An Inertial Extragradient Method for Solving Variational Inequalities Without Monotonicity

doi:10.1007/s40305-025-00665-1

Abstract

Abstract: In this work, we introduce an extragradient method with one-step inertial and self-adaptive step sizes to solve variational inequalities. We obtained weak convergence of the generated sequence under some appropriate conditions on both the iterative parameters and operator. In particular, our algorithm uses new explicit step sizes which plays an important role in improving the speed of the algorithm and weak convergence is obtained without using the monotonicity of the cost operator with one projection onto the feasible set and two evaluations of the operator per iteration. Finally, some numerical results show that our method is efficient and competitive with other methods in the literature.

Key words: Nonmonotone mapping, Tseng’s extragradient method, Variational inequality, Weak convergence

CLC Number:

47H09·47J20·65K15·90C25

Duong Viet Thong, Xiao-Huan Li, Hoang Van Thang, Luong Van Long. An Inertial Extragradient Method for Solving Variational Inequalities Without Monotonicity[J]. Journal of the Operations Research Society of China, 2026, 14(3): 1115-1139.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.jorsc.shu.edu.cn/EN/10.1007/s40305-025-00665-1

https://www.jorsc.shu.edu.cn/EN/Y2026/V14/I3/1115

[1] Abaidoo, R., Agyapong, E.K.: Financial development and institutional quality among emerging economies. J. Econ. Dev. 24, 198-216(2022)
[2] Alakoya, T.O., Mewomo, O.T., Shehu, Y.: Strong convergence results for quasimonotone variational inequalities. Math. Meth. Oper. Res. 95, 249-279(2022)
[3] Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3-11(2001)
[4] Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody. 12, 1164-1173(1976)
[5] Attouch, H., Cabot, A.: Convergence of a relaxed inertial proximal algorithm for maximally monotone operators. Math. Program. 184, 243-287(2020)
[6] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
[7] Baiocchi, C., Capelo, A.: Variational and quasivariational inequalities. Applications to free boundary problems. Wiley, New York (1984)
[8] Bǫt, R.I., Sedlmayer, M. Vuong, P.T.: A relaxed inertial forward-backward-forward algorithm for solving monotone inclusions with application to GANs (2017). arXiv:2003.07886
[9] Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradientlike approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635-646(2010)
[10] Ceng, L.C., Teboulle, M., Yao, J.C.: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J. Optim. Theory Appl. 146, 19-31(2010)
[11] Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 148, 318-335(2011)
[12] Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827-845(2011)
[13] Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119-1132(2012)
[14] Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281-295(1992)
[15] Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757-765(2015)
[16] Dong, Q.L., Lu, Y.Y., Yang, J.F.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217-2226(2016)
[17] Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, T.M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 87-102(2018)
[18] Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687-704(2018)
[19] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. I. Springer, New York (2003)
[20] Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417-437(2017)
[21] Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)
[22] Izuchukwu, C., Shehu, Y., Yao, J.C.: A simple projection method for solving quasimonotone variational inequality problems. Optim. Eng. 24, 915-938(2023)
[23] Izuchukwu, C., Shehu, Y., Yao, J.C.: New inertial forward-backward type for variational inequalities with quasi-monotonicity. J. Glob. Optim. 84, 441-464(2022)
[24] Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37-46(1990)
[25] Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Academic Press, New York (1980)
[26] Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399-412(2014)
[27] Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2001)
[28] Konnov, I.V.: Equilibrium formulations of relative optimization problems. Math. Meth. Oper. Res. 90, 137-152(2019)
[29] Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody. 12, 747-756(1976)
[30] Liu, H., Yang, J.: Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput. Optim. Appl. 77, 491-508(2020)
[31] Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control. Optim. 47, 1499-1515(2008)
[32] Maingé, P.E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146-168(2016)
[33] Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193-202(2015)
[34] Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502-520(2015)
[35] Reich, S., Thong, D.V., Cholamjiak, P., et al.: Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer. Algor. 88, 813-835(2021)
[36] Shehu, Y., Iyiola, O.S., Thong, D.V., Van, N.T.C.: An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems. Math. Meth. Oper. Res. 93, 213-242(2021)
[37] Shehu, Y., Liu, L., Mu, X., Dong, Q.L.: Analysis of versions of relaxed inertial projection and contraction method. Appl. Numer. Math. 165, 1-21(2021)
[38] Shehu, Y., Iyiola, O.S., Reich, S.: A modified inertial subgradient extragradient method for solving variational inequalities. Optim. Eng. 23, 421-449(2022)
[39] Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control. Optim. 37, 765-776(1999)
[40] Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308(1993)
[41] Thong, D.V., Hieu, D.V.: Modified Tseng’s extragradient algorithms for variational inequality problems. J. Fixed Point theory appl. 20, 152(2018).
[42] Thong, D.V., Vuong, P.T.: Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities. Optimziation 68, 2207-2226(2019)
[43] Thong, D.V., Dung, V.T., Tam, H.T.T.: A self adaptive projected gradient method for solving nonmonotone variational inequalities. J. Optim. Theory Appl. 206, 17(2025).
[44] Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38, 431-446(2000)
[45] Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399-409(2018)
[46] Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algorithms 81, 269-291(2019)
[47] Wang, F.H., Xu, H.K.: Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng’s extragradient method. Taiwan. J. Math. 16, 1125-1136(2012)
[48] Wang, K., Wang, Y., Iyiola, O.S., Shehu, Y.: Double inertial projection method for variational inequalities with quasi-monotonicity. Optimization 73, 707-739(2024)
[49] Wang, Zb., Chen, X., Yi, J., et al.: Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities. J. Glob. Optim. 82, 499522(2022)
[50] Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559-569(2014)
[51] Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algor. 80, 741-752(2019)
[52] Yang, J., Liu, H., Liu, Z.: Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 67, 2247-2258(2018)
[53] Yang, J., Liu, H.: A modified projected gradient method for monotone variational inequalities. J. Optim. Theory Appl. 179, 197-211(2018)
[54] Yao, Y., Postolache, M.: Iterative methods for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 155, 273-287(2012)
[55] Ye, M., He, Y.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60, 141-150(2015)

