Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 69, 2020 - Issue 10
292
Views
22
CrossRef citations to date
0
Altmetric
Articles

Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces

ORCID Icon, , & ORCID Icon
Pages 2279-2304 | Received 14 May 2019, Accepted 12 Oct 2019, Published online: 07 Nov 2019

References

  • Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. Berlin: Springer; 2003.
  • Kassay G, Reich S, Sabach S. Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J Optim. 2011;21:1319–1344. doi: 10.1137/110820002
  • Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York: Academic Press; 1980.
  • Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer; 2000.
  • Konnov IV. Equilibrium models and variational inequalities. Amsterdam: Elsevier; 2007.
  • Anh PK, Buong Ng, Hieu DV. Parallel methods for regularizing systems of equations involving accretive operators. Appl Anal. 2014;93:2136–2157. doi: 10.1080/00036811.2013.872777
  • Hartman P, Stampacchia G. On some non-linear elliptic differential-functional equations. Acta Math. 1966;115:271–310. doi: 10.1007/BF02392210
  • 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: 10.1007/s10589-016-9857-6
  • Wang YM, Xiao YB, Wang X, et al. Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems. J Nonlinear Sci Appl. 2016;9:1178–1192. doi: 10.22436/jnsa.009.03.44
  • Xiao YB, Sofonea M. Generalized penalty method for elliptic variational–hemivariational inequalities. Appl Math Optim. 2019. Available from: https://doi.org/10.1007/s00245-019-09563-4
  • Sofonea M, Xiao YB, Zeng SD. Generalized penalty method for history-dependent variational–hemivariational inequalities. 2019 (submitted).
  • Sofonea M, Xiao YB, Couderc M. Optimization problems for a viscoelastic frictional contact problem with unilateral constraints. Nonlinear Anal Real World Appl. 2019;50:86–103. doi: 10.1016/j.nonrwa.2019.04.005
  • Sofonea M, Xiao YB. Tykhonov well-posedness of elliptic variational–hemivariational inequalities. Electron J Differ Equ. 2019;2019:64. doi: 10.1186/s13662-019-2009-4
  • Sofonea M, Matei A, Xiao YB. Optimal control for a class of mixed variational problems. Z Angew Math Phys. 2019;70:127. doi:10.1007/s00033-019-1173-4
  • Vuong PT, Shehu Y. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer Algor. 2019;81:269–291. doi: 10.1007/s11075-018-0547-6
  • Anh TV. Line search methods for bilevel split pseudomonotone variational inequality problems. Numer Algor. 2019;81:1067–1087. doi: 10.1007/s11075-018-0583-2
  • Shehu Y, Iyiola O. On a modified extragradient method for variational inequality problem with application to industrial electricity production. J Ind Manag Optim. 2019;15:319–342.
  • Buranakorn K, Plubtieng S, Yuying T. New forward–backward splitting methods for solving pseudomonotone variational inequalities. Thai J Math. 2018;16:489–502.
  • 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
  • Shehu Y, Dong QL, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization. 2019;68:385–409. doi: 10.1080/02331934.2018.1522636
  • Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25:502–520. doi: 10.1137/14097238X
  • Malitsky Y. Golden ratio algorithms for variational inequalities. Math Program. 2019. doi:10.1007/s10107-019-01416-w
  • Malitsky Y. Proximal extrapolated gradient methods for variational inequalities. Optim Meth Softw. 2018;33:140–164. doi: 10.1080/10556788.2017.1300899
  • Maingé PE, Gobinddass ML. Convergence of one-step projected gradient methods for variational inequalities. J Optim Theory Appl. 2016;171:146–168. doi: 10.1007/s10957-016-0972-4
  • Hieu DV, Thong DV. A new projection method for a class of variational inequalities. Appl Anal. 2018. doi:10.1080/00036811.2018.1460816
  • Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekonom Mate Metody. 1976;12:747–756.
  • Popov LD. A modification of the Arrow–Hurwicz method for searching for saddle points. Mat Zametki. 1980;28:777–784.
  • 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
  • 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
  • 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: 10.1080/02331934.2010.539689
  • Ceng LC, Teboulle M, Yao J-C. 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
  • Vuong PT. On the weak convergence of the extragradient method for solving variational inequalities. J Optim Theory Appl. 2018;176:399–409. doi: 10.1007/s10957-017-1214-0
  • Khanh PD. A modified extragradient method for infinite-dimensional variational inequalities. Acta Math Vietnam. 2016;41:251–263. doi: 10.1007/s40306-015-0150-z
  • Khanh PD. Convergence rate of a modified extragradient method for pseudomonotone variational inequalities. Vietnam J Math. 2017;45:397–408. doi: 10.1007/s10013-016-0207-x
  • Khanh PD. A new extragradient method for strongly pseudomonotone variational inequalities. Numer Funct Anal Optim. 2016;37:1131–1143. doi: 10.1080/01630563.2016.1212372
  • 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
  • Verma RU. General system of strongly pseudomonotone nonlinear variational inequalities based on projection systems. J Inequal Pure Appl Math. 2007;8:9 pages.
  • Hieu DV, Quy PK, Vy LV. Explicit iterative algorithms for solving equilibrium problems. Calcolo. 2019;56:11. doi: 10.1007/s10092-019-0308-5
  • Hieu DV, Anh PK, Muu LD. Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl. 2019;73:913–932. doi: 10.1007/s10589-019-00093-x
  • Hieu DV, Cho YJ, Xiao Y-B. Golden ratio algorithms with new stepsize rules for variational inequalities. Math Meth Appl Sci. 2019. doi:10.1002/mma.5703
  • Hieu DV, Quy PK. An inertial modified algorithm for solving variational inequalities. RAIRO – Oper Res. 2019. doi:10.1051/ro/2018115
  • Gibali A, Hieu DV. A new inertial double-projection method for solving variational inequalities. J Fixed Point Theory Appl. 2019. doi:10.1007/s11784-019-0726-7
  • Hieu DV, Gibali A. Strong convergence of inertial algorithms for solving equilibrium problems. Optim Lett. 2019. doi:10.1007/s11590-019-01479-w
  • Xia Y, Wang J. A general methodology for designing globally convergent optimization neural networks. IEEE Trans Neural Netw. 1998;9:1331–1343. doi: 10.1109/72.728383
  • Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38:431–446. doi: 10.1137/S0363012998338806
  • Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York and Basel: Marcel Dekker; 1984.
  • Cottle RW, Yao J-C. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl. 1992;75:281–295. doi: 10.1007/BF00941468
  • Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc. 1967;73:591–597. doi: 10.1090/S0002-9904-1967-11761-0
  • Bot RI, Csetnek ER, Vuong PT. The forward-backward-forward method from discrete and continuous perspective for pseudo-monotone variational inequalities in Hilbert Spaces. 2018. Available from: https://arxiv.org/abs/1808.08084

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.