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A Direct Splitting Method for Nonsmooth Variational Inequalities

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Abstract

We propose a direct splitting method for solving a nonsmooth variational inequality in Hilbert spaces. The weak convergence is established when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which strongly explores the structure of the operator using projected subgradient-like techniques. The advantage of our method is that any nontrivial subproblem must be solved, like the evaluation of the resolvent operator. The necessity to compute proximal iterations is the main difficulty of other schemes for solving this kind of problem.

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Acknowledgements

The authors would like to thank CNPq. Research for this paper was partially supported by PRONEX, PROCAD-nf—UFG/UnB/IMPA and by project CAPES-MES-CUBA 226/2012 “Modelos de Otimização e Aplicações”. The second author was supported by a scholarship for his doctoral studies, granted by CAPES. The authors would like to extend gratitude toward the anonymous referees whose suggestions helped us to improve the presentation of this paper.

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Authors and Affiliations

  1. IME, Federal University of Goiás, Campus Samambaia, CEP 74001-970, Goiânia, GO, Brazil

    J. Y. Bello Cruz

  2. Federal Institute of Education, Science and Technology, Goiânia, GO, Brazil

    R. Díaz Millán

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  1. J. Y. Bello Cruz
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  2. R. Díaz Millán
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Correspondence to J. Y. Bello Cruz.

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Communicated by Alfredo N. Iusem.

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Bello Cruz, J.Y., Díaz Millán, R. A Direct Splitting Method for Nonsmooth Variational Inequalities. J Optim Theory Appl 161, 728–737 (2014). https://doi.org/10.1007/s10957-013-0478-2

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  • DOI: https://doi.org/10.1007/s10957-013-0478-2

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