Abstract
We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties. Namely, we assume that the operator of the variational inequality is continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Under the assumption that the dual solution set is not empty, we prove that our method converges to a solution of the variational inequality. Instead of the monotonicity assumption, we require the non-emptiness of the solution set of the dual formulation of the variational inequality. We provide numerical experiments illustrating the behaviour of our iterates. Moreover, we compare our new method with a recent similar one.
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Acknowledgments
R. Díaz Millán was partially supported by CNPq grant 200427/2015-6. This work was concluded while the second author was visiting the School of Information Technology and Mathematical Sciences at the University of South Australia. R. Díaz Millán would like to thank the great hospitality received during his visit, particularly to Regina S. Burachik and C. Yalçin Kaya. R. Díaz Millán would like to extend its gratitude to Prof. Ole Peter Smith for his valuable suggestions. The authors would like to thank the two referees and the editor, for their comments, which greatly improved the presentation of this paper.
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Burachik, R.S., Millán, R.D. A Projection Algorithm for Non-Monotone Variational Inequalities. Set-Valued Var. Anal 28, 149–166 (2020). https://doi.org/10.1007/s11228-019-00517-0
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DOI: https://doi.org/10.1007/s11228-019-00517-0
Keywords
- Variational inequality
- Projection algorithms
- Outer-semicontinuous operator
- Inner-semicontinuous operator