On the linear convergence of the alternating direction method of multipliers

We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two conve...

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Vydáno v:	Mathematical programming Ročník 162; číslo 1-2; s. 165 - 199
Hlavní autoři:	Hong, Mingyi, Luo, Zhi-Quan
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017 Springer Nature B.V
Témata:	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Convergence Convex analysis Convexity Decomposition Errors Feasibility Full Length Paper Inequality Lagrange multiplier Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Methods Multipliers Numerical Analysis Optimization Studies Texts Theoretical Error bound Linear convergence Dual ascent 49 Alternating directions of multipliers 90
ISSN:	0025-5610, 1436-4646
On-line přístup:	Získat plný text
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Abstract	We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth ℓ 1 regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth ... regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function. We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth ℓ 1 regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function.
Author	Luo, Zhi-Quan Hong, Mingyi
Author_xml	– sequence: 1 givenname: Mingyi surname: Hong fullname: Hong, Mingyi organization: Department of Industrial and Manufacturing Systems Engineering, Iowa State University – sequence: 2 givenname: Zhi-Quan surname: Luo fullname: Luo, Zhi-Quan email: luozq@umn.edu organization: Department of Electrical and Computer Engineering, University of Minnesota, School of Science and Engineering, The Chinese University of Hong Kong
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ContentType	Journal Article
Copyright	Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016 Mathematical Programming is a copyright of Springer, 2017.
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Keywords	Error bound Linear convergence Dual ascent 49 Alternating directions of multipliers 90
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References	De PierroARIusemANOn the convergence properties of Hildreth’s quadratic programming algorithmMath. Prog.1990473751105484010.1007/BF015808510712.90054 KontogiorgisSMeyerRRA variable-penalty alternating directions method for convex optimizationMath. Program.199883295316439630920.90118 BertsekasDPTsitsiklisJNParallel and Distributed Computation: Numerical Methods1989Englewood CliffsPrentice-Hall0743.65107 EcksteinJBertsekasDPOn the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operatorsMath. Program.199255293318116818310.1007/BF015812040765.90073 Goldfarb, D., Ma, S.: Fast multiple splitting algorithms for convex optimization. SIAM J. Optim. 22(2), 533–556 (2012) CottleRWDuvallSGZikanKA Lagrangian relaxation algorithm for the constrained matrix problemNav. Res. Logist. Q.198633557610.1002/nav.38003301060598.90070 Ventura, J.A., Hearn, D.W.: Computational Development of a Lagrangian Dual Approach for Quadratic Networks. 21(4), 469–485 (1991) BertsekasDPGafniEProjection methods for variational inequalities with application to the traffic assignment problemMath. Prog. Study19821713915965469710.1007/BFb01209650478.90071 HeBSTaoMYuanXMAlternating direction method with gaussian back substitution for separable convex programmingSIAM J. Optim.201222313340296885610.1137/1108223471273.90152 Zhang, H., Jiang, J.J., Luo, Z.-Q.: On the linear convergence of a proximal gradient method for a class of nonsmooth convex minimization problems. J. Oper. Res. Soc. Chin. 1(2), 163–186 (2013) GoldsteinAAConvex programming in hilbert spaceBull. Am. Math. Soc.19647070971016598210.1090/S0002-9904-1964-11178-20142.17101 LionsPLMercierBSplitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. Anal.19791696497955131910.1137/07160710426.65050 BoydSParikhNChuEPeleatoBEcksteinJDistributed optimization and statistical learning via the alternating direction method of multipliersFound. Trends Mach. Learn.20113112210.1561/22000000161229.90122(Michael Jordan, Editor in Chief) Deng, W., Yin, W.: On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers. J. Sci. Comput. 