An improved generalized Newton method for absolute value equations

Subject Areas : Statistics

Taher Lotfi ¹ (Professor, Department of Applied Mathematics (Numerical Analysis), Faculty of Basic Sciences, Islamic Azad University, Hamadan Branch, Iran)
Yaser Seif ² (Department of Applied Mathematics (Numerical Analysis), Faculty of Basic Sciences, Islamic Azad University, Hamadan Branch, Iran)

Received: 2019-12-14 Accepted : 2020-07-12 Published : 2021-03-01

Keywords: ماتریس اسکالر, معادلات قدرمطلقی, روش نیوتن, مقادیر منفرد,

Abstract :

In recent years the interest in studying the absolute value equation has been of great interest both theoretically and practically. The main reason for this is that various optimization problems, such as the complementary linear programming problem, can be written in the form of a robust equation that is easier to solve. The main purpose of this paper is to present a duplicate method for solving the equations of magnitude. Actually, in this paper, by introducing a scalar matrix, an improved generalized Newton method is proposed to solve the Absolute value equation. This new method is based on the methods of Mangserin [1] and Li [2]. In fact, if in the matrix A + αI-D, the value of the identity matrix coefficient is equal to zero, the Mangserin method and if the coefficient of the same matrix to one, the Li method is obtained. when all the singular values of the system matrix exceed one.

References:

 

1. O.L. Mangasarian, A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)

 

1.  C.X. Li, A modified generalized newton method for absolute value equations. J. Optim Theory Appl., 170:1055–1059 (2016)

 

1. J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b. Linear Multilinear A 52, 421–426 (2004)

 

1. O.L. Mangasarian, Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)

 

1. R.W. Cottle, G.B. Dantzig, Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

 

1. R.W.  Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem. Academic, San Diego (1992)

 

1. L. Caccetta, B. Qu, Zhou, A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)

 

1. S. L. Hu, Z. H. Huang, Q. Zhang, A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235, 1490–1501 (2011)

 

1. C. Zhang, Q. J. Wei, Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143, 391–403 (2009)

 

10. S .Ketabchi, H. Moosaei, An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side, Comput. Math. Appl. 64, 1882–1885 (2012)

 

11. N. Zainali, T. Lotfi, On developing a stable and quadratic convergent method for solving absolute value equation, J. Comput. and Appl. Math.,742-747 (2018)

 

12. F. Kh. Haghani, On Generalized Traub’s Method for Absolute Value Equations, J. Optim. Theo. Appl., 619-625 (2015)

 

13. J. Rohn, An algorithm for solving the absolute value equations. Electron. J. Linear Algebra 18, 589– 599 (2009)

 

14. B. Huang, Ch. Ma, Convergent conditions of the generalized Newton method for absolute value equation over second order cones, Appl. Math. Lett. 92, 151–157 (2019)

 

15. J.J., Zhang, The relaxed nonlinear PHSS-like iteration method for absolute value equations, Appl. Math. Comput. 265, 266–274 (2015)

 

16. O.L.Mangasarian, A hybrid algorithm for solving the absolute value equation, Optim. Lett. 9(7), 1469–1474 (2015)

 

17. J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett. 8, 35–44 (2014)

 

18. Noor, M.A., Iqbal, J., Al-Said, E., Residual iterative method for solving absolute value equations, Abstr. Appl. Anal. 2012 Article ID 406232, 9 (2012)

 

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

 

20. Rockafellar, R.T.: New applications of duality in convex programming. In Proceedings Fourth Conference on Probability, Brasov (1971)

 

21. F. Rahpeymaii, K. Amini, T. Allahviranloo, M. Rostamy Malkhalifeh, A new class of conjugate gradient methods for unconstrained smoot optimization and absolute value equations. Calcolo, 56:2 (2019)

 

22. F. Bu, F. Ma, The tensor splitting methods for solving tensor absolute value equation, J. Comput. Appl. Math., 39:178 (2020)