107
On the Unique Solvability of the Generalized Absolute Value Matrix Equation
Kai Xie, College of Mathematics and Statistics, Northwest Normal University, Lanzhou, P. R. China
Received: Jul. 13, 2021;       Accepted: Jul. 26, 2021;       Published: Aug. 2, 2021
DOI: 10.11648/j.ajam.20210904.12        View        Downloads  
Abstract
The generalized absolute value matrix equation has application in a variety of optimization problems, its unique solvability is still on the way. In this note, the unique solvability of the generalized absolute value matrix equation is considered. A new unique solvability of generalized absolute value matrix equation is given. The obtained result can be regarded as an extension of the absolute value equation to the generalized absolute value matrix equation. As an application, new convergence of matrix multisplitting Picard-iterative method is presented.
Keywords
Generalized Absolute Value Matrix Equation, Unique Solution, Singular Values
To cite this article
Kai Xie, On the Unique Solvability of the Generalized Absolute Value Matrix Equation, American Journal of Applied Mathematics. Vol. 9, No. 4, 2021, pp. 104-107. doi: 10.11648/j.ajam.20210904.12
Copyright
Copyright © 2021 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[ 1 ]
Rohn, J. (2004) A theorem of the alternatives for the equation Ax+B|x|=b. Linear and Multilinear Algebra, 52, 421-426.
[ 2 ]
Mangasarian, O. L. (2012) Primal-dual bilinear programming solution of the absolute value equation. Optim. Lett., 6, 1527-1533.
[ 3 ]
Mangasarian, O. L., & Meyer, R. R. (2006) Absolute value equations. Linear Algebra Appl., 419, 359-367.
[ 4 ]
Prokopyev, O. (2009) On equivalent reformulations for absolute value equations. Comput. Optim. Appl., 44, 363-372.
[ 5 ]
Dehghan, M., & Hajarian, M. (2008) An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl. Math. Comput., 202, 571-588.
[ 6 ]
Dehghan, M., & Hajarian, M. (2010) The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra Appl., 432, 1531-1552.
[ 7 ]
Dehghan, M., & Shirilord, A. (2019) A generalized modified Hermitian and skew--Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation. Appl. Math. Comput., 348, 632-651.
[ 8 ]
Dehghan, M., & Shirilord, A. HSS-like method for solving complex nonlinear Yang-Baxter matrix equation. Eng. Comput-Germany. (in press).
[ 9 ]
Hu, S. L., & Huang, Z. H. (2010) A note on absolute value equations. Optim. Lett., 4, 417-424.
[ 10 ]
Wu, S. L., & Guo, P. (2016) On the unique solvability of the absolute value equation. J. Optim. Theory Appl., 169, 705-712.
[ 11 ]
Wu, S. L., & Li, C. X. (2020) A note on unique solvability of the absolute value equation. Optim. Lett., 14, 1957-1960.
[ 12 ]
Rohn, J., Hooshyarbakhsh, V., & Farhadsefat, R. (2014) An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett., 8, 35-44.
[ 13 ]
Zhang, C., & Wei, Q. (2009) Global and finite convergence of a generalized newton method for absolute value equations. J. Optim. Theory Appl., 143, 391-403.
[ 14 ]
Caccetta, L., Qu, B., & Zhou, G. (2011) A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl., 48, 45-58.
[ 15 ]
Gu, X. M., Huang, T. Z., & Li, H. B., etc. (2017) Two CSCS-based iteration methods for solving absolute value equations. J. Appl. Anal Comput., 7, 1336-1356.
[ 16 ]
Li, C. X. (2016) A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl., 170, 1055-1059.
[ 17 ]
Mangasarian, O. L. (2009) A generalized Newton method for absolute value equations. Optim. Lett., 3, 101-108.
[ 18 ]
Mangasarian, O. L. (2009) Knapsack feasibility as an absolute value equation solvable by successive linear programming. Optim. Lett., 3, 161-170.
[ 19 ]
Noor, M. A., Iqbal, J., & Noor, K. I., etc. (2012) On an iterative method for solving absolute value equations. Optim. Lett., 6, 1027-1033.
[ 20 ]
Salkuyeh, D. K. (2014) The Picard-HSS iteration method for absolute value equations. Optim. Lett., 8, 2191-2202.
[ 21 ]
Tang, J., & Zhou, J. (2019) A quadratically convergent descent method for the absolute value equation Ax+B|x|=b. Oper. Res. Lett., 47, 229-234.
[ 22 ]
Rohn, J. (2009) On unique solvability of the absolute value equation. Optim. Lett., 3, 603-606.
[ 23 ]
Dehghan, M., & Shirilord, A. (2020) Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix euqation. Appl. Numer. Math., 158, 425-438.
Browse journals by subject