A parallel subgradient method extended to variational inequalities involving nonexpansive mappings

References

• Nadezhkina N, Takahashi W. Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM Optim. 2006;16:1230–1241. doi: 10.1137/050624315
   Web of Science ®Google Scholar
• Schu J. Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull Austral Math Soc. 1991;43:153–159. doi: 10.1017/S0004972700028884
   Web of Science ®Google Scholar
• Cho YJ, Argyros IK, Petrot N. Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput Math Appl. 2010;60:2292–2301. doi: 10.1016/j.camwa.2010.08.021
   Web of Science ®Google Scholar
• Takahashi S, Takahashi W. Viscosity approximation methods for equilibrium problems and fixed point in Hilbert space. J Math Anal Appl. 2007;331(1):506–515. doi: 10.1016/j.jmaa.2006.08.036
   Web of Science ®Google Scholar
• Maingés 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
• Nakajo N, Takahashi W. Strong convergence theorem for nonexpansive mappings and nonexpansive semigroups. J Math Anal Appl. 2003;279:372–379. doi: 10.1016/S0022-247X(02)00458-4
   Web of Science ®Google Scholar
• Kumam P, Katchang P. A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mapping. Nonlinear Anal Hybrid Syst. 2009;3(4):475–486. doi: 10.1016/j.nahs.2009.03.006
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PL, Luke DR. A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J Approx Theory. 2006;141:63–69. doi: 10.1016/j.jat.2006.01.003
   Web of Science ®Google Scholar
• Censor Y, Gibali A, Reich S, et al. Common solutions to variational inequalities. Set-Valued Var Anal. 2012;20:229–247. doi: 10.1007/s11228-011-0192-x
   Web of Science ®Google Scholar
• Anh PN, Le Thi HA. An Armijo-type method for pseudomonotone equilibrium problems and its applications. J Global Optim. 2013;57(3):803–820. doi: 10.1007/s10898-012-9970-8
   Web of Science ®Google Scholar
• Anh PN, Phuong NX. A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems. J Fixed Point Theory Appl. 2018;20(2):1–17. doi: 10.1007/s11784-018-0554-1
   Web of Science ®Google Scholar
• Jaiboon C, Kumam P. Strong convergence theorems for solving equilibrium problems and fixed point problems of ξ− strict pseudocontraction mappings by two hybrid projection method. J Comput Appl Math. 2010;234(3):722–732. doi: 10.1016/j.cam.2010.01.012
   Web of Science ®Google Scholar
• Anh PN, Anh TTH, Kuno T. Strong convergence theorems for variational inequalities on the solution set of Ky Fan inequalities. Acta Math Vietnamica. 2017;42:761–773. doi: 10.1007/s40306-017-0226-z
   Web of Science ®Google Scholar
• Anh PN, Muu LD, Nguyen VH, Strodiot JJ. Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J Optim Theory Appl. 2005;124:285–306. doi: 10.1007/s10957-004-0926-0
   Web of Science ®Google Scholar
• Anh PN, Thuy LQ, Anh TTH. Strong convergence theorem for the lexicographic Ky Fan inequality. Vietnam J Math. 2018;46(3):517–530. doi: 10.1007/s10013-017-0253-z
   Web of Science ®Google Scholar
• Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer-Verlag; 2000.
   Google Scholar
• Bertsekas DP, Gafni EM. Projection methods for variational inequalities with applications to the traffic assignment problem. Math Prog Study. 1982;17:139–159. doi: 10.1007/BFb0120965
   Web of Science ®Google Scholar
• Malitsky Y. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25:502–520. doi: 10.1137/14097238X
   Web of Science ®Google Scholar
• Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekon Mat Metody. 1976;12:747–756.
   Google Scholar
• Mann WR. Mean value methods in iteration. Proc Am Math Soc. 1950;4:506–510. doi: 10.1090/S0002-9939-1953-0054846-3
   Google Scholar
• Anh PN, Phuong NX. Linesearch methods for variational inequalities involving strict pseudocontractions. Optimization. 2015;64:1841–1854. doi: 10.1080/02331934.2014.906418
   Web of Science ®Google Scholar
• Xu HK. Viscosity approximation methods for nonexpansive mappings. Math Anal Appl. 2004;298:279–291. doi: 10.1016/j.jmaa.2004.04.059
   Web of Science ®Google Scholar
• Alber Y, Ryazantseva I. Nonlinear ill-posed problems of monotone type. Dordrecht: Springer, Academic Press; 2006.
   Google Scholar
• Ermoliev YM. On the method of generalized stochastic gradients and quasi-Fejér sequences. Cybernetics. 1969;5:208–220. doi: 10.1007/BF01071091
   Google Scholar
• Combettes PL, Quasi-Fejérian analysis of some optimization algorithms. In: Butnariu D, Reich S, Censor Y, editors. Inherently parallel algorithms in feasibility and optimization and their applications. Amsterdam: North-Holland; 2001. p. 115–152 (Studies in computational mathematics).
   Google Scholar
• Iusem AN, Svaiter BF, Teboulle M. Entropy-like proximal methods in convex programming. Math Oper Res. 1994;19:790–814. doi: 10.1287/moor.19.4.790
   Web of Science ®Google Scholar
• Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Program. 1994;63:123–145.
   Google Scholar
• Anh PN. A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization. 2013;62:271–283. doi: 10.1080/02331934.2011.607497
   Web of Science ®Google Scholar
• Anh PN. Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J Optim Theory Appl. 2012;154:303–320. doi: 10.1007/s10957-012-0005-x
   Web of Science ®Google Scholar
• Strodiot JJ, Vuong PT, Van NTT. A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J Global Optim. 2016;64:159–178. doi: 10.1007/s10898-015-0365-5
   Web of Science ®Google Scholar
• Vuong PT, Strodiot JJ, Nguyen VH. Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J Optim Theory Appl. 2012;155:605–627. doi: 10.1007/s10957-012-0085-7
   Web of Science ®Google Scholar
• Harker PT, Pang JS. A damped-Newton method for the linear complementarity problem. Lect Appl Math. 1990;26:265–284.
   Google Scholar