We introduce two new algorithms based upon the extragradient and inertial methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. Strong converge and weak convergence of the proposed algorithms are established under some mild assumptions. Numerical results show that the proposed algorithms are more efficient than some existing methods for equilibrium problems.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator damping. Set-valued. Anal. 9, 3–11 (2001)
Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces. SIAM J. Optim. 9, 773–782 (2004)
Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)
Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. European. J. Oper. Res. 227, 1–11 (2013)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–146 (1994)
Bot, R.I., Csetnek, E.R., Nimana, N.: Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data. Optim. Lett. 12, 17–33 (2018)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Dinh, B.V., Muu, L.D.: A projection algorithm for solving pseudomonotone equilibrium problems and it’s application to a class of bilevel equilibria. Optimization 64, 559–575 (2015)
Dong, Q.-L., Lu, Y.-Y., Yang, J.-F.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217–2226 (2016)
Dong, Q.-L., Yuan, H.B., Cho, Y.J., Rassias, Th.M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 87–102 (2018)
Dong, Q.-L., Cho, Y.J., Zhong, L.L., Rassias, Th.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)
Duc, P.M., Muu, L.D., Quy, N.V.: Solution-existence and algorithms with their convergence rate for strongly pseudomonotone equilibrium problems. Pacific J. Optim. 12, 833–845 (2016)
Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65, 1855–1866 (2016)
Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities, vol. III, pp 103–113. Academic Press, New York (1972)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algorithms. https://doi.org/10.1007/s11075-018-0532-0 (2018)
Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. https://doi.org/10.1007/s00186-018-0640-6 (2018)
Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems, Optimization. https://doi.org/10.1080/02331934.2018.1505886(2018)
Iofee, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)
Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)
Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)
Kassay, G., Miholca, M., Vinh, N.T.: Vector quasi-equilibrium problems for the sum of two multivalued mappings. J. Optim. Theory Appl. 169, 424–442 (2016)
Kassay, G., Hai, T.N., Vinh, N.T.: Coupling Popov’s algorithm with subgradient extragradient method for solving equilibrium problems. J. Nonlinear Convex Anal. 19, 959–986 (2018)
Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)
Li, M., Yao, Y.: Strong convergence of an iterative algorithm for λ-strictly pseudocontractive mappings in Hilbert spaces. Analele Ştiinţifice ale Universitatii Ovidius constanţa. Seria Mathematica 18, 219–228 (2010)
Maingé, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)
Maingé, P.E.: Strong convergence of projected subgradientmethods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16, 899–911 (2009)
Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris 255, 2897–2899 (1962)
Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4, Article 18 (2003)
Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Pham, K.A., Trinh, N.H.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algor. 76, 67–91 (2017)
Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms to equilibrium Problems. J. Glob. Optim. 52, 139–159 (2012)
Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Santos, P.S.M., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)
Shehu, Y., Iyiola, O.S.: Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J. Fixed Point Theory Appl. 19, 2483–2510 (2017)
Strodiot, J.J., Vuong, P.T., Van, N.T.T.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Global Optim. 64, 159–178 (2016)
Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag Optim. https://doi.org/10.3934/jimo.2018023 (2017)
Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)
Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)
Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)
Vuong, P.T., Strodiot, J.J., Hien, N.V.: Extragradient methods and linearsearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)
The authors would like to thank the editor and referee for careful reading, and constructive suggestions that allowed to improve significantly the presentation of this paper.
The first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 101.01-2017.08.
Dedicated to Professor Le Tuan Hoa on the occasion of his 60th birthday
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Vinh, N.T., Muu, L.D. Inertial Extragradient Algorithms for Solving Equilibrium Problems. Acta Math Vietnam 44, 639–663 (2019). https://doi.org/10.1007/s40306-019-00338-1
- Inertial method
- Extragradient algorithm
- Variational inequality
- Equilibrium problem
- Lipschitz-type inequality
Mathematics Subject Classification (2010)