ALGORITHMS FOR SOLVING STRONGLY PSEUDOMONOTONE EQUILIBRIUM PROBLEM ON HADAMARD MANIFOLDS
Main Article Content
Abstract
This paper is concerned with strongly pseudomonotone equilibrium problems on Hadamard manifolds. We have shown that sequences generated by the modified projection method converge to the unique solution of the strongly pseudomonotone equilibrium problem under given assumptions.
Keywords
Equilibrium problems, Hadamard manifolds, Strongly pseudomonotone, Modified projection method
Article Details
References
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[2] Azagra, D., Ferrera, J., Lóopez-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equa- tions on Riemannian manifolds. J. Funct. Anal. 220(2), 304-361 (2005)
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[18] Fan, J., Tan, B., Li, S.: An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds. Comp. Appl. Math. 40(2), 1-15 (2021)
[19] Hai, T.N.: Error bounds and stability of the projection method for strongly pseudomonotone equilibrium problems. Int. J. Comput. Math. 97(12), 2516-2530 (2020)
[20] Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80(4), 1413-1436 (2019)
[21] Khammahawong, K., Kumam, P., Chaipunya, P., Yao, J., Wen, C., Jirakitpuwapat, W.: An extragradient algorithm for strongly pseudomonotone equilibrium problems on Hadamard manifolds. Thai J. Math. 18, 350-371 (2020)
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[26] Duc,P.M.,Muu,L.D.,Quy,N.V.:Solution-existenceandalgorithmswiththeircon-vergence rate for strongly pseudomonotone equilibrium problems. Pacific J. Optim. 12(4), 833–845 (2016)
[27] Nguyen, L.V., Ansari, Q.H., Qin, X.: Linear conditioning, weak sharpness and finite conver- gence for equilibrium problems. J. Global Optim. 77(2), 405-424 (2020)
[28] Németh, S.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52(5), 1491- 1498 (2003)
[29] Pang, Jong-Shi: Study of equilibrium problem in Hadamard manifolds by extending some concepts of nonlinear analysis. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112, 1521-1537 (2018)
[30] Quoc, T.D., Muu, L.D.: Iterative methods for solving monotone equilibrium problems via dual gap functions. Comput. Optim. Appl 51(2), 709-728 (2012)
[31] Sakai, T.: Riemannian Geometry. American Mathematical Society, Providence (1996)
[32] Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Springer, Dordrecht (1994)
[33] Vuong, P., Strodiot, J.: A dynamical system for strongly pseudo-monotone equilibrium prob- lems. J. Optim. Theory Appl. 185(3), 767-784 (2020)
[34] Wang,X.,López,G.,Li,C.,Yao,J.-C.:EquilibriumproblemsonRiemannianmanifoldswith applications. J. Math. Anal. Appl. 473(2), 866-891 (2019)
[35] Yin,L.,Liu,H.,Yang,J.:Modifiedgoldenratioalgorithmsforpseudomonotoneequilibrium problems and variational inequalities. Appl. Math. 67, 273-296 (2022)
[2] Azagra, D., Ferrera, J., Lóopez-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equa- tions on Riemannian manifolds. J. Funct. Anal. 220(2), 304-361 (2005)
[3] Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomono- tone equilibrium problems. Numer. Algor. 76(1), 67-91 (2017)
[4] Anh, P.K., Hai, T.N.: Novel self-adaptive algorithms for non-lipschitz equilibrium problems with applications. J. Global Optim. 73(3), 637-657 (2019)
[5] Al-Homidan,S.,Ansari,Q.H.,Islam,M.:Existenceresultsandtwostepproximalpointalgo- rithm for equilibrium problems on Hadamard manifolds. Carpathian J. Math. 37(3), 393-406 (2021)
[6] Ansari, Q.H., Islam, M.: Explicit iterative algorithms for solving equilibrium problems on Hadamard manifolds. J. Nonlinear Convex Anal. 21(2), 425-439 (2020)
[7] Babu, F., Ali, A., Alkhaldi, A.H.: An extragradient method for non-monotone equilibrium problems on Hadamard manifolds with applications. Appl. Numer. Math. 180, 85-103 (2022)
[8] Bento, G., Cruz Neto, J., Melo, I.: Combinatorial convexity in Hadamard manifolds: exis- tence for equilibrium problems. J. Optim. Theory Appl. 195(3), 1087-1105 (2022)
