Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities
In this paper, we propose two hybrid subgradient extragradient methods for solving common solutions of variational inequalities problems (CSVIP). The first is a parallel algorithm which can be performed simultaneously while the second is a cyclic algorithm which is computed sequentially on each subp...
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| Veröffentlicht in: | Afrika mathematica Jg. 28; H. 5-6; S. 677 - 692 |
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| Abstract | In this paper, we propose two hybrid subgradient extragradient methods for solving common solutions of variational inequalities problems (CSVIP). The first is a parallel algorithm which can be performed simultaneously while the second is a cyclic algorithm which is computed sequentially on each subproblem in the family. The novelty of this paper is that we have designed the algorithms to develop possible practical numerical methods when the number of subproblems is large. The algorithms can be considered as improvements of some previously known results for CSVIPs. Numerical experiments are also performed to illustrate the efficiency of the proposed algorithms. |
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| AbstractList | In this paper, we propose two hybrid subgradient extragradient methods for solving common solutions of variational inequalities problems (CSVIP). The first is a parallel algorithm which can be performed simultaneously while the second is a cyclic algorithm which is computed sequentially on each subproblem in the family. The novelty of this paper is that we have designed the algorithms to develop possible practical numerical methods when the number of subproblems is large. The algorithms can be considered as improvements of some previously known results for CSVIPs. Numerical experiments are also performed to illustrate the efficiency of the proposed algorithms. |
| Author | Hieu, Dang Van |
| Author_xml | – sequence: 1 givenname: Dang Van surname: Hieu fullname: Hieu, Dang Van email: dv.hieu83@gmail.com organization: Department of Mathematics, Vietnam National University |
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| References | Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl. 24, 015015. doi:10.1088/0266-5611/24/1/015015 BurgerMKaltenbacherBRegularizing Newton-Kaczmart methods for nonlinear ill-posed problemsSIAM J. Numer. Anal.200644153182221737710.1137/0406137791112.65049 AlberYaIRyazantsevaINonlinear Ill-Posed Problems of Monotone Type2006DordrechtSpinger1086.47003 HieuDVA parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert spaceJ. Korean Math. Soc.201552373388331837210.4134/JKMS.2015.52.2.3731317.65151 TakahashiWNonlinear Functional Analysis2000YokohamaYokohama Publishers0997.47002 CensorYGibaliAReichSSabachSCommon solutions to variational inequalitiesSet Val. Var. Anal.201220229247291367710.1007/s11228-011-0192-x1296.47060 HaltmeierMKowarRLeitaoAScherzerOKaczmarz methods for regularizing nonlinear ill-posed equations. I. Convergence analysis.Inverse Probl. Imaging.20071289298228227010.3934/ipi.2007.1.2891123.65051 StarkHImage Recovery Theory and Applications1987OrlandoAcademic0627.94001 CensorYGibaliAReichSStrong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert spaceOptim. Methods Softw.2011264–5827845283780010.1080/10556788.2010.5515361232.58008 Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (Ed.), Advances in Imaging and Electron Physics, vol. 95. Academic, New York, pp. 155–270 (1996) KorpelevichGMThe extragradient method for finding saddle points and other problemsEkonomikai Matematicheskie Metody1976127477564511210342.90044 HieuDVParallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappingsJ. Appl. Math. Comput.2016 YamadaIButnariuDCensorYReichSThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappingsInherently Parallel Algorithms in Feasibility and Optimization and Their Applications2001AmsterdamElsevier47350410.1016/S1570-579X(01)80028-8 RockafellarRTOn the maximality of sums of nonlinear monotone operatorsTrans. Am. Math. Soc.1970149758828227210.1090/S0002-9947-1970-0282272-50222.47017 HartmanPStampacchiaGOn some