Augmented Lagrangian method for solving absolute value equation and its application in two-point boundary value problems
Subject Areas : StatisticsH. Moosaei 1 , S. Ketabchi 2 , M. T. Fooladi 3
1 - Department of Mathematics, Faculty of science, University of Bojnord, Bojnord, Iran
2 - Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
3 - Department of Mathematics, Faculty of science, University of Bojnord, Bojnord, Iran
Keywords: روش نیوتن تعمیم یافته, مسائل مقدار مرزی دو نقطه ای, دستگاه معادلات قدر مطلق, روش لاگرانژ بهبود یافته,
Abstract :
One of the most important topic that consider in recent years by researcher is absolute value equation (AVE). The absolute value equation seems to be a useful tool in optimization since it subsumes the linear complementarity problem and thus also linear programming and convex quadratic programming. This paper introduce a new method for solving absolute value equation. To do this, we transform absolute value equation to linear system and then demonstrate efficient augmented Lagrangian method to solve the linear system. Also this paper is considered a class of two-point boundary value problems and introduced a new method to solve them. In this paper is shown that this class of problems is equivalent to absolute value equation. To illustrate the feasibility and effectiveness our method we generate random problems and solve them also solve a class of two-point boundary value problems. In section numerical results, we consider the efficiency of the proposed method. Computational results show that convergence to high accuracy often occurs in short time.
[1] Roberts, S. M., & Shipman, J. S. (1972). Two-point boundary value problems: shooting methods.
[2] Aziz, A. K. (Ed.). (2014). Numerical solutions of boundary value problems for ordinary differential equations. Academic Press.
[3] Stynes, M., & Gracia, J. L. (2015). A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA Journal of Numerical Analysis, 35(2), 698-721.
[4] Mangasarian, O. L., & Meyer, R. R. (2006). Absolute value equations. Linear Algebra and Its Applications, 419(2-3), 359-367.
[5] Longquan, Y. O. N. G. (2010). Particle swarm optimization for absolute value equations. Journal of Computational Information Systems, 6(7), 2359-2366.
[6] Mangasarian, O. L. (2007). Absolute value equation solution via concave minimization. Optimization Letters, 1(1), 3-8.
[7] Mangasarian, O. L. (2009). A generalized Newton method for absolute value equations. Optimization Letters, 3(1), 101-108.
[8] Mangasarian, O. L. (2012). Primal-dual bilinear programming solution of the absolute value equation. Optimization Letters, 6(7), 1527-1533.
[9] Mangasarian, O. L. (2013). Absolute value equation solution via dual complementarity. Optimization Letters, 7(4), 625-630.
[10] Mangasarian, O. L. (2015). A hybrid algorithm for solving the absolute value equation. Optimization Letters, 9(7), 1469-1474.
[11] Moosaei, H., Ketabchi, S., Noor, M. A., Iqbal, J., & Hooshyarbakhsh, V. (2015). Some techniques for solving absolute value equations. Applied Mathematics and Computation, 268, 696-705.
[12] Noor, M. A., Iqbal, J., & Al-Said, E. (2012). Residual iterative method for solving absolute value equations. In Abstract and Applied Analysis (Vol. 2012). Hindawi.
[13] Noor, M. A., Iqbal, J., Khattri, S., & Al-Said, E. (2011). A new iterative method for solving absolute value equations. International Journal of Physical Sciences, 6(7), 1793-1797.
[14] Noor, M. A., Iqbal, J., Noor, K. I., & Al-Said, E. (2012). On an iterative method for solving absolute value equations. Optimization Letters, 6(5), 1027-1033.
[15] J Rohn, J. (2009). On unique solvability of the absolute value equation. Optimization Letters, 3(4), 603-606.
[16] Rohn, J. (2012). An algorithm for computing all solutions of an absolute value equation. Optimization Letters, 6(5), 851-856.
[17] Rohn, J. (2012). A theorem of the alternatives for the equation| Ax|−| B|| x|= b. Optimization Letters, 6(3), 585-591.
[18] Rohn, J., Hooshyarbakhsh, V., & Farhadsefat, R. (2014). An iterative method for solving Absolute value equations and sufficient conditions for unique solvability. Optimization Letters, 8(1), 35-44.
[19] Salkuyeh, D. K. (2014). The Picard–HSS iteration method for absolute value equations. Optimization Letters, 8(8), 2191-2202.
[20] Ketabchi, S., & Moosaei, H. (2012). Optimal error correction and methods of feasible directions. Journal of Optimization Theory and Applications, 154(1), 209-216.
[21] Ketabchi, S., & Moosaei, H. (2012). An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Computers & Mathematics with Applications, 64(6), 1882-1885.
[22] Ketabchi, S., & Moosaei, H. (2012). Minimum norm solution to the absolute value equation in the convex case. Journal of Optimization Theory and Applications, 154(3), 1080-1087.
[23] Moosaei, H., Ketabchi, S., & Jafari, H. (2015). Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afrika Matematika, 26(7-8), 1221-1228.
[24] Yong, L. (2015). Iteration Method for Absolute Value Equation and Applications in Two-point Boundary Value Problem of Linear Differential Equation. Journal of Interdisciplinary Mathematics, 18(4), 355-374.
[25] Allen Jr, R. C., Wing, G. M., & Scott, M. (1969). Solution of a certain class of nonlinear two-point boundary value problems. Journal of Computational Physics, 4(2), 250-257.
[26] Evtushenko, Y. G., Golikov, A. I., & Mollaverdy, N. (2005). Augmented Lagrangian method for large-scale linear programming problems. Optimization Methods and Software, 20(4-5), 515-524.
[27] Pardalos, P. M., Ketabchi, S., & Moosaei, H. (2014). Minimum norm solution to the positive semidefinite linear complementarity problem. Optimization, 63(3), 359-369.
[28] Mangasarian, O. L. (2004). A Newton method for linear programming. Journal of Optimization Theory and Applications, 121(1), 1-18.