[9] Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Tech- niques for Equilibria. Springer, Berlin (2019)
[10] Bento, G., Neto, J., Soares, P., Soubeyran, A.: A new regularization of equilibrium problems on hadamard manifolds: applications to theories of desires. Ann. Oper. Res. 316(2), 1301- 1318 (2022)
[11] Blum,E.,Oettli,W.:Fromoptimizationandvariationalinequalitiestoequilibriumproblems. Math. Student 62(1), 127-169 (1994)
[12] Chen, J., Liu, S., Chang, X.: Extragradient method and golden ratio method for equilibrium problems on Hadamard manifolds. Int. J. Comput. Math. 98(8), 1699-1712 (2021)
[13] Colao, V., L ópez, G., Marino, G., Mart n ́-M árquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388(1), 61-71 (2012)
[14] Cruz Neto, J., Jacinto, F., Soares, P., Souza, J.: On maximal monotonicity of bifunctions on Hadamard manifolds. J. Global Optim. 72(3), 591-601 (2018)
[15] Cruz Neto, J., Santos, P., Soares, P.: An extragradient method for equilibrium problems on Hadamard manifolds. Optim. Lett. 10(6), 1327?1336 (2016)
[16] Carmo, M.P.: Riemannian Geometry vol. 6. Birkhauser, Boston (1992)
[17] Duc,P.M.,Muu,L.D.,Quy,N.V.:Solution-existenceandalgorithmswiththeircon-vergence rate for strongly pseudomonotone equilibrium problems. Pacific J. Optim. 12(4), 833-845 (2016)
[18] Fan, J., Tan, B., Li, S.: An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds. Comp. Appl. Math. 40(2), 1-15 (2021)
[19] Hai, T.N.: Error bounds and stability of the projection method for strongly pseudomonotone equilibrium problems. Int. J. Comput. Math. 97(12), 2516-2530 (2020)
[20] Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80(4), 1413-1436 (2019)
[21] Khammahawong, K., Kumam, P., Chaipunya, P., Yao, J., Wen, C., Jirakitpuwapat, W.: An extragradient algorithm for strongly pseudomonotone equilibrium problems on Hadamard manifolds. Thai J. Math. 18, 350-371 (2020)
[22] Konnov, I.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)
[23] Li, C., López, G., Mart ín-Márquez, V.: Monotone vector fields and the proximal point algo- rithm on Hadamard manifolds. J. London Math. Soc. 79(3), 663-683 (2009)
[24] Ledyaev, Y., Zhu, Q.: Nonsmooth analysis on smooth manifolds. Trans. Amer. Math.Soc. 359(8), 3687-3732 (2007)
[25] Mastroeni, G.: On Auxiliary Principle for Equilibrium Problems. In: P. Daniele, Et.al.(eds.) Equilibrium Problems and Variational Models. Kluwer Academic Pub- lishers, Dordrecht, pp. 289-298 (2003)
[26] Duc,P.M.,Muu,L.D.,Quy,N.V.:Solution-existenceandalgorithmswiththeircon-vergence rate for strongly pseudomonotone equilibrium problems. Pacific J. Optim. 12(4), 833–845 (2016)
[27] Nguyen, L.V., Ansari, Q.H., Qin, X.: Linear conditioning, weak sharpness and finite conver- gence for equilibrium problems. J. Global Optim. 77(2), 405-424 (2020)
[28] Németh, S.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52(5), 1491- 1498 (2003)
[29] Pang, Jong-Shi: Study of equilibrium problem in Hadamard manifolds by extending some concepts of nonlinear analysis. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112, 1521-1537 (2018)
[30] Quoc, T.D., Muu, L.D.: Iterative methods for solving monotone equilibrium problems via dual gap functions. Comput. Optim. Appl 51(2), 709-728 (2012)
[31] Sakai, T.: Riemannian Geometry. American Mathematical Society, Providence (1996)
[32] Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Springer, Dordrecht (1994)
[33] Vuong, P., Strodiot, J.: A dynamical system for strongly pseudo-monotone equilibrium prob- lems. J. Optim. Theory Appl. 185(3), 767-784 (2020)
[34] Wang,X.,López,G.,Li,C.,Yao,J.-C.:EquilibriumproblemsonRiemannianmanifoldswith applications. J. Math. Anal. Appl. 473(2), 866-891 (2019)
[35] Yin,L.,Liu,H.,Yang,J.:Modifiedgoldenratioalgorithmsforpseudomonotoneequilibrium problems and variational inequalities. Appl. Math. 67, 273-296 (2022)