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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 14
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Research Article

The subgradient extragradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces

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Pages 4051-4081 | Received 16 Jul 2020, Accepted 02 May 2021, Published online: 05 Jun 2021

References

  • Muu LD, Oettli W. Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 1992;18:1159–1166.
  • Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Stud. 1994;63:123–145.
  • Stampacchia G. Formes bilineaires coercitives sur les ensembles convexes. C R Acad Sci Paris. 1964;258:4413–4416.
  • Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekon Math Metody. 1976;12:747–756. (In Russian).
  • Antipin AS. On a method for convex programs using a symmetrical modification of the Lagrange function. Ekon Math Metody. 1976;12:1164–1173.
  • Tran DQ, Dung ML, Nguyen VH. Extragradient algorithms extended to equilibrium problems. Optimization. 2008;57:749–776.
  • Anh PN, An LTH. The subgradient extragradient method extended to equilibrium problems. Optimization. 2015;64(2):225–248.
  • Anh PN, An LTH. New subgradient extragradient methods for solving monotone bilevel equilibrium problems. Optimization. 2019;68(11):2099–2124.
  • Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for variational inequality problems in Euclidean space. Optimization. 2012;61:1119–1132.
  • Fang C, Chen S. Some extragradient algorithms for variational inequalities. Advances in variational and hemivariational inequalities. Cham: Springer; 2015. 145–171. (Adv. Mech Math.; 33).
  • Hieu DV. New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl Anal. 2018;97(5):811–824. DOI: 10.1080/00036811.2017.1292350.
  • Hieu DV, Cho YJ, Xiao YB. Modified extragradient algorithms for solving equilibrium problems. Optimization. 2018;67:2003–2029.
  • Jolaoso LO, Taiwo A, Alakoya TO, et al. A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods in a reflexive Banach space. J Optim Theor Appl. 2020;185(3):744–766. DOI: 10.1007/s10957-020-01672-3.
  • Jolaoso LO, Alakoya TO, Taiwo A, et al. An inertial extragradient method via viscoscity approximation approach for solving equilibrium problem in Hilbert spaces. Optimization. 2020; DOI: 10.1080/02331934.2020.1716752.
  • Reich S, Sabach S. Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemporary Math. 2012;568:225–240.
  • Rehman H, Kumam P, Cho YJ, et al. Weak convergence of explicit extragradient algorithms for solving equilibrium problems. J Inequal Appl. 2019;1:1–25.
  • Vuong PT, Strodiot JJ, Nguyen VH. Extraradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J Optim Theor Appl. 2013;155:605–627.
  • Vuong PT. On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theor Appl. 2018;176(2):399–409.
  • Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theor Appl. 2011;148:318–335.
  • Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw. 2011;26:827–845.
  • Hieu DV. Halpern subgradient extragradient method extended to equilibrium problems. Rev R Acad Cienc Exactas F'i,s Nat Ser A Math RACSAM. 2017;111:823–840.
  • Yang J, Liu H. The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim Lett. 2019; DOI: 10.1007/s1150-019-01474-1.
  • Yang J, Liu H, Liu ZX. Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization. 2018;67(12):2247–2258.
  • Takahashi W, Zembayashi K. Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009;70:45–57.
  • Hieu DV, Strodiot J-J. Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces. J Fixed Point Theory Appl. 2018;20:131.
  • Cioranescu I. Geometry of Banach spaces, duality mappings and nonlinear problems. Dordrecht: Kluwer; 1990.
  • Reich S. Book review: geometry of banach spaces, duality mappings and nonlinear problems. Bull Am Math Soc. 1992;26:367–371.
  • Takahashi Y, Hashimoto K, Kato M. On sharp uniform convexity, smoothness, and strong type, cotype inequalities. J Nonlinear Convex Anal. 2002;3:267–281.
  • Alber YI. Metric and generalized projections in Banach spaces: properties and applications. In: Kartsatos AG, editors. Theory and applications of nonlinear operators of accretive and monotone type. 1996. p. 15–50.
  • Alber YI, Reich S. An iterative method for solving a class of nonlinear operator in Banach spaces. Pan Am Math J. 1994;4:39–54.
  • Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim. 2002;13:938–945.
  • Nakajo K. Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces. Appl Math Comput. 2015;271:251–258.
  • Xu HK. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991;16:1127–1138.
  • Tiel JV. Convex analysis: an introducinisurtory text. New York: Wiley; 1984.
  • Xu HK. Iterative algorithms for nonlinear operators. J Lond Math Soc. 2002;66:240–256.
  • Maingé PE. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008;16:899–912.
  • Raeisi M, Eskandani GZ. A hybrid extragradient method for a general split equality problem involving resolvents and pseudomonotone bifunctions in Banach spaces. Calcolo. 2019;56: 15. Article no. 4.
  • Hieu DV. Modified subgradient extragradient algorithm for pseudomonotone equilibrium problems. Bul Kor Math Soc. 2018;55(5):1503–1521.
  • Jolaoso LO, Aphane M. A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems. Fixed Point Theory Appl. 2020; 2020:107. DOI: 10.1186/s13663-020-00676-y.
  • Jun Y, Liu H. The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim Lett. 2019;13:1–14.
  • Dadashi V, Iyiola OS, Shehu Y. The subgradient extragradient method for pseudomonotone equilibrium. Optim. 2020;69:901–923.
  • Nash JF. Equilibrium points in n-person games. Proc Natl Acad Sci. 1950;36:48–49.
  • Nash JF. Non-cooperative games. Ann Math. 1951;54:286–295.
  • Bauschke HH, Borwein JM. On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996;38:367–426.
  • Deutsch F. Best approximation in inner product spaces. New York: Springer; 2001.
  • Alber YI, Ryazantseva I. Nonlinear ill-posed problems of monotone type. Dordrecht: Springer; 2006.

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