A new hybrid method for solving variational inequalities

1Malitsky, Yu.V
2Semenov, VV
1Taras Shevchenko National University of Kyiv
2V.М. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2014, 4:49-55
https://doi.org/10.15407/dopovidi2014.04.049
Section: Information Science and Cybernetics
Language: English
Abstract: 

We introduce a new method for solving variational inequalities with monotone and Lipschitz-continuous operators acting in a Hilbert space. The iterative process based on the well-known projection method and the hybrid (or outer approximations) method. However, we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by a slightly different choice of sets in the hybrid method. We prove the strong convergence of the sequences generated by our method.

Keywords: hybrid method, variational inequalities
References: 

1. Facchinei F., Pang J.-S. Finite-dimensional variational inequalities and complementarity problem. Vol. 2. New York: Springer, 2003.
2. Bauschke H. H., Combettes P. L. Convex analysis and monotone operator theory in Hilbert spaces. Berlin: Springer, 2011. https://doi.org/10.1007/978-1-4419-9467-7
3. Korpelevich G. M. Ekonom. i Matem. Metody, 1976, 12: 747–756.
4. Khobotov E. N. USSR Comput. Math. Math. Phys., 1989, 27: 120–127. https://doi.org/10.1016/0041-5553(87)90058-9
5. Nadezhkina N., Takahashi W. SIAM J. Optim., 2006, 16: 1230–1241. https://doi.org/10.1137/050624315
6. Iusem A. N., Nasri M. J. of Global Optim., 2011, 50: 59–76. https://doi.org/10.1007/s10898-010-9613-x
7. Voitova T. A., Denisov S. V., Semenov V. V. Zh. Obch. Prykl. Mat., 2011, No. 1(104): 10–23.
8. Lyashko S. I., Semenov V. V., Voitova T. A. Kibern Syst. Anal., 2011, 47: 631–639. https://doi.org/10.1007/s10559-011-9343-1
9. Apostol R. Ya., Grynenko A. A., Semenov V. V. Zh. Obch. Prykl. Mat., 2012, No. 1(107): 3–14.
10. Nakajo K., Takahashi W. J. Math. Anal. Appl., 2003, 279: 372–379. https://doi.org/10.1016/S0022-247X(02)00458-4
11. Censor Y., A. Gibali A., Reich S. J. of Optim. Theory Appl., 2011, 148:  318–335. https://doi.org/10.1007/s10957-010-9757-3
12. Censor Y., Gibali A., Reich S. Optimiz. Methods and Software, 2011, 26: 827–845. https://doi.org/10.1080/10556788.2010.551536