Two-step inertial Tseng's extragradient method for variational inequality and hierarchical fixed-point problems

Ghadah ALbeladi; Rehan Ali; Mohammad Farid; Ghadah ALbeladi; Rehan Ali; Mohammad Farid

doi:10.3934/math.2026795

AIMS Mathematics

2026, Volume 11, Issue 7: 19567-19592. doi: 10.3934/math.2026795

Previous Article Next Article

Research article Special Issues

Two-step inertial Tseng's extragradient method for variational inequality and hierarchical fixed-point problems

1.
Department of Mathematics, College of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
2.
Department of Mathematics, Central University of Kashmir, Ganderbal, Jammu and Kashmir-191131, India
3.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia

Received: 08 March 2026 Revised: 28 May 2026 Accepted: 10 June 2026 Published: 02 July 2026
MSC : 47H05, 47H10, 49J35, 47J25

We developed novel inertial Tseng-type extragradient methods, in both two-step and one-step formulations based on the Krasnosel'skiǐ-Mann iteration framework. The proposed schemes were designed to solve variational inequality problems (VIPs) and hierarchical fixed-point problems (HFPPs) governed by nonexpansive and quasi-nonexpansive operators in Hilbert spaces. The obtained sequences were shown to converge weakly to a shared solution, together with additional supporting results, and its applicability was demonstrated through an example. This formulation generalizes and brings together several known methods and outcomes in the literature.
- hierarchical fixed-point problem,
- variational inequality,
- iterative methods,
- Tseng's extragradient method,
- quasi-nonexpansive mapping
Citation: Ghadah ALbeladi, Rehan Ali, Mohammad Farid. Two-step inertial Tseng's extragradient method for variational inequality and hierarchical fixed-point problems[J]. AIMS Mathematics, 2026, 11(7): 19567-19592. doi: 10.3934/math.2026795

Related Papers:

Abstract

We developed novel inertial Tseng-type extragradient methods, in both two-step and one-step formulations based on the Krasnosel'skiǐ-Mann iteration framework. The proposed schemes were designed to solve variational inequality problems (VIPs) and hierarchical fixed-point problems (HFPPs) governed by nonexpansive and quasi-nonexpansive operators in Hilbert spaces. The obtained sequences were shown to converge weakly to a shared solution, together with additional supporting results, and its applicability was demonstrated through an example. This formulation generalizes and brings together several known methods and outcomes in the literature.

