On the unique solvability and numerical study of absolute value equations

Achache Mohamed

doi:10.33993/jnaat482-1182

Authors

Achache Mohamed University Ferhat Abbas of Sétif, Algeria

DOI:

https://doi.org/10.33993/jnaat482-1182

Keywords:

absolute value equations, complementarity, generalized Newton method, global convergence, optimization

Abstract views: 418

Abstract

The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), \(Ax-B\vert x\vert =b\), when the condition \(\Vert A^{-1}\Vert <\frac{1}{\left\Vert B\right\Vert }\) holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where \(B=I\).

Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition \(\Vert A^{-1}\Vert <\frac{1}{4\Vert B\Vert }\), that the algorithm converges linearly global to the unique solution of the AVE.

Numerical results are reported to show the efficiency of the proposed method and to compare with an available method.

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References

M. Achache, and N. Hazzam, Solving absolute value equations via complementarity and interior point methods. Journal of Nonlinear Functional Analysis. (2018), pp. 1–10, https://doi.org/10.23952/jnfa.2018.39 DOI: https://doi.org/10.23952/jnfa.2018.39

L. Abdellah, M. Haddou and T. Migot, Solving absolute value equation using complementarity and smoothing functions. Journal of Computational and Applied Mathematics.327(2018), pp. 196–207, https://doi.org/10.1016/j.cam.2017.06.019 DOI: https://doi.org/10.1016/j.cam.2017.06.019

L. Caccetta, B. Qu and G. Zhou, A globally and quadratically convergent method for absolute value equations. Computational Optimization and Applications. 48(2011), pp. 45–58, https://doi.org/10.1007/s10589-009-9242-9 DOI: https://doi.org/10.1007/s10589-009-9242-9

R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, New-York (1992).

F.K. Haghani, On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. (166)(2015), pp. 619–625, https://doi.org/10.1007/s10957-015-0712-1 DOI: https://doi.org/10.1007/s10957-015-0712-1

S.L. Hu and Z.H. Huang, A note on absolute value equations. Optimization Letters. 4(2010), pp. 417–424, https://doi.org/10.1007/s11590-009-0169-y DOI: https://doi.org/10.1007/s11590-009-0169-y

O.L. Mangasarian, A generalized Newton method for absolute value equations. Optimization Letters. 3(2009), pp. 101–108, https://doi.org/10.1007/s11590-008-0094-5 DOI: https://doi.org/10.1007/s11590-008-0094-5

O.L. Mangasarian, Linear complementarity as absolute value equation solution. Optimization Letters. 8(2014), pp. 1529–1534,https://doi.org/10.1007/s11590-013-0656-z DOI: https://doi.org/10.1007/s11590-013-0656-z

O.L. Mangasarian and R.R. Meyer, Absolute value equations. Linear Algebra and its Applications. 419(2006), pp. 359–367, https://doi.org/10.1016/j.laa.2006.05.004 DOI: https://doi.org/10.1016/j.laa.2006.05.004

M.A. Noor, J. Iqbal, K. I. Noor and A. Al-Said, On a iterative method for solving absolute value equations. Optimization Letters. 6(2012), pp. 1027–1033, https://doi.org/10.1007/s11590-011-0332-0 DOI: https://doi.org/10.1007/s11590-011-0332-0

C. X. Li, A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170(2016), pp. 1055–1059, https://doi.org/10.1007/s10957-016-0956-4 DOI: https://doi.org/10.1007/s10957-016-0956-4

T. Lotfi and H. Veiseh, A note on unique solvability of the absolute value equation. Journal of Linear and Topological Algebra. 2 (2)(2013), pp. 77–81.

B.T. Polyak, Introduction to Optimization, Optimization Software Inc, Publications Division, New York, (1987).

O. Prokopyev. On equivalent reformulations for absolute value equations. Computational Optimization and Applications. 44(2009), pp. 363–372, https://doi.org/10.1007/s10589-007-9158-1 DOI: https://doi.org/10.1007/s10589-007-9158-1

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equation. Journal of Industrial and Management Optimization. 9 (4)(2013), pp. 789–798,https://doi.org/10.3934/jimo.2013.9.789 DOI: https://doi.org/10.3934/jimo.2013.9.789

J. Rohn, On unique solvability of the absolute value equations. Optimization Letters. 3(2009), pp. 603–606, https://doi.org/10.1007/s11590-009-0129-6 DOI: https://doi.org/10.1007/s11590-009-0129-6

J. Rohn, An algorithm for computing all solutions of an absolute value equation. Optimization Letters. 6(2012), pp. 851–856, https://doi.org/10.1007/s11590-011-0305-3 DOI: https://doi.org/10.1007/s11590-011-0305-3

J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b. Linear and Multilinear Algebra. 52(2004), pp. 421–426, https://doi.org/10.1080/0308108042000220686 DOI: https://doi.org/10.1080/0308108042000220686