Abstract
We suggest an iterative method for solving absolute value equation Ax − |x| = b, where \({A\in R^{n\times n}}\) is symmetric matrix and \({b\in R^{n}}\), coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method.
References
Burden R.L., Faires J.D.: Numerical Analysis, 7th edn. The PWS Publishing Company, Boston (2006)
Chung S.J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 (1989)
Cottle R.W., Dantzig G.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)
Cottle R.W., Pang J.S., Stone R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)
Frommer A., Szyld D.B.: H-splittings and two stage iterative methods. Numer. Math. 63, 345–356 (1992)
Horst R., Pardalos P., Thoai N.V.: Introduction to Global Optimization. Kluwer, Dodrecht (1995)
Mangasarian O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)
Mangasarian O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)
Mangasarian O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)
Mangasarian O.L.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465–485 (1977)
Mangasarian O.L.: The linear complementarity problem as a separable bilinear program. J. Glob. Optim. 6, 153–161 (1995)
Mangasarian O.L., Meyer R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)
Noor M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)
Noor M.A.: Extended general variational inequalities. Appl. Math. Lett. 22, 182–185 (2009)
Noor M.A., Noor K.I., Rassias T.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)
Noor, M.A., Iqbal, J., Khattri, S., Al-Said, E.: A new iterative method for solving absolute value equations. Int. J. Phy. Sci. 6 (2011)
Pardolas P.M., Rassias Th.M., Khan A.A.: Nonlinear Analysis and Variational Analysis. Springer, Berlin (2010)
Rohn J.: A theorem of the alternatives for the equation Ax + B|x| = b, Lin. Mult. Algeb. 52, 421–426 (2004)
Rohn, J.: An algorithm for computing all solutions of an absolute vale equation. Optim. Lett. doi:10.1007/s11590-011-0305-3
Ujevic N.: A new iterative method for solving linear systems. Appl. Math. Comput. 179, 725–730 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Noor, M.A., Iqbal, J., Noor, K.I. et al. On an iterative method for solving absolute value equations. Optim Lett 6, 1027–1033 (2012). https://doi.org/10.1007/s11590-011-0332-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0332-0