References

[1]	H. Abass, A. Adamu, O. Oyewole, M. Aphane, A golden ratio technique for equilibrium problem in reflexive Banach spaces, Demonstr. Math., 58 (2025), 20250178. http://dx.doi.org/10.1515/dema-2025-0178 doi: 10.1515/dema-2025-0178
[2]	G. AlNemer, R. Ali, K. R. Kazmi, Inertial KM-type extragradient scheme for solving a variational inequality and a hierarchical fixed point problems, J. Inequal. Appl., 2021 (2021), 38. http://dx.doi.org/10.1186/s13660-021-02565-3 doi: 10.1186/s13660-021-02565-3
[3]	F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38 (2000), 1102–1119. http://dx.doi.org/10.1137/S0363012998335802 doi: 10.1137/S0363012998335802
[4]	F. Alvarez, H. Attouch, An inertial proximalmethod for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. http://dx.doi.org/10.1023/A:1011253113155 doi: 10.1023/A:1011253113155
[5]	F. Alvarez, Weak convergence of a relaxed and an inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optimiz., 14 (2004), 773–782. http://dx.doi.org/10.1137/S1052623403427859 doi: 10.1137/S1052623403427859
[6]	R. I. Bot, E. R. Csetnek, C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472–487. https://doi.org/10.1016/j.amc.2015.01.017 doi: 10.1016/j.amc.2015.01.017
[7]	H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Elsever, 5 (1973).
[8]	C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. http://dx.doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[9]	A. Cabot, Proximal point algorithm controlled by a slowly vanishing term: Application to hierarchical minimization, SIAM J. Optimiz., 15 (2005), 555–572. http://dx.doi.org/10.1137/S105262340343467X doi: 10.1137/S105262340343467X
[10]	L. C. Ceng, D. Ghosh, H. U. Rahaman, X. Zhao, Composite Tseng-type extragradient algorithms with adaptive inertial correction strategy for solving bilevel split pseudomonotone VIP under split common fixed-point constraint, J. Comput. Appl. Math., 470 (2025), 116683. http://dx.doi.org/10.1016/j.cam.2025.116683 doi: 10.1016/j.cam.2025.116683
[11]	Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optimiz. Theory App., 148 (2011), 318–335. http://dx.doi.org/10.1007/s10957-010-9757-3 doi: 10.1007/s10957-010-9757-3
[12]	Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Method. Softw., 26 (2011), 827–845. https://doi.org/10.1080/10556788.2010.551536 doi: 10.1080/10556788.2010.551536
[13]	Y. Censor, A. Gibali, S. Reich, Extension of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119–1132. https://doi.org/10.1080/02331934.2010.539689 doi: 10.1080/02331934.2010.539689
[14]	P. L. Combettes, L. E. Glaudin, Quasi-nonexpansive iterations on the affine hull of orbits: From Mann's mean value algorithm to inertial methods, SIAM J. Optim., 27 (2017), 2356–2380. http://dx.doi.org/10.1137/17M112806X doi: 10.1137/17M112806X
[15]	V. T. Dung, P. K. Anh, D. V. Thong, Convergence of two-step inertial tseng's extragradient methods for quasimonotone variational inequality problems, Commun. Nonlinear Sci., 136 (2024), 108110. http://dx.doi.org/10.1016/j.cnsns.2024.108110 doi: 10.1016/j.cnsns.2024.108110
[16]	F. Facchinei, J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, New York: Springer-Verlag, 2003.
[17]	M. Farid, The subgradient extragradient method for solving mixed equilibrium problems and fixed point problems in Hilbert spaces, J. Appl. Numer. Optim., 1 (2019), 335–345. http://dx.doi.org/10.23952/jano.1.2019.3.10 doi: 10.23952/jano.1.2019.3.10
[18]	M. Farid, P. Peeyada, R. Ali, W. Cholamjiak, Extragradient method with inertial iterative technique for pseudomonotone split equilibrium and fixed point, J. Anal., 32 (2024), 1463–1485. http://dx.doi.org/10.1007/s41478-023-00695-z doi: 10.1007/s41478-023-00695-z
[19]	G. Fichera, Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 34 (1963), 138–142.
[20]	K. Goebel, W. A. Kirk, Topics in metric fixed point theory, cambridge studies in advanced mathematics, UK, Cambridge: Cambridge University Press, 1990.
[21]	K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, New York: Marcel Dekker, 1984.
[22]	P. Hartman, G. Stampacchia, On some non-linear elliptic differential-functional equation, Acta Math., 115 (1996), 271–310. http://dx.doi.org/10.1007/BF02392210 doi: 10.1007/BF02392210
[23]	Y. Huang, L. You, G. Cai, Q. L. Dong, Subgradient extragradient algorithm with double inertial steps for solving variational inequality problems and fixed point problems in Hilbert spaces, Rend. Circ. Mat. Palermo, 74 (2025), 129. http://dx.doi.org/10.1007/s12215-025-01250-4 doi: 10.1007/s12215-025-01250-4
