Shrinking inertial extragradient methods for solving split equilibrium and fixed point problems. (English) Zbl 1498.47132
Summary: This paper presents two shrinking inertial extragradient algorithms for finding a solution of the split equilibrium and fixed point problems involving nonexpansive mappings and pseudomonotone bifunctions that satisfy Lipschitz-type continuous in the setting of real Hilbert spaces. The strong convergence theorems of the introduced algorithms are showed either with or without the prior knowledge of the Lipschitz-type constants of bifunctions under some constraint qualifications of the scalar sequences. Some numerical experiments are performed to demonstrate the computational effectiveness of the established algorithms.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 65K15 | Numerical methods for variational inequalities and related problems |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
split equilibrium problems; split fixed point problems; pseudomonotone bifunction; nonexpansive mapping; shrinking method; inertial method; extragradient method; strong convergenceReferences:
| [1] | P.N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization 62 (2013) 271-283. · Zbl 1290.90084 |
| [2] | Q.H. Ansari, N. Nimana, N. Petrot, Split hierarchical variational inequality problems and related problems, Fixed Point Theory Appl. 2014 (2014) Article no. 208. · Zbl 1325.47109 |
| [3] | C. Byrne, Y. Censor, A. Gibali, S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012) 759-775. · Zbl 1262.47073 |
| [4] | H. Iiduka, Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings, Math. Program. 159 (2016) 509-538. · Zbl 1351.65035 |
| [5] | M. Suwannaprapa, N. Petrot, S. Suantai, Weak convergence theorems for split feasibility problems on zeros of the sum of monotone operators and fixed point sets in Hilbert spaces, Fixed Point Theory Appl. 2017 (2017) Article no. 6. · Zbl 1461.47036 |
| [6] | W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510. · Zbl 0050.11603 |
| [7] | S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274-276. · Zbl 0423.47026 |
| [8] | W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008) 276-286. · Zbl 1134.47052 |
| [9] | E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994) 127-149. · Zbl 0888.49007 |
| [10] | G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res. 227 (2013) 1-11. · Zbl 1292.90315 |
| [11] | P. Daniele, F. Giannessi, A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003. · Zbl 1030.00031 |
| [12] | L.D. Muu, W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA 18 (1992) 1159-1166. · Zbl 0773.90092 |
| [13] | A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom. 15 (1999) 91-100. · Zbl 0974.65066 |
| [14] | G.M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody 12 (1976) 747-756. · Zbl 0342.90044 |
| [15] | D.Q. Tran, L.M. Dung, V.H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008) 749-776. · Zbl 1152.90564 |
| [16] | F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces, SIAM J. Optim. 9 (2004) 773-782. · Zbl 1079.90096 |
| [17] | F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator damping, Set-valued. Anal. 9 (2001) 3-11. · Zbl 0991.65056 |
| [18] | D.V. Hieu, An inertial-like proximal algorithm for equilibrium problems, Math. Meth. Oper. Res. 88 (2018) 399-415. · Zbl 06989576 |
| [19] | A. Moudafi, Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure and Appl. Math. 4 (2003) Article no. 18. · Zbl 1175.90413 |
| [20] | N.T. Vinh, L.D. Muu, Inertial extragradient algorithms for solving equilibrium problems, Acta Math. Vietnam. 44 (2019) 639-663. · Zbl 07115063 |
| [21] | D.V. Hieu, P.K. Quy, L.T. Hong, L.V. Vy, Accelerated hybrid methods for solving pseudomonotone equilibrium problems, Adv. Comput. Math. 46 (2020) Article no. 58. · Zbl 1455.65084 |
| [22] | B.V. Dinh, D.X. Son, T.V. Anh, Extragradient-proximal methods for split equilibrium and fixed point problems in Hilbert spaces, Vietnam J. Math. 45 (4) (2017) 651-668. · Zbl 06807624 |
| [23] | N. Petrot, M. Rabbani, M. Khonchaliew, V. Dadashi, A new extragradient algorithm for split equilibrium problems and fixed point problems, J. Ineq. Appl. 2019 (2019) doi:10.1186/s13660-019-2086-7. · Zbl 1499.90160 |
| [24] | M. Prangprakhon, N. Nimana, N. Petrot, A sequential constraint method for solving variational inequality over the intersection of fixed point sets, Thai J. Math. 18 (3) (2020) 1105-1123. · Zbl 1489.65096 |
| [25] | G. Mastroeni, On Auxiliary Principle for Equilibrium Problems, In: P. Daniele, F. Giannessi and A. Maugeri, (eds.) Equilibrium Problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 2003. · Zbl 1069.49009 |
| [26] | S. Karamardian, S. Schaible, J.P. Crouzeix, Characterizations of generalized monotone maps, J. Optim. Theory Appl. 76 (1993) 399-413. · Zbl 0792.90070 |
| [27] | T.D. Quoc, P.N. Anh, L.D. Muu, Dual extragradient algorithms extended to equilibrium problems, J. Glob. Optim. 52 (2012) 139-159. · Zbl 1258.90088 |
| [28] | M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90 (1996) 31-43. · Zbl 0903.49006 |
| [29] | A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics 2057, Springer-Verlag, Berlin, Heidelberg, Germany, 2012. · Zbl 1256.47043 |
| [30] | K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984. · Zbl 0537.46001 |
| [31] | D.S. Kim, B.V. Dinh, Parallel extragradient algorithms for multiple set split equilibrium problems in Hilbert spaces, Numer. Algorithms. 77 (2018) 741-761. · Zbl 1436.65077 |
| [32] | I.V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2000. · Zbl 0982.49009 |
| [33] | P.N. Anh, H.A. Le Thi, An Armijo-type method for pseudomonotone equilibrium problems and its applications, J. Global. Optim. 57 (2013) 803-820. · Zbl 1285.65040 |
| [34] | J. Contreras, M. Klusch, J.B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, EEE Trans. Power Syst. 19 (2004) 195-206. |
| [35] | S. Suantai, N. Petrot, M. Khonchaliew, Inertial extragradient methods for solving split equilibrium problems, Mathematics 9 (16) (2021) 1884. |
| [36] | H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011 · Zbl 1218.47001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
