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A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces

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Abstract

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution.

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Acknowledgments

JYBC and RDM were partially supported by project CAPES-MES-CUBA 226/2012. JYBC was partially supported by CNPq grants 303492/2013-9, 474160/2013-0 and 202677/2013-3 and by project UNIVERSAL FAPEG/CNPq. RDM was supported by his scholarship for his doctoral studies, granted by CAPES. This work was completed while the first author was visiting the University of British Columbia. The author is very grateful for the warm hospitality. The authors would like to thank to Professor Dr. Heinz H. Bauschke, Professor Dr. Ole Peter Smith and anonymous referees whose suggestions helped us to improve the presentation of this paper.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, Federal University of Goiás, Goiânia, Avenida Esperana, s/n Campus Samambaia, Goiânia, GO, 74690-900, Brazil

    J. Y. Bello Cruz

  2. Mathematics, Federal Institute of Education, Science and Technology, Goiânia, G.O., 74055-110, Brazil

    R. Díaz Millán

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Cruz, J.Y.B., Millán, R.D. A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces. J Glob Optim 65, 597–614 (2016). https://doi.org/10.1007/s10898-015-0397-x

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