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Strong convergence of a self-adaptive inertial Tseng’s extragradient method for pseudomonotone variational inequalities and fixed point problems. (English) Zbl 1485.65072

Summary: In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

MSC:

65K15 Numerical methods for variational inequalities and related problems
47J25 Iterative procedures involving nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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[1] G. Fichera, Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 34 (1963), no. 8, 138-142. · Zbl 0128.18305
[2] G. Stampacchia, Formes bilinearies coercitives sur les ensembles convexes, C. R. Math. Acad. Sci. Paris 258 (1964), 4413-4416. · Zbl 0124.06401
[3] T. O. Alakoya, A. Taiwo, and O. T. Mewomo, On system of split generalised mixed equilibrium and fixed point problems for multivalued mappings with no prior knowledge of operator norm, Fixed Point Theory 23 (2022), no. 1, 45-74. · Zbl 1518.65055
[4] G. N. Ogwo, T. O. Alakoya, and O. T. Mewomo, Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems, Optimization 2021 (2021), https://doi.org/10.1080/02331934.2021.1981897. · Zbl 07664566
[5] G. N. Ogwo, C. Izuchukwu, and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms 88 (2021), no. 3, 1419-1456. · Zbl 07411111
[6] G. N. Ogwo, T. O. Alakoya, and O. T. Mewomo, Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces, Demonstr. Math. (2021), https://doi.org/10.1515/dema-2020-0119.
[7] Y. Censor, A. Gibali, and S. Reich, Strong convergence of subgradient extragradient methods for variational inequality problems in Hilbert space, Optim. Methods Softw. 26 (2011), 827-845. · Zbl 1232.58008
[8] Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 148 (2011), 318-335. · Zbl 1229.58018
[9] G. N. Ogwo, C. Izuchukwu, Y. Shehu, and O. T. Mewomo, Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems, J. Sci. Comput. 90 (2021), 10, https://doi.org/10.1007/s10915-021-01670-1. · Zbl 1510.47092
[10] C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl. 9 (2017), no. 2, 223-248. · Zbl 06828873
[11] S. Reich, D. V. Thong, P. Cholamjiak, and L. V. Long, Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space, Numer. Algorithms, 88 (2021), 813-835. · Zbl 1486.65069
[12] S. H. Khan, T. O. Alakoya, and O. T. Mewomo, Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach spaces, Math. Comput. Appl. 25 (2020), 54.
[13] T. O. Alakoya, A. Taiwo, O. T. Mewomo, and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat. 67 (2021), no. 1, 1-31. · Zbl 1472.65077
[14] M. Sibony, Methodes iteratives pour les equation set en equations aux derives partielles nonlinearesde type monotone, Calcolo 7 (1970), 65-183. · Zbl 0225.35010
[15] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonom. Mat. Methody 12 (1976), 747-756. · Zbl 0342.90044
[16] A. S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekonom. Math. Methody 12 (1976), no. 6, 1164-1173. · Zbl 0368.90115
[17] G. Cai, A. Gibali, O. S Iyiola, and Y. Shehu, A new double projection method for solving variational inequality in Banach space, J. Optim. Theory Appl. 178 (2018), 219-239. · Zbl 06931888
[18] V. T. Duong, V. T. Nguyen, and V. H. Dang, Accelerated hybrid and shrinking projection methods for variational inequality problems, Optimization 68 (2019), no. 5, 981-998. · Zbl 07068093
[19] R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 163 (2014), 399-412. · Zbl 1305.49012
[20] P. T. Vuong, On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities, J. Optim. Theory Appl. 176 (2018), 399-409. · Zbl 1442.47052
[21] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2000), 431-446. · Zbl 0997.90062
[22] G. Cai, Q. L. Dong, and Y. Peng, Strong convergence theorems for inertial Tseng’s extragradient method for solving variational inequality problems and fixed point problems, Optim. Lett. 15 (2021), 1457-1474.
[23] D. V. Thong and D. V. Hieu, Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algorithms 82 (2019), 761-789. · Zbl 1441.47079
[24] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 4 (1964), no. 5, 1-17. · Zbl 0147.35301
[25] T. O. Alakoya and O. T. Mewomo, Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems, Comput. Appl. Math. 2021 (2021), https://doi.org/10.1007/s40314-021-01749-3. · Zbl 1499.65251
[26] T. O. Alakoya, A. O. E. Owolabi, and O. T. Mewomo, An inertial algorithm with a self-adaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions, J. Nonlinear Var. Anal. 5 (2021), 803-829.
[27] T. O. Alakoya, A. O. E. Owolabi, and O. T. Mewomo, Inertial algorithm for solving split mixed equilibrium and fixed point problems for hybrid-type multivalued mappings with no prior knowledge of operator norm, J. Nonlinear Convex Anal. (Special Issue of J.C. Yao) (2021), (to appear). · Zbl 1468.65065
