Two strong convergence subgradient extragradient methods for solving variational inequalities in Hilbert spaces
In this paper we are focused on solving monotone and Lipschitz continuous variational inequalities in real Hilbert spaces. Motivated by several recent results related to the subgradient extragradient method (SEM), we propose two SEM extensions which do not require the knowledge of the Lipschitz constant associated with the variational inequality operator. Under mild and standard conditions, we establish the strong convergence of our schemes. Primary numerical examples demonstrate the potential of our algorithms as well as compare their performances to several related results.
KeywordsProjection and contraction method Subgradient extragradient method Mann type method Viscosity method Variational inequality problem
Mathematics Subject Classification47H09 47H10 47J20 47J25
The authors would like to thank the referees for their comments on the manuscript which helped in improving earlier version of this paper.
- 1.Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat. Metody 12, 1164–1173 (1976)Google Scholar
- 10.Dong, Q.L., Gibali, A., Jiang, D.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algorithms. (2018) https://doi.org/10.1007/s11075-017-0467-x
- 13.Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols. I and II. Springer, New York (2003)Google Scholar
- 20.Hieu, D.V., Thong, D.V.: A new projection method for a class of variational inequalities. Appl. Anal. (2018). https://doi.org/10.1080/00036811.2018.1460816
- 41.Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms. (2018) https://doi.org/10.1007/s11075-018-0527-x
- 43.Reich, S.: Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)Google Scholar