Adaptive inertial subgradient extragradient methods for finding minimum-norm solutions of pseudomonotone variational inequalities. (English) Zbl 1524.47102
Summary: In this paper, four modified inertial subgradient extragradient methods with a new non-monotonic step size criterion are investigated for pseudomonotone variational inequality problems in real Hilbert spaces. Our algorithms employ two different step sizes in each iteration to update the values of iterative sequences, and they work well without the prior information about the Lipschitz constant of the operator. Strong convergence theorems of the proposed iterative schemes are established under some suitable and mild conditions. Some numerical examples are provided to demonstrate the computational efficiency and advantages of the proposed methods over other known ones.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H05 | Monotone operators and generalizations |
Keywords:
variational inequality; inertial method; extragradient method; pseudomonotone mapping; non-Lipschitz operator; real Hilbert spaces; strong convergenceReferences:
| [1] | Q. H. M. J. C. Ansari Islam Yao, Nonsmooth variational inequalities on Hadamard manifolds, Appl. Anal., 99, 340-358 (2020) · Zbl 1431.49003 · doi:10.1080/00036811.2018.1495329 |
| [2] | G. Q. L. Y. Cai Dong Peng, Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators, J. Optim. Theory Appl., 188, 447-472 (2021) · Zbl 1473.65075 · doi:10.1007/s10957-020-01792-w |
| [3] | L. C. A. X. J. C. Ceng Petruşel Qin Yao, Pseudomonotone variational inequalities and fixed points, Fixed Point Theory, 22, 543-558 (2021) · Zbl 1489.47088 · doi:10.24193/fpt-ro.2021.2.36 |
| [4] | L. C. A. X. J. C. Ceng Petruşel Qin Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70, 1337-1358 (2021) · Zbl 1486.47105 · doi:10.1080/02331934.2020.1858832 |
| [5] | Y. A. S. Censor Gibali Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148, 318-335 (2011) · Zbl 1229.58018 · doi:10.1007/s10957-010-9757-3 |
| [6] | Y. A. S. Censor Gibali Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26, 827-845 (2011) · Zbl 1232.58008 · doi:10.1080/10556788.2010.551536 |
| [7] | S. V. V. V. L. M. Denisov Semenov Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybernet. Systems Anal., 51, 757-765 (2015) · Zbl 1331.49010 · doi:10.1007/s10559-015-9768-z |
| [8] | Q. L. Y. J. L. L. T. M. Dong Cho Zhong Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70, 687-704 (2018) · Zbl 1390.90568 · doi:10.1007/s10898-017-0506-0 |
| [9] | Q. L. Dong, S. He and L. Liu, A general inertial projected gradient method for variational inequality problems, Comput. Appl. Math., 40 (2021), 168, 24 pp. · Zbl 1476.65115 |
| [10] | S. M. F. Grad Lara, Solving mixed variational inequalities beyond convexity, J. Optim. Theory Appl., 190, 565-580 (2021) · Zbl 1475.90111 · doi:10.1007/s10957-021-01860-9 |
| [11] | B. S. He, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim., 35, 69-76 (1997) · Zbl 0865.90119 |
| [12] | D. V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 111, 823-840 (2017) · Zbl 1378.65136 · doi:10.1007/s13398-016-0328-9 |
| [13] | D. V. A. Hieu Gibali, Strong convergence of inertial algorithms for solving equilibrium problems, Optim. Lett., 14, 1817-1843 (2020) · Zbl 1459.90213 · doi:10.1007/s11590-019-01479-w |
| [14] | O. S. Iyiola and Y. Shehu, Inertial version of generalized projected reflected gradient method, J. Sci. Comput., 93 (2022), 24, 38 pp. · Zbl 1519.47083 |
| [15] | L. O. T. O. A. O. T. Jolaoso Alakoya Taiwo Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 70, 387-412 (2021) · Zbl 1459.65097 · doi:10.1080/02331934.2020.1716752 |
| [16] | G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Èkonom. i Mat. Metody, 12, 747-756 (1976) · Zbl 0342.90044 |
| [17] | R. S. Kraikaew Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163, 399-412 (2014) · Zbl 1305.49012 · doi:10.1007/s10957-013-0494-2 |
| [18] | H. M. S. D. V. V. T. N. T. Linh Reich Thong Dung Lan, Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems, Numer. Algorithms, 89, 1695-1721 (2022) · Zbl 1505.47084 · doi:10.1007/s11075-021-01169-8 |
