An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator. (English) Zbl 1530.47077
Summary: In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 65K15 | Numerical methods for variational inequalities and related problems |
Keywords:
bilevel variational inequality problems; subgradient extragradient algorithm; non-Lipschitz continuous; strong convergenceReferences:
| [1] | Dafermos, S., Traffic equilibrium and variational inequalities. Transp Sci, 42-54 (1980) |
| [2] | Nagurney, A.; Ramanujam, P., Transportation network policy modeling with goal targets and generalized penalty functions. Transp Sci, 3-13 (1996) · Zbl 0849.90055 |
| [3] | Liang, Y. C.; Fan, Q. N.; Shen, P. P., A hybrid newton method for stochastic variational inequality problems and application to traffic equilibrium. Asia-Pac J Oper Res, 1-25 (2021) · Zbl 1481.90300 |
| [4] | Barbagallo, A.; Scilla, G., Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model. J Math Anal Appl, 1118-1134 (2018) · Zbl 1377.49008 |
| [5] | Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math Program, 161-220 (1990) · Zbl 0734.90098 |
| [6] | Wang, D. Q.; Zhao, T. Y.; Ceng, L. C.; Yin, J.; Fu, Y. X., Strong convergence results for variational inclusions, systems of variational inequalities and fixed point problems using composite viscosity implicit methods. Optimization, 1-36 (2021) |
| [7] | Yang, J., Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities. Appl Anal, 1067-1078 (2021) · Zbl 07328937 |
| [8] | Zhao, T. Y.; Wang, D. Q.; Ceng, L. C.; He, L.; Wang, C. Y.; Fan, H. L., Quasi-inertial Tseng’s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators. Numer Funct Anal Optim, 69-90 (2020) · Zbl 07336635 |
| [9] | Ceng, L. C.; Zhu, L. J.; Yin, T. C., On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems. Open Math, 1770-1784 (2022) · Zbl 1505.49012 |
| [10] | Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., Pseudomonotone variational inequalities and fixed points. Fixed Point Theory, 543-558 (2021) · Zbl 1489.47088 |
| [11] | Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization, 1337-1358 (2021) · Zbl 1486.47105 |
| [12] | Ceng, L. C.; Shang, M. J., Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization, 715-740 (2021) · Zbl 07339862 |
| [13] | Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C., A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems. Fixed Point Theory, 93-108 (2020) · Zbl 1477.47060 |
| [14] | Reich, S.; Thong, D. V.; Cholamjiak, P.; Van Long, L., Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer Algorithms, 1-23 (2021) |
| [15] | Cholamjiak, P.; Thong, D. V.; Cho, Y. J., A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl Math, 217-245 (2020) · Zbl 1473.65076 |
| [16] | Korpelevich, G. M., The extragradient method for finding saddle points and other problems. Matecon, 747-756 (1976) · Zbl 0342.90044 |
| [17] | Tseng, P., A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim, 431-446 (2000) · Zbl 0997.90062 |
| [18] | Censor, Y.; Gibali, A.; Reich, S., The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl, 318-335 (2011) · Zbl 1229.58018 |
| [19] | Yang, J.; Liu, H., Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algorithms, 741-752 (2019) · Zbl 1493.47107 |
| [20] | Hieu, D. V.; Anh, P. K.; Muu, L. D., Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl, 913-932 (2019) · Zbl 07066034 |
| [21] | Facchinei, F.; Pang, J. S., Finite-dimensional variational inequalities and complementarity problems, Vols. I and II (2003), Springer: Springer New York · Zbl 1062.90002 |
| [22] | Mordukhovich, B., Variational analysis and applications (2018), Springer: Springer Cham · Zbl 1402.49003 |
| [23] | Duc, P. M.; Muu, L. D., A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization, 1855-1866 (2016) · Zbl 1352.65162 |
| [24] | Moudafi, A., Proximal methods for a class of bilevel monotone equilibrium problems. J Global Optim, 287-292 (2010) · Zbl 1190.90125 |