[28] E. C. Godwin, C. Izuchukwu, and O. T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital. 14 (2021), no. 2, 379-401. · Zbl 1494.47106
[29] B. Tan and X. Qin, Strong convergence of an inertial Tseng’s extra gradient algorithm for pseudomonotone variational inequalities with applications to optimal control problems, 2020, arXiv: arXiv:2007.11761v1 [math.OC].
[30] L. C. Ceng and J. C. Yao, Strong Convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10 (2006), 1293-1303. · Zbl 1110.49013
[31] L. C. Ceng, A. Petrusel, X. Qin, and J. C. Yao, A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory 21 (2020), 93-108. · Zbl 1477.47060
[32] L. C. Ceng, A. Petrusel, X. Qin, and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization 70 (2021), 1337-1358. · Zbl 1486.47105
[33] L. C. Ceng, A. Petrusel, J. C. Yao, and Y. Yao, Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions, Fixed Point Theory 20 (2019), 113-133. · Zbl 1430.49004
[34] L. C. Ceng, A. Petrusel, J. C. Yao, and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory 19 (2018), 487-501. · Zbl 1406.49010
[35] L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization 70 (2021), 715-740. · Zbl 07339862
[36] T. Y. Zhao, D. Q. Wang, L. C. Ceng, L. He, C. Y. Wang, and H. L. Fan, Quasi-inertial Tsengas extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer. Funct. Anal. Optim. 42 (2020), 69-90. · Zbl 07336635
[37] H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimisation problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim. 19 (2008), 1881-1893. · Zbl 1176.47064
[38] N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241. · Zbl 1143.47047
[39] C. Izuchukwu and Y. Shehu, Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity, J. Appl. Numer. Optim. 2 (2020), 249-277.
[40] B. Tan and S. Y. Cho, Inertial extragradient methods for solving pseudomonotone variational inequalities with non-Lipschitz mappings and their optimisation applications, Appl. Set-Valued Anal. Optim. 3 (2021), 165-192.
[41] P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47 (2008), 1499-1515. · Zbl 1178.90273
[42] P. E. Maingé, Projected subgradient techniques and viscosity methods for optimisation with variational inequality constraints, European J. Oper. Res. 205 (2010), 501-506. · Zbl 1188.49014
[43] T. O. Alakoya, L. O. Jolaoso, and O. T. Mewomo, Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat. 32, (2021), 897-923. · Zbl 1488.65149
[44] A. Taiwo, T. O. Alakoya, and O. T. Mewomo, Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with applications, Asian-Eur. J. Math. 14 (2021), no. 8, 2150137, p.31. · Zbl 1479.47074
[45] F. U. Ogbuisi and O. T. Mewomo, Solving split monotone variational inclusion problem and fixed point problem for certain multivalued maps in Hilbert spaces, Thai J. Math. 19 (2021), no. 2, 503-520. · Zbl 1484.47165
[46] M. A. Olona, T. O. Alakoya, A. O. E. Owolabi, and O. T. Mewomo, Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math. 54 (2021), 47-67. · Zbl 1468.65065
[47] M. A. Olona, T. O. Alakoya, A. O. E. Owolabi, and O. T. Mewomo, Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings, J. Nonlinear Funct. Anal. 2021 (2021), Art. ID 10, p. 21. · Zbl 1468.65065
[48] A. Gibali, S. Reich, and R. Zalas, Outer Approximation methods for solving variational inequalities in Hilbert spaces, Optimization 66 (2017), 417-437. · Zbl 1367.58006
[49] E. Kopecklá and S. Reich, A note on alternating projections in Hilbert space, J. Fixed Point Theory Appl. 12 (2012), 41-47. · Zbl 1264.41033
[50] C. E. Chidume and S. Maruster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math. 234 (2010), 861-882. · Zbl 1191.65064
[51] S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal. 75 (2012), 742-750. · Zbl 1402.49011
[52] K. Goebel and W. A. Kirk, On Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. · Zbl 0708.47031
[53] G. Marino and H. K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), 43-52. · Zbl 1095.47038
[54] D. V. Thong, Viscosity approximation methods for solving fixed point problems and split common fixed point problems, J. Fixed Point Theory Appl. 19 (2017), 1481-1499. · Zbl 1453.47021
[55] R. W. Cottle and J. C. Yao, Pseudomonotone complementary problems in Hilbert space, J. Optim. Theory Appl. 75 (1992), 281-295. · Zbl 0795.90071
[56] X. Chen, Z-b. Wang, and Z-y. Chen, A new method for solving variational inequalities and fixed points problems of demi-contractive pappings in Hilbert spaces, J. Sci. Comput. 85 (2020), no. 18. · Zbl 1460.49005
[57] L. C. Ceng, A. Petruşel, and J. C. Yao, On Mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities, Mathematics 7 (2019), no. 10, 925, . · doi:10.3390/math7100925
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