A preconditioner for solving large-scale variational inequality problems by a semismooth inexact approach

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.

Keywords:

• Large-scale variational inequality problems
• Semismooth nonlinear system
• Inexact Newton method
• LSQR method
• Incomplete LU preconditioner

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.

Additional information

Notes on contributors

Federica Tinti

Email: [email protected]