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Convergence of One-Step Projected Gradient Methods for Variational Inequalities

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Abstract

In this paper, we revisit the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method. A main feature of the method is that it formally requires only one projection step onto the feasible set and one evaluation of the involved mapping per iteration. Contrary to what was done so far, we establish the convergence of the method in a more general setting that allows us to use varying step-sizes without any requirement of additional projections. A linear convergence rate is obtained, when A is assumed to be strongly monotone. Preliminary numerical experiments are also performed.

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Acknowledgments

The authors do wish to thank the editor in chief (Franco Giannessi) and the two anonymous referees for their constructive suggestions that greatly improved this manuscript.

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Authors and Affiliations

  1. D.S.I., L.A.M.I.A., Université des antilles, 97233, Martinique, F.W.I., France

    P. E. Maingé

  2. UMR Espace-Dev., I.R.D., Université de Guyane, 97300, Cayenne, France

    M. L. Gobinddass

Authors
  1. P. E. Maingé
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  2. M. L. Gobinddass
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Correspondence to P. E. Maingé.

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Communicated by Alfredo N. Iusem.

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Maingé, P.E., Gobinddass, M.L. Convergence of One-Step Projected Gradient Methods for Variational Inequalities . J Optim Theory Appl 171, 146–168 (2016). https://doi.org/10.1007/s10957-016-0972-4

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  • DOI: https://doi.org/10.1007/s10957-016-0972-4

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