non-linear elliptic diferential-functional equationsActa Math.196611527131020653710.1007/BF023922100142.38102 HarkerPTPangJ-SA damped-newton method for the linear complementarity problemLect. Appl. Math.19902626528410662870699.65054 KimTHXuHKStrong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroupsNonlinear Anal.20066411401152219681410.1016/j.na.2005.05.0591090.47059 CensorYChenWCombettesPLDavidiRHermanGTOn the effectiveness of projection methods for convex feasibility problems with linear inequality constraintsComput. Optim. Appl.20111244.90155 HieuDVAnhPKMuuLDModified hybrid projection methods for finding common solutions to variational inequality problemsComput. Optim. Appl.201606723685 AnhPKHieuDVParallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-nonexpansive mappingsJ. Appl. Math. Comput.201548241263334060510.1007/s12190-014-0801-61325.47128 CensorYGibaliAReichSThe subgradient extragradient method for solving variational inequalities in Hilbert spaceJ. Optim. Theory Appl.2011148318335278056610.1007/s10957-010-9757-31229.58018 AnhPKHieuDVParallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problemsVietnam J. Math.20151347.47038 HieuDVMuuLDAnhPKParallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappingsNumer. Algorithms201673197217353953810.1007/s11075-015-0092-506627372 BauschkeHHBorweinJMOn projection algorithms for solving convex feasibility problemsSIAM Rev.199638367426140959110.1137/S00361445932517100865.47039 SolodovMVSvaiterBFForcing strong convergence of proximal point iterations in Hilbert spaceMath. Progr.200087189202173466510.1007/s1010799001130971.90062 CezaroADHaltmeierMLeitaoAScherzerOOn steepest-descent-Kaczmarz method for regularizing systems of nonlinear ill-posed equationsAppl. Math. Comput.200820259660724356941157.65032 AnhPKBuongNHieuDVParallel methods for regularizing systems of equations involving accretive operatorsAppl. Anal.20149321362157324038110.1080/00036811.2013.8727771297.47066 HaltmeierMKowarRLeitaoAScherzerOKaczmarz methods for regularizing nonlinear ill-posed equation. II. ApplicationsInverse Probl. 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| References_xml | – reference: AnhPKHieuDVParallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problemsVietnam J. Math.20151347.47038 – reference: KimTHXuHKStrong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroupsNonlinear Anal.20066411401152219681410.1016/j.na.2005.05.0591090.47059 – reference: HieuDVAnhPKMuuLDModified hybrid projection methods for finding common solutions to variational inequality problemsComput. Optim. Appl.201606723685 – reference: HaltmeierMKowarRLeitaoAScherzerOKaczmarz methods for regularizing nonlinear ill-posed equation. II. ApplicationsInverse Probl. Imaging.20071507523230897610.3934/ipi.2007.1.5071135.65026 – reference: HieuDVParallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappingsJ. Appl. Math. Comput.2016 – reference: CensorYChenWCombettesPLDavidiRHermanGTOn the effectiveness of projection methods for convex feasibility problems with linear inequality constraintsComput. Optim. Appl.20111244.90155 – reference: CensorYGibaliAReichSSabachSCommon solutions to variational inequalitiesSet Val. Var. Anal.201220229247291367710.1007/s11228-011-0192-x1296.47060 – reference: TakahashiWNonlinear Functional Analysis2000YokohamaYokohama Publishers0997.47002 – reference: CensorYGibaliAReichSStrong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert spaceOptim. Methods Softw.2011264–5827845283780010.1080/10556788.2010.5515361232.58008 – reference: HieuDVA parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert spaceJ. Korean Math. Soc.201552373388331837210.4134/JKMS.2015.52.2.3731317.65151 – reference: HarkerPTPangJ-SA damped-newton method for the linear complementarity problemLect. Appl. Math.19902626528410662870699.65054 – reference: RockafellarRTOn the maximality of sums of nonlinear monotone operatorsTrans. Am. Math. Soc.1970149758828227210.1090/S0002-9947-1970-0282272-50222.47017 – reference: SolodovMVSvaiterBFForcing strong convergence of proximal point iterations in Hilbert spaceMath. Progr.200087189202173466510.1007/s1010799001130971.90062 – reference: StarkHImage Recovery Theory