The inertial iterative extragradient methods for solving pseudomonotone equilibrium programming in Hilbert spaces

Springer Science and Business Media LLC - Tập 2022 Số 1 - 2022
Habib ur Rehman1, Poom Kumam1,2, Ioannis K. Argyros3, Wiyada Kumam4, Meshal Shutaywi5
1Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand
2Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
3Department of Mathematical Sciences, Cameron University, Lawton, USA
4Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, Thailand
5Department of Mathematics College of Science & Arts, King Abdulaziz University, Rabigh, Saudi Arabia

Tóm tắt

AbstractIn this paper, we present new iterative techniques for approximating the solution of an equilibrium problem involving a pseudomonotone and a Lipschitz-type bifunction in Hilbert spaces. These techniques consist of two computing steps of a proximal-type mapping with an inertial term. Improved simplified stepsize rules that do not involve line search are investigated, allowing the method to be implemented more quickly without knowing the Lipschitz-type constants of a bifunction. The iterative sequences converge weakly on a specific solution to the problem when the control parameter conditions are properly specified. The numerical tests were carried out, and the results demonstrated the applicability and quick convergence of innovative approaches over earlier ones.

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Tài liệu tham khảo

Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22(3), 265 (1954)

Attouch, F.A.H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Var. Anal. 9, 3–11 (2001)

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. CMS Books in Mathematics. Springer, Berlin (2017)

Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90(1), 31–43 (1996)

Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)

Blum, E.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

Browder, F., Petryshyn, W.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20(2), 197–228 (1967)

Ceng, L.-C., Yao, J.-C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214(1), 186–201 (2008)

Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148(2), 318–335 (2010)

Cournot, A.A.: Recherches sur les Principes Mathématiques de la Théorie des Richesses. Hachette, Paris (1838)

Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2002)

Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. 88(3), 399–415 (2018)

Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67(11), 2003–2029 (2018)

Konnov, I.: Equilibrium Models and Variational Inequalities, vol. 210. Elsevier, Amsterdam (2007)

Liu, L., Cho, S.Y., Yao, J.-C.: Convergence analysis of an inertial Tseng’s extragradient algorithm for solving pseudomonotone variational inequalities and applications. J. Nonlinear Var. Anal. 5(4), 627–644 (2021)

Lyashko, S.I., Semenov, V.V.: A new two-step proximal algorithm of solving the problem of equilibrium programming. In: Optimization and Its Applications in Control and Data Sciences, pp. 315–325. Springer, Berlin (2016)

Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Equilibrium Problems and Variational Models. Nonconvex Optimization and Its Applications, pp. 289–298. Springer, Boston (2003)

Muangchoo, K., ur Rehman, H., Kumam, P.: Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications. J. Appl. Math. Comput. 67(1–2), 891–917 (2021)

Muu, L., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal., Theory Methods Appl. 18(12), 1159–1166 (1992)

Nash, J.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

Nash, J.F., et al.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36(1), 48–49 (1950)

Ogbuisi, F., Iyiola, O., Ngnotchouye, J., Shumba, T.: On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems. J. Nonlinear Funct. Anal. 2021(1), Article ID 4 (2021)

Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–598 (1967)

Peypouquet, J.: Convex Optimization in Normed Spaces: Theory, Methods and Examples. Springer, Berlin (2015)

Polyak, B.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)

Popov, L.D.: A modification of the Arrow–Hurwicz method for search of saddle points. Math. Notes Acad. Sci. USSR 28(5), 845–848 (1980)

Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

ur Rehman, H., Gibali, A., Kumam, P., Sitthithakerngkiet, K.: Two new extragradient methods for solving equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115(2), 75 (2021)

ur Rehman, H., Kumam, P., Cho, Y.J., Suleiman, Y.I., Kumam, W.: Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw. 36(1), 82–113 (2020)

ur Rehman, H., Kumam, P., Dong, Q.-L., Cho, Y.J.: A modified self-adaptive extragradient method for pseudomonotone equilibrium problem in a real Hilbert space with applications. Math. Methods Appl. Sci. 44(5), 3527–3547 (2020)

ur Rehman, H., Kumam, P., Gibali, A., Kumam, W.: Convergence analysis of a general inertial projection-type method for solving pseudomonotone equilibrium problems with applications. J. Inequal. Appl. 2021(1), 63 (2021)

ur Rehman, H., Pakkaranang, N., Kumam, P., Cho, Y.J.: Modified subgradient extragradient method for a family of pseudomonotone equilibrium problems in real a Hilbert space. J. Nonlinear Convex Anal. 21(9), 2011–2025 (2020)

Van Hieu, D., Duong, H.N., Thai, B.: Convergence of relaxed inertial methods for equilibrium problems. J. Appl. Numer. Optim. 3(1), 215–229 (2021)

Wang, S., Zhang, Y., Ping, P., Cho, Y., Guo, H.: New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces. Filomat 33(6), 1677–1693 (2019)

Yang, J.: The iterative methods for solving pseudomontone equilibrium problems. J. Sci. Comput. 84(3), 50 (2020)

Yao, Y., Iyiola, O.S., Shehu, Y.: Subgradient extragradient method with double inertial steps for variational inequalities. J. Sci. Comput. 90(2), 71 (2022)

Yao, Y., Li, H., Postolache, M.: Iterative algorithms for split equilibrium problems of monotone operators and fixed point problems of pseudo-contractions. Optimization, 1–19 (2020). https://doi.org/10.1080/02331934.2020.1857757

Zhao, X., Köbis, M.A., Yao, Y., Yao, J.-C.: A projected subgradient method for nondifferentiable quasiconvex multiobjective optimization problems. J. Optim. Theory Appl. 190(1), 82–107 (2021)

Zhu, L.-J., Yao, Y., Postolache, M.: Projection methods with linesearch technique for pseudomonotone equilibrium problems and fixed point problems. UPB Sci. Bull., Ser. A, Appl. Math. Phys. 83(1), 3–14 (2021)

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Tập 2022 Số 1

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