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A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces

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Abstract

In this paper, we study the variational inequalities involving monotone and Lipschitz continuous mapping in Banach spaces. A new and simple iterative method, which combines Halpern’s technique and the subgradient extragradient idea, is given. Under mild and standard assumptions, we establish the strong convergence of our algorithm in a uniformly smooth and convex Banach spaces. We also present a modification of our method using a line-search approach, this enable to obtain strong convergence in real and reflexive Banach spaces, without the prior knowledge of the Lipschitz constant. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms. Our results generalize and extend some of the existing works in Hilbert spaces to Banach spaces as well as provide an extension from weak to strong convergence.

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Acknowledgements

We are very grateful to the anonymous referee and Associate Editor whose insightful comments helped to considerably improve an earlier version of this paper. The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing (cstc2017jcyjAX0006), Science and Technology Project of Chongqing Education Committee (Grant No. KJ KJ1703041), the University Young Core Teacher Foundation of Chongqing (020603011714), Talent Project of Chongqing Normal University (Grant No. 02030307-00024). The second author work is supported by the EU FP7 IRSES program STREVCOMS, Grant No. PIRSES-GA-2013-612669. The fourth author is grateful to the Alexander von Humboldt Foundation and the Institute of Mathematics of the University of Würzburg, Germany for their support and hospitality.

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Authors and Affiliations

  1. School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Gang Cai

  2. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  3. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA

    Olaniyi S. Iyiola

  4. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

Authors
  1. Gang Cai
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  2. Aviv Gibali
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  3. Olaniyi S. Iyiola
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  4. Yekini Shehu
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Correspondence to Aviv Gibali.

Additional information

Communicated by Alfredo N. Iusem.

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Cai, G., Gibali, A., Iyiola, O.S. et al. A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces. J Optim Theory Appl 178, 219–239 (2018). https://doi.org/10.1007/s10957-018-1228-2

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