66(3), 889–916 (2012) LuoZ-QTsengPOn the convergence of the coordinate descent method for convex differentiable minimizationJ. Optim. Theory Appl.199272735114176410.1007/BF009399480795.90069 HeBSYuanXMOn the O(1/n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/n)$$\end{document} convergence rate of the Douglas–Rachford alternating direction methodSIAM J. Numer. Anal.201250700709291428210.1137/1108369361245.90084 MonteiroRSvaiterBIteration-complexity of block-decomposition algorithms and the alternating direction method of multipliersSIAM J. Optim.201323475507303311610.1137/1108494681267.90181 GlowinskiRNumerical Methods for Nonlinear Variational Problems1984New YorkSpringer10.1007/978-3-662-12613-40536.65054 OrtegaJMRheinboldtWCIterative Solution of Nonlinear Equations in Several Variables1970New YorkAcademic Press0241.65046 BertsekasDPHoseinPATsengPRelaxation methods for network flow problems with convex arc costsSIAM J. Control Optim.1987251219124390504210.1137/03250670641.90036 Ma, S.: Alternating proximal gradient method for convex minimization. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0150-0 GabayDFortinMGlowinskiRApplication of the method of multipliers to varuational inequalitiesAugmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problem1983AmsterdamNorth-Holland29933110.1016/S0168-2024(08)70034-1 BertsekasDPNonlinear Programming1999BelmontAthena Scientific1015.90077 YangJFZhangYAlternating direction algorithms for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_1$$\end{document}-problems in compressive sensingSIAM J. Sci. Comput.201133250278278319410.1137/0907777611256.65060 RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/97814008731730193.18401 LuoZ-QTsengPOn the Linear convergence of descent methods for convex essentially smooth minimizationSIAM J. Control Optim.199230408425114907610.1137/03300250756.90084 GlowinskiRLe TallecPAugmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics1989PhiladelphiaSIAM Studies in Applied Mathematics10.1137/1.97816119708380698.73001 ZeniosSAMulveyJMRelaxation techniques for strictly convex network problemsAnn. Oper. Res.1986551753894803310.1007/BF02739237 Boley, D.: Local Linear Convergence of the Alternating Direction Method of Multipliers on Quadratic or Linear Programs. SIAM J. Optim. 23(4), 2183–2207 (2013) LuoZ-QTsengPOn the convergence rate of dual ascent methods for strictly convex minimizationMath. Oper. Res.199318846867125168310.1287/moor.18.4.8460804.90103 Zhou, Z., Li, X., Wright, J., Candes, E.J., Ma, Y.: Stable principal component pursuit. In: Proceedings of 2010 IEEE International Symposium on Information Theory (2010) DouglasJRachfordHHOn the numerical solution of the heat conduction problem in 2 and 3 space variablesTrans. Am. Math. Soc.1956824214398419410.1090/S0002-9947-1956-0084194-40070.35401 GabayDMercierBA dual algorithm for the solution of nonlinear variational problems via finite-element approximationsComput. Math. Appl.19762174010.1016/0898-1221(76)90003-10352.65034 Goldstein, T., O’Donoghue, B., Setzer, S.: Fast alternating direction optimization methods. SIAM J. Imaging Sci, 7(3), 1588–1623 (2014) MangasarianOLShiauT-HLipschitz continuity of solutions of linear inequalities, programs and complementarity problemsSIAM J. Control Optim.19872558359588518710.1137/03250330613.90066 OhuchiAKajiILagrangian dual coordinatewise maximization algorithm for network transportation problems with quadratic costsNetworks19841451553010.1002/net.32301404040585.90062 Levitin, E.S., Poljak, B.T.: Constrained minimization methods. Z. Vycisl. Mat. i Mat. Fiz. 6, 787–823 (1965). English translation in USSR Comput. Math. Phys. 6, 1–50 (1965) BregmanLMThe relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programmingUSSR