Two Bregman projection methods for solving variational inequalities

References

• Hartman P, Stampacchia G. On some non-linear elliptic differential-functional equations. Acta Math. 1966;115:271–310. doi: https://doi.org/10.1007/BF02392210
   Web of Science ®Google Scholar
• Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York (NY): Academic Press; 1980.
   Google Scholar
• Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. Berlin: Springer; 2003.
   Google Scholar
• Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer; 2000.
   Google Scholar
• Konnov IV. Equilibrium models and variational inequalities. Amsterdam: Elsevier; 2007.
   Google Scholar
• Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekon Mat Metody. 1976;12:747–756.
   Google Scholar
• Antipin AS. On a method for convex programs using a symmetrical modification of the Lagrange function. Ekon Mat Metody. 1976;12:1164–1173.
   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: https://doi.org/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 Methods Softw. 2011;26:827–845. doi: https://doi.org/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. 2012;61:1119–1132. doi: https://doi.org/10.1080/02331934.2010.539689
   Web of Science ®Google Scholar
• Popov LD. A modification of the Arrow-Hurwicz method for searching for saddle points. Mat Zametki. 1980;28:777–784.
   Google Scholar
• Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25:502–520. doi: https://doi.org/10.1137/14097238X
   Web of Science ®Google Scholar
• Malitsky YV. Golden ratio algorithms for variational inequalities. Math Program. 2019. doi:https://doi.org/10.1007/s10107-019-01416-w
   Web of Science ®Google Scholar
• Denisov SV, Semenov VV, Stetsyuk PL. Bregman extragradient method with monotone rule of step adjustment. Cybern Syst Anal. 2019;55:377–383. doi: https://doi.org/10.1007/s10559-019-00144-5
   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: https://doi.org/10.1007/s10898-010-9613-x
   Web of Science ®Google Scholar
• Gibali A. A new Bregman projection method for solving variational inequalities in Hilbert spaces. Pure Appl Funct Anal. 2018;3:403–415.
   Google Scholar
• Semenov VV. A version of the mirror descent method to solve variational inequalities. Cybern Syst Anal. 2017;53:234–243. doi: https://doi.org/10.1007/s10559-017-9923-9
   Web of Science ®Google Scholar
• Hieu DV, Cholamjiak P. Modified extragradient method with Bregman distance for variational inequalities. Appl Anal. 2020. doi:https://doi.org/10.1080/00036811.2020.1757078
   Web of Science ®Google Scholar
• Hieu DV, Thong DV. New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J Global Optim. 2018;70:385–399. doi: https://doi.org/10.1007/s10898-017-0564-3
   Web of Science ®Google Scholar
• Hieu DV, Anh PK, Muu LD. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl. 2017;66:75–96. doi: https://doi.org/10.1007/s10589-016-9857-6
   Web of Science ®Google Scholar
• Khanh PD. A new extragradient method for strongly pseudomonotone variational inequalities. Numer Funct Anal Optim. 2016;37:1131–1143. doi: https://doi.org/10.1080/01630563.2016.1212372
   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: https://doi.org/10.1007/s10898-013-0042-5
   Web of Science ®Google Scholar
• Rockafellar RT. On the maximality of sums of nonlinear monotone operators. Trans Amer Math Soc. 1970;149:75–88. doi: https://doi.org/10.1090/S0002-9947-1970-0282272-5
   Web of Science ®Google Scholar
• Ceng LC, Teboulle M, Yao JC. Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J Optim Theory Appl. 2010;146:19–31. doi: https://doi.org/10.1007/s10957-010-9650-0
   Web of Science ®Google Scholar
• Rockafellar RT. Monotone operators and the proximal point algorithm. SIAM J Control Optim. 1976;14:877–898. doi: https://doi.org/10.1137/0314056
   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: https://doi.org/10.1007/s10957-017-1214-0
   PubMed Web of Science ®Google Scholar
• Hieu DV, Cho YJ, Xiao YB, et al. Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces. Vietnam J Math. 2020. doi:https://doi.org/10.1007/s10013-020-00447-7
   Web of Science ®Google Scholar
• Hieu DV, Cho YJ, Xiao YB, et al. Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces. Optimization. 2019;69:2279–2304. doi:https://doi.org/10.1080/02331934.2019.1683554
   Web of Science ®Google Scholar
• Hieu DV, Strodiot JJ, Muu LD. An explicit extragradient algorithm for solving variational inequalities. J Optim Theory Appl. 2020;185:476–503. doi: https://doi.org/10.1007/s10957-020-01661-6
   Web of Science ®Google Scholar
• Bregman LM. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys. 1967;7:200–217. doi: https://doi.org/10.1016/0041-5553(67)90040-7
   Google Scholar
• Reich S, Sabach S. Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp Math. 2012;568:225–240. doi: https://doi.org/10.1090/conm/568/11285
   Google Scholar
• Reich S, Sabach S. Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 2010;73:122–135. doi: https://doi.org/10.1016/j.na.2010.03.005
   Web of Science ®Google Scholar
• Reich S, Sabach S. Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer Funct Anal Optim. 2010;31:22–44. doi: https://doi.org/10.1080/01630560903499852
   Web of Science ®Google Scholar
• Beck A. First-order methods in optimization. Philadelphia (PA): Society for Industrial and Applied Mathematics; 2017.
   Google Scholar
• Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York (NY): Marcel Dekker; 1984.
   Google Scholar
• Cottle RW, Yao JC. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl. 1992;75:281–295. doi: https://doi.org/10.1007/BF00941468
   Web of Science ®Google Scholar
• Reich S, Sabach S. A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J Nonlinear Convex Anal. 2009;10:471–485.
   Web of Science ®Google Scholar
• Hieu DV, Anh PK, Muu LD. Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl. 2019;73:913–932. doi: https://doi.org/10.1007/s10589-019-00093-x
   Web of Science ®Google Scholar