| [19] | H. J. Liu Yang, Weak convergence of iterative methods for solving quasimonotone variational inequalities, Comput. Optim. Appl., 77, 491-508 (2020) · Zbl 07342388 · doi:10.1007/s10589-020-00217-8 |
| [20] | Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25, 502-520 (2015) · Zbl 1314.47099 · doi:10.1137/14097238X |
| [21] | S. D. V. P. L. V. Reich Thong Cholamjiak Long, Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space, Numer. Algorithms, 88, 813-835 (2021) · Zbl 1486.65069 · doi:10.1007/s11075-020-01058-6 |
| [22] | S. P. Saejung Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75, 742-750 (2012) · Zbl 1402.49011 · doi:10.1016/j.na.2011.09.005 |
| [23] | Y. Q. L. D. Shehu Dong Jiang, Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optimization, 68, 385-409 (2019) · Zbl 1431.49009 · doi:10.1080/02331934.2018.1522636 |
| [24] | Y. Shehu, Q. L. Dong and L. L. Liu, Fast alternated inertial projection algorithms for pseudo-monotone variational inequalities, J. Comput. Appl. Math., 415 (2022), 114517, 16 pp. · Zbl 1524.49022 |
| [25] | Y. A. Shehu Gibali, New inertial relaxed method for solving split feasibilities, Optim. Lett., 15, 2109-2126 (2021) · Zbl 1475.90061 · doi:10.1007/s11590-020-01603-1 |
| [26] | Y. O. S. Shehu Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algorithms, 76, 259-282 (2017) · Zbl 06788836 · doi:10.1007/s11075-016-0253-1 |
| [27] | Y. O. S. Shehu Iyiola, Projection methods with alternating inertial steps for variational inequalities: weak and linear convergence, Appl. Numer. Math., 157, 315-337 (2020) · Zbl 1445.49004 · doi:10.1016/j.apnum.2020.06.009 |
| [28] | Y. Shehu, O. S. Iyiola, X. H. Li and Q. L. Dong, Convergence analysis of projection method for variational inequalities, Comput. Appl. Math., 38 (2019), 161, 21 pp. · Zbl 1438.47117 |
| [29] | M. B. Solodov Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., 87, 189-202 (2000) · Zbl 0971.90062 · doi:10.1007/s101079900113 |
| [30] | B. S. Y. Tan Cho, Inertial extragradient methods for solving pseudomonotone variational inequalities with non-Lipschitz mappings and their optimization applications, Appl. Set-Valued Anal. Optim., 3, 165-192 (2021) |
| [31] | D. V. Thong, V. T. Dung and L. V. Long, Inertial projection methods for finding a minimum-norm solution of pseudomonotone variational inequality and fixed-point problems, Comput. Appl. Math., 41 (2022), 254, 25 pp. · Zbl 1502.47086 |
| [32] | D. V. D. V. T. M. Thong Hieu Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Optim. Lett., 14, 115-144 (2020) · Zbl 1433.49016 · doi:10.1007/s11590-019-01511-z |
| [33] | D. V. Thong, Y. Shehu and O. S. Iyiola, A new iterative method for solving pseudomonotone variational inequalities with non-Lipschitz operators, Comput. Appl. Math., 39 (2020), 108, 24 pp. · Zbl 1449.47115 |
| [34] | D. V. Y. O. S. Thong Shehu Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algorithms, 84, 795-823 (2020) · Zbl 1439.47047 · doi:10.1007/s11075-019-00780-0 |
| [35] | D. V. P. T. Thong Vuong, Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities, Optimization, 68, 2207-2226 (2019) · Zbl 1502.47085 · doi:10.1080/02331934.2019.1616191 |
| [36] | P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38, 431-446 (2000) · Zbl 0997.90062 · doi:10.1137/S0363012998338806 |
| [37] | P. T. Y. Vuong Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms, 81, 269-291 (2019) · Zbl 1415.47011 · doi:10.1007/s11075-018-0547-6 |
| [38] | J. H. Yang Liu, Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer. Algorithms, 80, 741-752 (2019) · Zbl 1493.47107 · doi:10.1007/s11075-018-0504-4 |
| [39] | J. H. Z. Yang Liu Liu, Modified subgradient extragradient algorithms for solving monotone variational inequalities, Optimization, 67, 2247-2258 (2018) · Zbl 1415.49010 · doi:10.1080/02331934.2018.1523404 |
| [40] | Y. Yao, O. S. Iyiola and Y. Shehu, Subgradient extragradient method with double inertial steps for variational inequalities, J. Sci. Comput.90 (2022), 71, 29 pp. · Zbl 07458297 |
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.
