Modified Tseng's extragradient methods for solving pseudo-monotone variational inequalities

References

• Fichera G. Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincei VIII Ser Rend Cl Sci Fis Mat Nat. 1963;34:138–142.
   Google Scholar
• Fichera G. Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincei Mem Cl Sci Fis Mat Nat Sez I VIII Ser. 1964;7:91–140.
   Google Scholar
• Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. New York: Springer; 2003. (Springer Series in Operations Research, Vols. I and II).
   Google Scholar
• Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York: Academic Press; 1980.
   Google Scholar
• Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer-Verlag; 2001.
   Google Scholar
• Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335. doi: 10.1007/s10957-010-9757-3
   PubMed Web of Science ®Google Scholar
• Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw. 2011;26:827–845. doi: 10.1080/10556788.2010.551536
   Web of Science ®Google Scholar
• Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optimization. 2011;61:1119–1132. doi: 10.1080/02331934.2010.539689
   Web of Science ®Google Scholar
• Khanh PD, Vuong PT. Modified projection method for strongly pseudomonotone variational inequalities. J Global Optim. 2014;58:341–350. doi: 10.1007/s10898-013-0042-5
   Web of Science ®Google Scholar
• Kim DS, Khanh PD, Vuong PT. Qualitative properties of strongly pseudomonotone variational inequalities. Opt Lett. 2016;10:341–350. doi: 10.1007/s11590-014-0777-z
   Web of Science ®Google Scholar
• Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25:502–520. doi: 10.1137/14097238X
   Web of Science ®Google Scholar
• Malitsky YV, Semenov VV. A hybrid method without extrapolation step for solving variational inequality problems. J Glob Optim. 2015;61:193–202. doi: 10.1007/s10898-014-0150-x
   Web of Science ®Google Scholar
• Solodov MV, Svaiter BF. A new projection method for variational inequality problems. SIAM J Control Optim. 1999;37:765–776. doi: 10.1137/S0363012997317475
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Modified subgradient extragradient method for inequality variational problems. Numer Algorithms. 2018;79:597–610. doi: 10.1007/s11075-017-0452-4
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J Comput Appl Math. 2018;341:80–98. doi: 10.1016/j.cam.2018.03.019
   Web of Science ®Google Scholar
• Vuong PT. On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theory Appl. 2018;176:399–409. doi: 10.1007/s10957-017-1214-0
   PubMed Web of Science ®Google Scholar
• Vuong PT, Shehu Y. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer Algorithms. 2019;81:269–291. doi: 10.1007/s11075-018-0547-6
   Web of Science ®Google Scholar
• Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekonomika i Mat Metody. 1976;12:747–756.
   Google Scholar
• Antipin AS. On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat Metody. 1976;12:1164–1173.
   Google Scholar
• Ceng LC, Teboulle M, Yao Y. Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J Optim Theory Appl. 2010;146:19–31. doi: 10.1007/s10957-010-9650-0
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. New York: Springer; 2011. (CMS Books in Mathematics).
   Google Scholar
• Tseng P. A modified Forward-Backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38:431–446. doi: 10.1137/S0363012998338806
   Web of Science ®Google Scholar
• Boţ RI, Csetnek ER, Vuong PT. The Forward-Backward-Forward method from discrete and continuous perspective for pseudo-monotone variational inequalities in hilbert spaces. 2018. arXiv:1808.08084.
   Google Scholar
• Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker; 1984.
   Google Scholar
• Denisov SV, Semenov VV, Chabak LM. Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal. 2015;51:757–765. doi: 10.1007/s10559-015-9768-z
   Web of Science ®Google Scholar
• Iusem AN, Gárciga Otero R. Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer Funct Anal Optim. 2001;22:609–640. doi: 10.1081/NFA-100105310
   Web of Science ®Google Scholar
• Iusem AN, Nasri M. Korpelevich's method for variational inequality problems in Banach spaces. J Global Optim. 2011;50:59–76. doi: 10.1007/s10898-010-9613-x
   Web of Science ®Google Scholar
• Cottle RW, Yao JC. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl. 1992;75:281–295. doi: 10.1007/BF00941468
   Web of Science ®Google Scholar
• Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc. 1967;73:591–597. doi: 10.1090/S0002-9904-1967-11761-0
   Web of Science ®Google Scholar
• Liu LS. Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach space. J Math Anal Appl. 1995;194:114–125. doi: 10.1006/jmaa.1995.1289
   Web of Science ®Google Scholar
• Xu HK. Iterative algorithms for nonlinear operators. J Lond Math Soc. 2002;66:240–256. doi: 10.1112/S0024610702003332
   Web of Science ®Google Scholar
• Reich S. Constructive techniques for accretive and monotone operators. Applied nonlinear analysis. New York: Academic Press; 1979. p. 335–345.
   Google Scholar
• Mann WR. Mean value methods in iteration. Proc Am Math Soc. 1953;4:506–510. doi: 10.1090/S0002-9939-1953-0054846-3
   Web of Science ®Google Scholar
• Maingé PE. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim. 2008;47:1499–1515. doi: 10.1137/060675319
   Web of Science ®Google Scholar
• Yang J, Liu H. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algorithms. 2019;80:741–752. doi: 10.1007/s11075-018-0504-4
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Weak and strong convergence theorems for variational inequality problems. Numer Algorithms. 2018;78:1045–1060. doi: 10.1007/s11075-017-0412-z
   Web of Science ®Google Scholar
• Wang FH, Xu HK. Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng's extragradient method. Taiwanese J Math. 2012;16:1125–1136. doi: 10.11650/twjm/1500406682
   Web of Science ®Google Scholar