Low Computational Cost Algorithms for Solving Variational Inequalities over The Fixed Point Set
Main Article Content
Abstract
In this paper, we discuss several extragradient-like algorithms for solving variational inequalities over the fixed point set of a nonexpansive mapping. Under the assumptions that the involving mapping is pseudomonotone and Lipschitz continuous, the sequence generated by our algorithms converges to a desired solution. Compared with the original extragradient algorithm, the new ones have an advantage: they do not require to compute any projection onto the feasible set. This feature helps to reduce the computational cost of our methods. Moreover, to implement the new algorithms, we do not need to know the Lipschitz constant of the involving mapping. Also, we extend these results to the equilibrium problem and present some numerical experiments to verify the efficiency of the new algorithms.
Keywords
Variational inequality, Lipschitz continuity, pseudomonotonicity, extragradient algorithm
Article Details
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References
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finding saddle points and other problems, Ekon. Mat.
Metody, vol. 12, pp. 207-216, 1922.
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methods for accretive variational inequalities over the
set of common fixed points of nonexpansive
semigroups, Optimization, vol. 65, pp. 1553-1567,
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https://doi.org/10.1080/02331934.2016.1166501
finding saddle points and other problems, Ekon. Mat.
Metody, vol. 12, pp. 207-216, 1922.
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pseudomonotone non-Lipschitz variational inequalities,
Int. J. Comput. Math., vol. 98, no. 9, pp. 1807-1820,
2020.
https://doi.org/10.1080/00207160.2020.1847279
[3] T. N. Hai, Two modified extragradient algorithms for
solving variational inequalities, J. Global Optim., vol.
24, pp. 91-106, 2020.
http://doi.org/10.1007/s10898-020-00895-y
[4] H. Iiduka, A new iterative algorithm for the variational
inequality problem over the fixed point set of a firmly
nonexpansive mapping, Optimization, vol. 59, no. 6,
pp. 819-885, 2010.
https://doi.org/10.1080/02331930902884158
[5] H. Iiduka, Fixed point optimization algorithm and its
application to power control in CDMA data networks,
Math. Program., vol. 133, pp. 227-242, 2012.
https://doi.org/10.1007/s10107-010-0427-x
[6] H. Iiduka, I. Yamada, An ergodic algorithm for the
power-control games for CDMA data networks, J.
Math. Model. Algorithms, vol. 8, no. 1, pp. 1-18, 2019.
https://doi.org/10.1007/s10852-008-9099-4
[7] N. T. T. Thuy, Regularization methods and iterative
methods for variational inequality with accretive
operator, Acta Math. Vietnam., vol. 41, pp. 55-68,
2016.
https://doi.org/10.1134/S0965542512110103
[8] P. K. Anh, N. T. Vinh,, Self-adaptive gradient
projection algorithms for variational inequalities involving non-Lipschitz continuous operators, Numer
Algorithms, pp. 983-1001, 2019.
https://doi.org/10.1007/s11075-018-0578-z
[9] J. Y. Bello Cruz, A. N. Iusem, Convergence of direct
methods for paramonotone variational inequalities,
Comput. Optim. Appl., vol. 46, pp. 247-263, 2010.
https://doi.org/10.1007/s10589-009-9246-5
[10] N. Buong, N. S. Ha, N. T. T. Thuy, A new explicit
iteration method for a class of variational inequalities,
Numer Algorithms, vol. 18, pp. 467-481, 2016.
https://doi.org/10.1007/s11075-015-0056-9
[11] P. D. Khanh, P. T. Vuong, Modified projection method
for strongly pseudomonotone variational inequalities, J.
Global Optim., vol. 58, pp. 341-350, 2014.
https://doi.org/10.1007/s10898-013-0042-5
[12] G. M. Korpelevich, The extragradient method for
finding saddle points and other problems, Ekon. Mat.
Metody, vol. 12, pp. 207-216, 1922.
[13] N. T. T. Thuy, P. T. Hieu, J. J. Strodiot, Regularization
methods for accretive variational inequalities over the
set of common fixed points of nonexpansive
semigroups, Optimization, vol. 65, pp. 1553-1567,
2016.
https://doi.org/10.1080/02331934.2016.1166501