| [25] | Ceng, L. C.; Köbis, E.; Zhao, X. P., On general implicit hybrid iteration method for triple hierarchical variational inequalities with hierarchical variational inequality constraints. Optimization, 1961-1986 (2020) · Zbl 1447.49012 |
| [26] | Ceng, L. C.; Yao, J. C.; Shehu, Y., On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints. J Ineq Appl (2022) · Zbl 1511.47070 |
| [27] | He, L.; Cui, Y. L.; Ceng, L. C.; Zhao, T. Y.; Wang, D. Q.; Hu, H. Y., Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule. J Inequal Appl, 1-37 (2021) · Zbl 1504.49016 |
| [28] | Suantai, S.; Pholasa, N.; Cholamjiak, P., The modified inertial relaxed CQ algorithm for solving split feasibility problems. J Ind Manag Optim (2018) |
| [29] | Thong, D. V.; Li, X. H.; Dong, Q. L.; Cho, Y. Je.; Rassias, T. M., A projection and contraction method with adaptive step sizes for solving bilevel Pseudo-Monotone variational inequality problems. Optimization, 1-24 (2020) |
| [30] | Tan, B.; Li, S.; Qin, X., An accelerated extragradient algorithm for bilevel pseudomonotone variational inequality problems with application to optimal control problems. Rev R Acad Cienc Exactas Fís Nat Ser A Math RACSAM (2021) · Zbl 1487.47115 |
| [31] | Hieu, D. V.; Thong, D. V.; Moudai, A., Regularization projection method for solving bilevel variational inequality problem. Optim Lett, 205-229 (2021) · Zbl 1461.49012 |
| [32] | Dinh, B. V.; Muu, L. D., Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities. Acta Math Vietnamica, 529-540 (2013) · Zbl 1306.49053 |
| [33] | Thong, D. V.; Hieu, D. V., A strong convergence of modified subgradient extragradient method for solving bilevel pseudomonotone variational inequality problems. Optimization, 1313-1334 (2020) · Zbl 07201649 |
| [34] | Tan, B.; Cho, S. Y., Two projection-based methods for bilevel pseudomonotone variational inequalities involving non-Lipschitz operators. Revista de la Real Acad de Ciencias Exactas, Físicas y Nat Ser A Mat, 1-20 (2022) · Zbl 1503.47098 |
| [35] | Ceng, L. C.; Ghosh, D.; Shehu, Y.; Yao, J. C., Triple-adaptive subgradient extragradient with extrapolation procedure for bilevel split variational inequality. J Inequal Appl, 1-22 (2023) |
| [36] | Tan, B.; Cho, S. Y., Two adaptive modified subgradient extragradient methods for bilevel pseudomonotone variational inequalities with applications. Commun Nonlinear Sci Numer Simul (2022) · Zbl 07469350 |
| [37] | Saejung, S.; Yotkaew, P., Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal Theory Methods Appl, 742-750 (2012) · Zbl 1402.49011 |
| [38] | He, Y., A new double projection algorithm for variational inequalities. J Comput Appl Math, 166-173 (2006) · Zbl 1081.65066 |
| [39] | Tan, B.; Li, S.; Qin, X., Self-adaptive inertial single projection methods for variational inequalities involving non-Lipschitz and Lipschitz operators with their applications to optimal control problems. Appl Numer Math, 219-241 (2021) · Zbl 07398303 |
| [40] | Dong, Q. L.; Jiang, D.; Gibali, A., A modified subgradient extragradient method for solving the variational inequality problem. Numer Algorithms, 927-940 (2018) · Zbl 06967417 |
| [41] | Gibali, A.; Hieu, D. V., A new inertial double-projection method for solving variational inequalities. J Fixed Point Theory Appl, 1-21 (2019) · Zbl 07152628 |
| [42] | Shehu, Y.; Iyiola, O. S., Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Appl Numer Math, 315-337 (2020) · Zbl 1445.49004 |
| [43] | Maingé, P. E., A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim, 1499-1515 (2008) · Zbl 1178.90273 |
| [44] | Harker, P. T.; Pang, J. S., A damped-Newton method for the linear complementarity problem. Lect Appl Math, 265-284 (1990) · Zbl 0699.65054 |
| [45] | Hieu, V. D.; Anh, P. K.; Muu, L. D., Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl, 75-96 (2017) · Zbl 1368.65103 |
| [46] | Solodov, M. V.; Svaiter, B. F., A new projection method for variational inequality problems. SIAM J Control Optim, 765-776 (1999) · Zbl 0959.49007 |
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