Self-adaptive inertial subgradient extragradient scheme for pseudomonotone variational inequality problem

References

[1] F. Alvarez, “Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in hilbert space,” SIAM J. Optim., vol. 14, no. 3, pp. 773–782, 2004. https://doi.org/10.1137/s1052623403427859.Search in Google Scholar

[2] F. Alvarez and H. Attouch, “An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping,” Set-Valued Anal., vol. 9, nos 1–2, pp. 3–11, 2001. https://doi.org/10.1023/a:1011253113155.10.1023/A:1011253113155Search in Google Scholar

[3] H. Attouch, J. Peypouquet, and P. Redont, “A dynamical approach to an inertial forward-backward algorithm for convex minimization,” SIAM J. Optim., vol. 24, no. 1, pp. 232–256, 2014. https://doi.org/10.1137/130910294.Search in Google Scholar

[4] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, New York. NY, John Wiley & Sons, 1984.Search in Google Scholar

[5] C. Baiocchi, Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems, Wiley, Capelo, 1984.Search in Google Scholar

[6] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408, Springer, New York, 2011.10.1007/978-1-4419-9467-7Search in Google Scholar

[7] C. Yair, A. Gibali, and S. Reich, “Strong convergence of subgradient extragradient methods for the variational inequality problem in hilbert space,” Optim. Methods Software, vol. 26, nos 4–5, pp. 827–845, 2011. https://doi.org/10.1080/10556788.2010.551536.Search in Google Scholar

[8] Y. Censor, A. Gibali, and S. Reich, “The subgradient extragradient method for solving variational inequalities in hilbert space,” J. Optim. Theor. Appl., vol. 148, no. 2, pp. 318–335, 2011. https://doi.org/10.1007/s10957-010-9757-3.Search in Google Scholar PubMed PubMed Central

[9] T. Chamnarnpan, S. Phiangsungnoen, and P. Kumam, “A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems,” Afr. Mat., vol. 26, nos 1–2, pp. 87–98, 2015. https://doi.org/10.1007/s13370-013-0187-x.Search in Google Scholar

[10] T. Chamnarnpan, N. Wairojjana, and P. Kumam, “Hierarchical fixed points of strictly pseudo contractive mappings for variational inequality problems,” SpringerPlus, vol. 2, no. 1, p. 540, 2013. https://doi.org/10.1186/2193-1801-2-540.Search in Google Scholar PubMed PubMed Central

[11] P. Cholamjiak, D. V. Thong, and Y. J. Cho, “A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems,” Acta Appl. Math., vol. 169, pp. 217–245, 2019, https://doi.org/10.1007/s10440-019-00297-7.Search in Google Scholar

[12] J. Deepho, W. Kumam, and P. Kumam, “A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems,” J. Math. Model. Algorithm. Oper. Res., vol. 13, no. 4, pp. 405–423, 2014. https://doi.org/10.1007/s10852-014-9261-0.Search in Google Scholar

[13] S. V. Denisov, V. V. Semenov, and L. M. Chabak, “Convergence of the modified extragradient method for variational inequalities with non-lipschitz operators,” Cybern. Syst. Anal., vol. 51, no. 5, pp. 757–765, 2015. https://doi.org/10.1007/s10559-015-9768-z.Search in Google Scholar

[14] Q.-L. Dong, Y.-Y. Lu, and J. Yang, “The extragradient algorithm with inertial effects for solving the variational inequality,” Optimization, vol. 65, no. 12, pp. 2217–2226, 2016. https://doi.org/10.1080/02331934.2016.1239266.Search in Google Scholar

[15] Q. L. Dong, Y. J. Cho, L. L. Zhong, and T. M. Rassias, “Inertial projection and contraction algorithms for variational inequalities,” J. Global Optim., vol. 70, no. 3, pp. 687–704, 2018. https://doi.org/10.1007/s10898-017-0506-0.Search in Google Scholar

