Abstract
In this paper, we study strong convergence of the algorithm for solving classical variational inequalities problem with Lipschitz-continuous and monotone mapping in real Hilbert space. The algorithm is inspired by Tseng’s extragradient method and the viscosity method with a simple step size. A strong convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Finally, we provide some numerical experiments to show the efficiency and advantage of the proposed algorithm.
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Hartman, P., Stampacchia, G.: On some linear elliptic differential equations. Acta Math. 115, 271–310 (1966)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problem. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)
Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)
Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)
Malitsky, Yu.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)
Tseng, P.: A modified forward-backward splitting method for maximal monotone mapping. SIAM J. Control. Optim. 38, 431–446 (2000)
Solodov, M.V., Svaiter, B.F.: A new projection method for monotone variational inequalities. SIAM. J. Control. Optim. 37, 765–776 (1999)
Malitsky, Yu.V.: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)
Mainge, P.E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)
Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)
Duong, V.T., Dang V.H.: Weak and strong convergence theorems for variational inequality problems. Numer Algor, https://doi.org/10.1007/s11075-017-0412-z
Mainge, P.E.: The viscosity approximation process for quasi-nonexpansive mapping in Hilbert space. Comput. Math. Appl. 59, 74–79 (2010)
Rapeepan, K., Satit, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 163, 399–412 (2014)
Yekini, S., Olaniyi, S.I.: Strong convergence result for monotone variational inequalities. Numer Algor 76, 259–282 (2017)
Gibali, A.: A new non-Lipschitzian method for solving variational inequalities in Euclidean spaces. J. Nonlinear Anal. Optim. 6, 41–51 (2015)
Moudafi, A.: Viscosity methods for fixed points problems. Journal of Mathematical Analysis and Applications. 241, 46–55 (2000)
Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody. 12(6), 1164–1173 (1976)
Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66 (2), 240–256 (2002)
Acknowledgements
The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped to improve the original version of this paper.
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The Project was supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2017JM1014).
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Yang, J., Liu, H. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algor 80, 741–752 (2019). https://doi.org/10.1007/s11075-018-0504-4
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DOI: https://doi.org/10.1007/s11075-018-0504-4