# American Institute of Mathematical Sciences

April  2008, 4(2): 393-406. doi: 10.3934/jimo.2008.4.393

## Two numerical schemes for general variational inequalities

 1 Department of Mathematics and Statistics,Thompson Rivers University, 900 McGill Road, PO Box 3010, Kamloops, BC V2C 5N3, Canada, Canada

Received  March 2007 Revised  March 2008 Published  April 2008

In this paper, we compare between the forward-backward splitting method and the extra-gradient method for solving general variational inequalities. It is known that both of these methods are predictor-corrector methods. They use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods would be better than the forward-backward splitting methods for general variational inequalities. We suggest some new step selection procedure independent of the Lipschitz constant. This is a very desirable circumstance when the operator approximates a differential operator. We prove its convergence in Hilbert spaces of any dimension. Our proof is simple as compared with other methods.
Citation: P. Smoczynski, Mohamed Aly Tawhid. Two numerical schemes for general variational inequalities. Journal of Industrial & Management Optimization, 2008, 4 (2) : 393-406. doi: 10.3934/jimo.2008.4.393
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