[16] Q. L. Dong, Y. J. Cho, L. L. Zhong, and T. M. Rassias, “Inertial projection and contraction algorithms for variational inequalities,” J. Global Optim., vol. 70, no. 3, pp. 687–704, 2018. https://doi.org/10.1007/s10898-017-0506-0.Search in Google Scholar

[17] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Science & Business Media, New York, 2007.Search in Google Scholar

[18] K. Goebel and S. Reich, Uniform Convexity. Hyperbolic Geometry, and Nonexpansive, Marcel Dekker Inc, U.S.A., New York, 1984.Search in Google Scholar

[19] C. Jaiboon and P. Kumam, “An extragradient approximation method for system of equilibrium problems and variational inequality problems,” Thai J. Math., vol. 7, no. 1, pp. 77–104, 2012.Search in Google Scholar

[20] S. Kesornprom and P. Cholamjiak, “Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications,” Optimization, vol. 68, no. 12, pp. 2369–2395, 2019. https://doi.org/10.1080/02331934.2019.1638389.Search in Google Scholar

[21] P. D. Khanh and P. T. Vuong, “Modified projection method for strongly pseudomonotone variational inequalities,” J. Global Optim., vol. 58, no. 2, pp. 341–350, 2014. https://doi.org/10.1007/s10898-013-0042-5.Search in Google Scholar

[22] E. N. Khobotov, “Modification of the extra-gradient method for solving variational inequalities and certain optimization problems,” USSR Comput. Math. Math. Phys., vol. 27, no. 5, pp. 120–127, 1987. https://doi.org/10.1016/0041-5553(87)90058-9.Search in Google Scholar

[23] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31, SIAM, New York, 1980.Search in Google Scholar

[24] I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495, Springer Science & Business Media, Berlin, 2001.10.1007/978-3-642-56886-2Search in Google Scholar

[25] G. M. Korpelevich, “The extragradient method for finding saddle points and other problems,” Matecon, vol. 12, pp. 747–756, 1976.Search in Google Scholar

[26] W. Kumam, H. Piri, and P. Kumam, “Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings,” Appl. Math. Comput., vol. 248, pp. 441–455, 2014. https://doi.org/10.1016/j.amc.2014.09.118.Search in Google Scholar

[27] Y. V. Malitsky, “Projected reflected gradient methods for variational inequalities,” SIAM J. Optim., vol. 25, no. 1, 502–520, 2015.10.1137/14097238XSearch in Google Scholar

[28] Y. V. Malitsky and V. V. Semenov, “A hybrid method without extrapolation step for solving variational inequality problems,” J. Global Optim., vol. 61, no. 1, pp. 193–202, 2015. https://doi.org/10.1007/s10898-014-0150-x.Search in Google Scholar

[29] P. Marcotte, “Application of khobotovâs algorithm to variational inequalities and network equilibrium problems,” INFOR Inf. Syst. Oper. Res., vol. 29, no. 4, pp. 258–270, 1991. https://doi.org/10.1080/03155986.1991.11732174.Search in Google Scholar

[30] M. A. Noor, “Some developments in general variational inequalities,” Appl. Math. Comput., vol. 152, no. 1, pp. 199–277, 2004. https://doi.org/10.1016/s0096-3003(03)00558-7.Search in Google Scholar

[31] P. Ochs, T. Brox, and T. Pock, “Ipiasco: inertial proximal algorithm for strongly convex optimization,” J. Math. Imag. Vis., vol. 53, no. 2, pp. 171–181, 2015. https://doi.org/10.1007/s10851-015-0565-0.Search in Google Scholar

[32] P. Ochs, Y. Chen, T. Brox, and T. Pock, “Ipiano: inertial proximal algorithm for nonconvex optimization,” SIAM J. Imag. Sci., vol. 7, no. 2, pp. 1388–1419, 2014. https://doi.org/10.1137/130942954.Search in Google Scholar

[33] E. U. Ofoedu, “Strong convergence theorem for uniformly l-lipschitzian asymptotically pseudocontractive mapping in real banach space,” J. Math. Anal. Appl., vol. 321, no. 2, pp. 722–728, 2006. https://doi.org/10.1016/j.jmaa.2005.08.076.Search in Google Scholar