An Inertial Extragradient Method for Solving Variational Inequalities Without Monotonicity

PDF

Knowledge

Abstract

Cite this article

share this article

Related Articles 14

Recommended Articles

Metrics

Comments

[1]	Duong Viet Thong, Xiao-Huan Li, Pham Thi Hong Tham, Nguyen Hai Duong. Algorithms for Finding Minimum-Norm Solutions of Quasimonotone Variational Inequalities [J]. Journal of the Operations Research Society of China, 2026, 14(3): 1061-1087.
[2]	Jun Yang. A Strong Convergence Result for Solving Split Variational Inequality Problem [J]. Journal of the Operations Research Society of China, 2025, 13(2): 630-649.
[3]	Hui Zhang, Yu-Hong Dai. Mirror Frameworks for Relatively Lipschitz and Monotone-Like Variational Inequalities [J]. Journal of the Operations Research Society of China, 2025, 13(1): 83-113.
[4]	Tong-Tong Shang, Guo-Ji Tang. Expected Residual Minimization Method for Stochastic Tensor Variational Inequalities [J]. Journal of the Operations Research Society of China, 2024, 12(4): 1048-1071.
[5]	Xing-Ju Cai, Ke Guo, Fan Jiang, Kai Wang, Zhong-Ming Wu, De-Ren Han. The Developments of Proximal Point Algorithms [J]. Journal of the Operations Research Society of China, 2022, 10(2): 197-239.
[6]	Erik Alex Papa Quiroz, Paulo Roberto Oliveira. Proximal Methods with Bregman Distances to Solve VIP on Hadamard Manifolds with Null Sectional Curvature [J]. Journal of the Operations Research Society of China, 2021, 9(3): 499-523.
[7]	You-Cai Zhang, Ke Guo, Tao Wang. Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces [J]. Journal of the Operations Research Society of China, 2021, 9(1): 195-206.
[8]	Hai-Lin Sun, Xiao-Jun Chen. Two-Stage Stochastic Variational Inequalities: Theory, Algorithms and Applications [J]. Journal of the Operations Research Society of China, 2021, 9(1): 1-32.
[9]	Hong-Jin He · Kai Wang · Xing-Ju Cai ·De-Ren Han. An LQP-Based Two-Step Method for Structured Variational Inequalities [J]. Journal of the Operations Research Society of China, 2017, 5(3): 301-317.
[10]	Jiao-Jiao Hou · Chen Ling · Hong-Jin He. A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors [J]. Journal of the Operations Research Society of China, 2017, 5(1): 45-.
[11]	Ming-Lu Ye. A Cutting Hyperplane Projection Method for Solving Generalized Quasi-Variational Inequalities [J]. Journal of the Operations Research Society of China, 2016, 4(4): 483-.
[12]	Bing-Sheng He. PPA-Like Contraction Methods for Convex Optimization: A Framework Using Variational Inequality Approach [J]. Journal of the Operations Research Society of China, 2015, 3(4): 391-.
[13]	Xiang-Feng Wang. On the Convergence Rate of a Class of Proximal-Based Decomposition Methods for Monotone Variational Inequalities [J]. Journal of the Operations Research Society of China, 2015, 3(3): 347-.
[14]	?Zhong-Ping Wan ·? Jia-Wei Chen . On Bilevel Variational Inequalities [J]. Journal of the Operations Research Society of China, 2013, 1(4): 483-510.