The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces
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In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach.
KeywordsMaximal monotone operator Glowinski–Le Tallec splitting method Equilibrium problem Nash equilibrium Global convergence
The authors would like to thank the Associate Editor and the two anonymous referees for their useful remks, comments and suggestions that allowed to improve substantially the original version of this paper. This work was mostly carried out when the first author was a PhD student working at the Institute for Computational Science and Technology—Ho Chi Minh City, Vietnam. This research was supported by this Institute and partly, for the first author, by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant 101.01-2017.315 and the Austrian Science Foundation (FWF), Grant P26640-N25.
- 1.Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpansiveness properties of the marginal mappings in generalized variational inequalities involving co-coercive operartors. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol. 77, pp. 89–111. Springer, Boston (2005)Google Scholar
- 7.Chen, G., Rockafellar, R.T.: Convergence and Structure of Forward–backward Splitting Methods. Technical Report, Department of Applied Mathematics, University of Washington (1990)Google Scholar
- 8.Chen, G., Rockafellar, R.T.: Extended Forward–backward Splitting Methods and Convergence. Technical Report, Department of Applied Mathematics, University of Washington (1990)Google Scholar
- 9.Chen, G., Rockafellar, R.T.: Forward–Backward Splitting Methods in Lagrangian Optimization. Technical Report, Department of Applied Mathematics, University of Washington (1992)Google Scholar
- 13.Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)Google Scholar