Abstract
The numerical solution of a large-scale variational inequality problem can be obtained using the generalization of an inexact Newton method applied to a semismooth nonlinear system. This approach requires a sparse and large linear system to be solved at each step. In this work we obtain an approximate solution of this system by the LSQR algorithm of Paige and Saunders combined with a convenient preconditioner that is a variant of the incomplete LU–factorization. Since the computation of the factorization of the preconditioning matrix can be very expensive and memory consuming, we propose a preconditioner that admits block-factorization. Thus the direct factorization is only applied to submatrices of smaller sizes. Numerical experiments on a set of test-problems arising from the literature show the effectiveness of this approach.
Acknowledgements
The authors are very grateful to anonymous referees who stimulated them to improve the paper with their comments. This research was supported by the Italian Ministry for Education, University and Research (MIUR), FIRB Project RBAU01JYPN: ‘Parallel Algorithms and Numerical Nonlinear Optimization’.
Notes
†A C k -function is called an L C k -function if its k-th derivative is locally Lipschitz-continuous.
†In this case the solution of the inner system is computed at the machine precision.