Two numerical schemes for general variational inequalities

P. Smoczynski; Mohamed Aly Tawhid

doi:10.3934/jimo.2008.4.393

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2008, Volume 4, Issue 2: 393-406. Doi: 10.3934/jimo.2008.4.393

This issue Previous Article Higher-order symmetric duality in multiobjective programming with invexity Next Article

Two numerical schemes for general variational inequalities

P. Smoczynski^1, and
Mohamed Aly Tawhid^1,

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

Received: February 28, 2007

Revised: February 29, 2008

Published: March 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 65K05, 90C33; Secondary: 65K10.

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