[24]	O. S. Iyiola, Y. Shehu, Convergence results of two-step inertial proximal point algorithm, Appl. Numer. Math., 182 (2022), 57–75. https://doi.org/10.1016/j.apnum.2022.07.013 doi: 10.1016/j.apnum.2022.07.013
[25]	K. R. Kazmi, R. Ali, M. Furkan, Krasnoselski-Mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem, Numer. Algorithms, 77 (2018), 289–308. http://dx.doi.org/10.1007/s11075-017-0316-y doi: 10.1007/s11075-017-0316-y
[26]	K. R. Kazmi, R. Ali, M. Furkan, Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings, Numer. Algorithms, 79 (2018), 499–527. http://dx.doi.org/10.1007/s11075-017-0448-0 doi: 10.1007/s11075-017-0448-0
[27]	G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747–756.
[28]	C. A. Micchelli, L. Shen, Y. Xu, Proximity algorithms for image models: Denoising, Inverse Probl., 27 (2011), 045009. http://dx.doi.org/10.1088/0266-5611/27/4/045009 doi: 10.1088/0266-5611/27/4/045009
[29]	A. Moudafi, P. E. Mainge, Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory A., 2006 (2006), 95453. http://dx.doi.org/10.1155/FPTA/2006/95453 doi: 10.1155/FPTA/2006/95453
[30]	A. Moudafi, Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl., 23 (2007), 1635–1640. http://dx.doi.org/10.1088/0266-5611/23/4/015 doi: 10.1088/0266-5611/23/4/015
[31]	A. Moudafi, P. E. Mainge, Strong convergence of an iterative method for hierarchical fixed-point problems, Pac. J. Optim., 3 (2007), 529–538.
[32]	N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optimiz. Theory App., 128 (2006), 191–201. http://dx.doi.org/10.1007/s10957-005-7564-z doi: 10.1007/s10957-005-7564-z
[33]	Z. Y. Peng, D. Li, Y. Zhao, R. L. Liang, An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator, Commun. Nonlinear Sci., 127 (2023), 107549. http://dx.doi.org/10.1016/j.cnsns.2023.107549 doi: 10.1016/j.cnsns.2023.107549
[34]	H. U. Rahman, Z. Y. Peng, J. C. Yao, Approximate subgradient extragradient methods for solving variational inequality problems: Convergence analysis and applications in signal and image processing, Commun. Nonlinear Sci., 152 (2026), 109211. http://dx.doi.org/10.1016/j.cnsns.2025.109211 doi: 10.1016/j.cnsns.2025.109211
[35]	Z. Y. Peng, L. O. Jolaoso, Y. Shehu, J. C. Yao, An extrapolated projection and contraction algorithm with past iterates, Numer. Algorithms, 2026. http://dx.doi.org/10.1007/s11075-026-02318-7 doi: 10.1007/s11075-026-02318-7
[36]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 5 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
[37]	B. T. Polyak, Introduction to optimization, New York: Optimization Software, Publications Division, 1987.
[38]	C. Poon, J. Liang, Geometry of first-order methods and adaptive acceleration, arXiv Preprint, 2020. Available from: https://arXiv.org/abs/2003.03910.
[39]	C. Poon, J. Liang, Trajectory of alternating direction method of multipliers and adaptive acceleration, arXiv Preprint, 2019. Available from: https://arXiv.org/abs/1906.10114.
[40]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, B. Am. Math. Soc., 73 (1967), 591–597. https://doi.org/10.1090/S0002-9904-1967-11761-0 doi: 10.1090/S0002-9904-1967-11761-0
[41]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431–446. http://dx.doi.org/10.1137/S0363012998338806 doi: 10.1137/S0363012998338806
[42]	D. V. Thong, D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78 (2018), 1045–1060. http://dx.doi.org/10.1007/s11075-017-0412-z doi: 10.1007/s11075-017-0412-z
[43]	D. V. Thong, D. V. Hieu, Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems, Optimization, 67 (2018), 83–102. http://dx.doi.org/10.1080/02331934.2017.1377199 doi: 10.1080/02331934.2017.1377199
[44]	D. V. Thong, D. V. Hieu, Modified Tseng's extragradient algorithm for variational inequality problems, J. Fix. Point Theory A., 20 (2018), 152. http://dx.doi.org/10.1007/s11784-018-0634-2 doi: 10.1007/s11784-018-0634-2
[45]	W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optimiz. Theory App., 118 (2003), 417–428. http://dx.doi.org/10.1023/A:1025407607560 doi: 10.1023/A:1025407607560
[46]	I. Yamada, N. Ogura, Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Func. Anal. Opt., 25 (2004), 619–655. https://doi.org/10.1081/NFA-200045815 doi: 10.1081/NFA-200045815
[47]	Q. Yang, J. Zhao, Generalized KM theorem and their applications, Inverse Probl., 22 (2006), 833–844. http://dx.doi.org/10.1088/0266-5611/22/3/006 doi: 10.1088/0266-5611/22/3/006
[48]	Y. Yao, Y. C. Liou, Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed-point problems, Numer. Func. Anal. Opt., 24 (2008), 015015. http://dx.doi.org/10.1088/0266-5611/24/1/015015 doi: 10.1088/0266-5611/24/1/015015

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)