and Applications1987OrlandoAcademic0627.94001 – reference: Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl. 24, 015015. doi:10.1088/0266-5611/24/1/015015 – reference: AnhPKHieuDVParallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-nonexpansive mappingsJ. Appl. Math. Comput.201548241263334060510.1007/s12190-014-0801-61325.47128 – reference: KorpelevichGMThe extragradient method for finding saddle points and other problemsEkonomikai Matematicheskie Metody1976127477564511210342.90044 – reference: YamadaIButnariuDCensorYReichSThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappingsInherently Parallel Algorithms in Feasibility and Optimization and Their Applications2001AmsterdamElsevier47350410.1016/S1570-579X(01)80028-8 – reference: CezaroADHaltmeierMLeitaoAScherzerOOn steepest-descent-Kaczmarz method for regularizing systems of nonlinear ill-posed equationsAppl. Math. Comput.200820259660724356941157.65032 – reference: CensorYGibaliAReichSThe subgradient extragradient method for solving variational inequalities in Hilbert spaceJ. Optim. Theory Appl.2011148318335278056610.1007/s10957-010-9757-31229.58018 – reference: AlberYaIRyazantsevaINonlinear Ill-Posed Problems of Monotone Type2006DordrechtSpinger1086.47003 – reference: Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (Ed.), Advances in Imaging and Electron Physics, vol. 95. Academic, New York, pp. 155–270 (1996) – reference: HieuDVMuuLDAnhPKParallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappingsNumer. Algorithms201673197217353953810.1007/s11075-015-0092-506627372 – reference: AnhPKBuongNHieuDVParallel methods for regularizing systems of equations involving accretive operatorsAppl. Anal.20149321362157324038110.1080/00036811.2013.8727771297.47066 – reference: HartmanPStampacchiaGOn some non-linear elliptic diferential-functional equationsActa Math.196611527131020653710.1007/BF023922100142.38102 – reference: BauschkeHHBorweinJMOn projection algorithms for solving convex feasibility problemsSIAM Rev.199638367426140959110.1137/S00361445932517100865.47039 – reference: HaltmeierMKowarRLeitaoAScherzerOKaczmarz methods for regularizing nonlinear ill-posed equations. I. Convergence analysis.Inverse Probl. Imaging.20071289298228227010.3934/ipi.2007.1.2891123.65051 – reference: BurgerMKaltenbacherBRegularizing Newton-Kaczmart methods for nonlinear ill-posed problemsSIAM J. Numer. Anal.200644153182221737710.1137/0406137791112.65049 – volume: 115 start-page: 271 year: 1966 ident: 473_CR15 publication-title: Acta Math. doi: 10.1007/BF02392210 – volume: 26 start-page: 265 year: 1990 ident: 473_CR20 publication-title: Lect. Appl. 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Soc. doi: 10.1090/S0002-9947-1970-0282272-5 – volume: 38 start-page: 367 year: 1996 ident: 473_CR5 publication-title: SIAM Rev. doi: 10.1137/S0036144593251710 – volume: 12 start-page: 747 year: 1976 ident: 473_CR22 publication-title: Ekonomikai Matematicheskie Metody – ident: 473_CR28 doi: 10.1088/0266-5611/24/1/015015 – volume-title: Nonlinear Functional Analysis year: 2000 ident: 473_CR26 – start-page: 473 volume-title: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications year: 2001 ident: 473_CR27 doi: 10.1016/S1570-579X(01)80028-8 – volume: 26 start-page: 827 issue: 4–5 year: 2011 ident: 473_CR9 publication-title: Optim. Methods Softw. doi: 10.1080/10556788.2010.551536 – volume: 73 start-page: 197 year: 2016 ident: 473_CR19 publication-title: Numer. Algorithms doi: 10.1007/s11075-015-0092-5 – volume: 1 start-page: 289 year: 2007 ident: 473_CR13 publication-title: Inverse Probl. 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Imaging. doi: 10.3934/ipi.2007.1.507 – volume: 64 start-page: 1140 year: 2006 ident: 473_CR21 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2005.05.059 – volume: 48 start-page: 241 year: 2015 ident: 473_CR3 publication-title: J. Appl. Math. Comput. doi: 10.1007/s12190-014-0801-6 – year: 2016 ident: 473_CR18 publication-title: J. Appl. Math. Comput. doi: 10.1007/s12190-015-0980-9 |
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| Title | Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities |
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