A self-adaptive extragradient method for fixed-point and pseudomonotone equilibrium problems in Hadamard spaces
Tóm tắt
In this work, we study a self-adaptive extragradient algorithm for approximating a common solution of a pseudomonotone equilibrium problem and fixed-point problem for multivalued nonexpansive mapping in Hadamard spaces. Our proposed algorithm is designed in such a way that its step size does not require knowledge of the Lipschitz-like constants of the bifunction. Under some appropriate conditions, we establish the strong convergence of the algorithm without prior knowledge of the Lipschitz constants. Furthermore, we provide a numerical experiment to demonstrate the efficiency of our algorithm. This result extends and complements recent results in the literature.
Tài liệu tham khảo
Aremu, K.O., Izuchukwu, C., Mebawondu, A.A., Mewomo, O.T.: A viscosity type proximal point algorithm for monotone equilibrium problem and fixed point problem in a Hadamard space. Asian-Eur. J. Math. 14, 2150058 (2021). https://doi.org/10.1142/S1793557121500583
Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 133, 195–218 (2008)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Bokodisa, A.T., Jolaoso, L.O., Aphane, M.: A parallel hybrid Bregman subgradient extragradient method for a system of pseudomonotone equilibrium and fixed point problems. Symmetry 3, 216 (2021). https://doi.org/10.3390/sym13020216
Chaipunya, P., Kumam, P.: On the proximal point method in Hadamard spaces. Optimization 66, 1647–1665 (2017)
Colao, V., Lopez, G., Marino, G., Martín-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388, 61–77 (2012)
Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23(3), 395–419 (2003)
Dehghan, H., Rooin, J.: Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces (2014). arXiv:1410.1137v1 [math.FA]
Dhompongsa, S., Kaewkhao, A., Panyanak, B.: Lim’s theorem for multivalued mappings in CAT(0) spaces. J. Math. Anal. Appl. 312(2), 478–487 (2005)
Dhompongsa, S., Kirk, W.A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007)
Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65(4), 762–772 (2006)
Dhompongsa, S., Panyanak, B.: On △-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)
Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities, III, pp. 103–113. Academic Press, New York (1972)
Hieu, D.V.: Common solutions to pseudomonotone equilibrium problems. Bull. Iranian Math. Soc. 42(5), 1207–1219 (2016)
Hieu, D.V., Muu, L.D., Ahn, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016). https://doi.org/10.1007/s11075-015-0092-5
Hieu, D.V., Thai, B.H., Kumam, P.: Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. Adv. Oper. Theory 5, 1684–1717 (2020)
Isiogugu, F.O.: Demiclosedness principle and approximation theorems for certain classes of multivalued mappings in Hilbert spaces. Fixed Point Theory Appl. 2013, 61 (2013)
Iusem, A.N., Mohebbi, V.: Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces. Comput. Appl. Math. 39(2), 1–22 (2020). https://doi.org/10.1007/s40314-020-1076-1
Izuchukwu, C., Aremu, K.O., Mebawondu, A.A., Mewomo, O.T.: A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space. Appl. Gen. Topol. 20(1), 193–210 (2019)
Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: An inertial extragradient method via viscoscity approximation approach for solving equilibrium problem in Hilbert spaces. Optimization (2020). https://doi.org/10.1080/02331934.2020.1716752
Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems. Rend. Circ. Mat. Palermo, II Ser. 69, 711–735 (2020)
Jolaoso, L.O., Aphane, M.: A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems. Fixed Point Theory Appl. 2020, 9 (2020). https://doi.org/10.1186/s13663-020-00676-y
Jolaoso, L.O., Lukumon, G.A., Aphane, M.: Convergence theorem for system of pseudomonotone equilibrium and split common fixed point problems in Hilbert spaces. Boll. Unione Mat. Ital. (2021). https://doi.org/10.1007/s40574-020-00271-4
Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659–673 (1995)
Kakavandi, B.A., Amini, M.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)
Khammahawong, K., Kumam, P., Chaipunya, P.: An extragadient algorithm for strongly pseudomonotone equilibrium problems on Hadamard manifolds. Thai J. Math. 18(1), 350–371 (2020)
Khatibzadeh, H., Mohebbi, V.: Approximating solutions of equilibrium problems in Hadamard spaces. Miskolc Math. Notes 20, 281–297 (2019)
Kimura, Y., Kishi, Y.: Equilibrium problems and their resolvents in Hadamard spaces. J. Nonlinear Convex Anal. 19(9), 1503–1513 (2018)
Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68(12), 3689–3696 (2008)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Èkon. Mat. Metody 12, 747–756 (1976)
Kumam, P., Chaipunya, P.: Equilibrium problems and proximal algorithms in Hadamard spaces (2018). arXiv:1807.10900v1 [math.oc]
Li, C., Wang, J.H.: Newton’s method on Riemannian manifolds: Smale’s point estimation theory under the condition. IMA J. Numer. Anal. 26(2), 228–251 (2006)
Lim, T.C.: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, 179–182 (1976)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)
Ranjbar, S., Khatibzadeh, H.: Convergence and w-convergence of modified Mann iteration for a family of asymptotically nonexpansive type mappings in complete CAT(0) spaces. Fixed Point Theory 17, 151–158 (2016)
Rehman, H., Kumam, P., Cho, Y.J., Yordsorn, P.: Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J. Inequal. Appl. 2019, 282 (2019)
Rehman, H., Kumam, P., Je Cho, Y., Suleiman, Y.I., Kumam, W.: Modified popovs explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw. 36(1), 82–113 (2021)
Rehman, H.U., Kumam, P., Dong, Q.L., Peng, Y., Deebani, W.: A new Popovs subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space. Optimization 70(12), 2675–2710 (2021)
Trans, D.Q., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Yang, J., Liu, H.: A modified projected gradient method for monotone variational inequalities. J. Optim. Theory Appl. 179, 197–211 (2018)