Augmented Lagrangian Method for Finding Minimum Norm Solution to the Absolute Value Equation

Document Type : Research Article

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran.

2 Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran.

Abstract

‎In this paper‎, ‎we give an algorithm to compute the minimum 1-norm solution to the absolute value equation (AVE)‎. ‎The augmented Lagrangian method is investigated for solving this problems‎ . ‎This approach leads to an unconstrained minimization problem with once differentiable convex objective function‎. ‎We propose a quasi-Newton method for solving unconstrained optimization problem‎. ‎Computational results show that convergence to high accuracy often occurs in just a few iterations‎.

Keywords


1. Cottle, R.W. and Dantzig, G. Complementary pivot theory of mathemat ical programming, Linear Algebra Appl, 1 (1968), 103-125.
2. Cottle, R.W, Pang, J.S. and Stone, R.E. The linear complementarity problem, Academic Press, New York, 1992.
3. Evtushenko, Yu.G., Golikov, A.I. and Mollaverdi, N. Augmented lagrangian method for large-scale linear programming problems, Optimization Methods and Software, 20 (2005), 515-524.
4. Kanzow, C., Qi, H. and Qi, L. On the minimum norm solution of linear programs, J. Optim. Theory Appl, 116 (2003), 333-345.
5. Ketabchi, S. and Moosaei, H. Minimum norm solution to the absolute value equation in the convex case, J. Optim. Theory Appl, 154 (2012), 1080-1087.
6. Mangasarian, O.L. A Newton method for linear programming, J. Optim. Theory Appl, 121 (2004), 1-18.
7. Mangasarian, O.L. and Meyer, R.R. Absolute value equations, Linear Algebra Appl, 419 (2006), 359-367.
8. Moosaei, H., Ketabchi, S., Noor, M., Iqbal, J. and Hooshyarbakhsh, V. Some techniques for solving absolute value equations, Applied Mathematics and Computation, 268 (2015), 696-705.
9. Pardalos, P.M., Ketabchi, S. and Moosaei, H. Minimum norm solution to the positive semidefinite linear complementarity problem, Optimization, 63(3) (2014), 359-369.
10. Pardalos, P.M. and Rosen, J.B. Global optimization approach to the linear complementarity problem, SIAM Journal on Scientific and Statistical Computing, 9 (1988), 341-353.
11. Pardalos, P.M. and Ye, Y. Class of linear complementarity problems solvable in polynomial time, Linear Algebra and its Applications, 152 (1991), 3-17.
12. Prokopyev, O. On equivalent reformulations for absolute value equations, Computational Optimization and Applications, 44 (3) (2009), 363-372.
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