[34] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Am. Math. Soc., vol. 73, no. 4, pp. 591–597, 1967. https://doi.org/10.1090/s0002-9904-1967-11761-0.Search in Google Scholar

[35] Y. Shehu and P. Cholamjiak, “Iterative method with inertial for variational inequalities in hilbert spaces,” Calcolo, vol. 56, no. 1, p. 4, 2019. https://doi.org/10.1007/s10092-018-0300-5.Search in Google Scholar

[36] Y. Shehu, Q.-L. Dong, and D. Jiang, “Single projection method for pseudo monotone variational inequality in hilbert spaces,” Optimization, vol. 68, no. 1, pp. 385–409, 2019. https://doi.org/10.1080/02331934.2018.1522636.Search in Google Scholar

[37] M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM J. Contr. Optim., vol. 37, no. 3, pp. 765–776, 1999. https://doi.org/10.1137/s0363012997317475.Search in Google Scholar

[38] K. Sombut, D. Kitkuan, A. Padcharoen, and P. Kumam, “Weak convergence theorems for a class of split variational inequality problems,” 2018 International Conf. on Control, Artificial Intelligence, Robotics & Optimization (ICCAIRO), IEEE, 2018, pp. 277–282.10.1109/ICCAIRO.2018.00053Search in Google Scholar

[39] D. Sun, “A projection and contraction method for the nonlinear complementarity problem and its extensions,” Mathematica Numerica Sinica, vol. 16, pp. 183–194, 1994.Search in Google Scholar

[40] D. V. Thong and D. Van Hieu, “Inertial extragradient algorithms for strongly pseudomonotone variational inequalities,” J. Comput. Appl. Math., vol. 341, pp. 80–98, 2018. https://doi.org/10.1016/j.cam.2018.03.019.Search in Google Scholar

[41] D. V. Thong and D. Van Hieu, “Modified subgradient extragradient method for variational inequality problems,” Numer. Algorithm., vol. 79, no. 2, pp. 597–610, 2018. https://doi.org/10.1007/s11075-017-0452-4.Search in Google Scholar

[42] Habib ur Rehman, P. Kumam, Y. J. Cho, and P. Yordsorn, “Weak convergence of explicit extragradient algorithms for solving equilibrium problems,” J. Inequalities Appl., vol. 2019, no. 1, pp. 1–25, 2019. https://doi.org/10.1186/s13660-019-2233-1.Search in Google Scholar

[43] D. Van Hieu, “New inertial algorithm for a class of equilibrium problems,” Numer. Algorithm., vol. 24, no. 1, pp. 1–24, 2018, https://doi.org/10.3846/mma.2019.001.Search in Google Scholar

[44] D. Van Hieu and P. Cholamjiak, “Modified extragradient method with bregman distance for variational inequalities,” Hist. Anthropol., pp. 1–16, 2020, https://doi.org/10.1080/00036811.2020.1757078.Search in Google Scholar

[45] J. Yang, H. Liu, and Z. Liu, “Modified subgradient extragradient algorithms for solving monotone variational inequalities,” Optimization, vol. 67, no. 12, pp. 2247–2258, 2018. https://doi.org/10.1080/02331934.2018.1523404.Search in Google Scholar

[46] Y. Yao, G. Marino, and L. Muglia, “A modified korpelevich’s method convergent to the minimum-norm solution of a variational inequality,” Optimization, vol. 63, no. 4, pp. 559–569, 2014. https://doi.org/10.1080/02331934.2012.674947.Search in Google Scholar

[47] S. E. Yimer, P. Kumam, A. G. Gebrie, and R. Wangkeeree, “Inertial method for bilevel variational inequality problems with fixed point and minimizer point constraints,” Mathematics, vol. 7, no. 9, p. 841, 2019. https://doi.org/10.3390/math7090841.Search in Google Scholar