Comput. Math. Math. Phys.1967720021721561710.1016/0041-5553(67)90040-70186.23807 TaoMYuanXMRecovering low-rank and sparse components of matrices from incomplete and noisy observationsSIAM J. Optim.2011215781276548910.1137/1007818941218.90115 TsengPBertsekasDPRelaxation methods for problems with strictly convex costs and linear constraintsMath. Oper. Res.199116462481112046410.1287/moor.16.3.4620755.90067 WangXFYuanXMThe linearized alternating direction method of multipliers for dantzig selectorSIAM J. Sci. Comput.20123427922811302372610.1137/1108335431263.90061 Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Prog. 125(2), 263–295 (2010) IusemANOn dual convergence and the rate of primal convergence of bregman’s convex programming methodSIAM J. Control Optim.19911401423111252710.1137/08010250753.90051 EcksteinJSvaiterBFGeneral projective splitting methods for sums of maximal monotone operatorsSIAM J. Control Optim.201048787811248609410.1137/0706988161194.49038 Goldfarb, D., Ma, S., Scheinberg, K.: Fast alternating linearization methods for minimizing the sum of two convex functions. Math. Prog. A. 141(1,2), 349–382 (2013) YuanMLinYModel selection and estimation in regression with grouped variablesJ. R. Stat. Soc. Ser. B (Statistical Methodology)2006684967221257410.1111/j.1467-9868.2005.00532.x1141.62030 PangJ-SA posteriori error bounds for the linearly-constrained variational inequality problemMath. Oper. Res.19871247448490641910.1287/moor.12.3.474 TsengPBertsekasDPRelaxation methods for problems with strictly convex separable costs and linear constraintsMath. Prog.19873830332190376910.1007/BF025920170636.90072 Eckstein, J.: Splitting Methods for Monotone Operators with Applications to Parallel Optimization. Ph.D. Thesis, Operations Research Center, MIT (1989) HoffmanAJOn approximate solutions of systems of linear inequalitiesJ. Res. Nat. Bur. Stand.1952492632655127510.6028/jres.049.027 TsengPDual ascent methods for problems with strictly convex costs and linear constraints: a unified approachSIAM J. Control Optim.199028214242103598010.1137/03280110692.49025 Chen, C., He, B. , Yuan, X., Ye, Y.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Prog. 155(1), 57–79 (2013) PangJ-SOn the Convergence of Dual Ascent Methods for Large Scale Linearly Constrained Optimization Problems1984DallasThe University of Texas, School of Management LinYYPangJ-SIterative methods for large convex quadratic programs: a surveySIAM J. Control Optim.19871838341187706910.1137/03250230624.90083 DP Bertsekas (1034_CR3) 1987; 25 BS He (1034_CR24) 2012; 22 DP Bertsekas (1034_CR4) 1989 D Gabay (1034_CR16) 1983 1034_CR45 P Tseng (1034_CR44) 1990; 28 J Eckstein (1034_CR15) 2010; 48 DP Bertsekas (1034_CR1) 1999 J-S Pang (1034_CR41) 1987; 12 1034_CR48 R Glowinski (1034_CR19) 1989 AR Pierro De (1034_CR10) 1990; 47 PL Lions (1034_CR30) 1979; 16 1034_CR54 Z-Q Luo (1034_CR34) 1993; 18 A Ohuchi (1034_CR38) 1984; 14 P Tseng (1034_CR47) 1991; 16 SA Zenios (1034_CR53) 1986; 5 1034_CR11 1034_CR52 1034_CR13 J-S Pang (1034_CR40) 1984 DP Bertsekas (1034_CR2) 1982; 17 JM Ortega (1034_CR39) 1970 S Boyd (1034_CR6) 2011; 3 J Douglas (1034_CR12) 1956; 82 M Yuan (1034_CR51) 2006; 68 RT Rockafellar (1034_CR42) 1970 D Gabay (1034_CR17) 1976; 2 JF Yang (1034_CR50) 2011; 33 1034_CR21 BS He (1034_CR25) 2012; 50 1034_CR20 1034_CR23 Z-Q Luo (1034_CR33) 1992; 30 J Eckstein (1034_CR14) 1992; 55 1034_CR29 RW Cottle (1034_CR9) 1986; 33 AA Goldstein (1034_CR22) 1964; 70 S Kontogiorgis (1034_CR28) 1998; 83 LM Bregman (1034_CR7) 1967; 7 YY Lin (1034_CR31) 1987; 18 P Tseng (1034_CR46) 1987; 38 M Tao (1034_CR43) 2011; 21 AJ Hoffman (1034_CR26) 1952; 49 OL Mangasarian (1034_CR36) 1987; 25 Z-Q Luo (1034_CR32) 1992; 72 1034_CR5 AN Iusem (1034_CR27) 1991; 1 1034_CR35 R Glowinski (1034_CR18) 1984 XF Wang (1034_CR49) 2012; 34 1034_CR8 R Monteiro (1034_CR37) 2013; 23
References_xml	– reference: ZeniosSAMulveyJMRelaxation techniques for strictly convex network problemsAnn. Oper. Res.1986551753894803310.1007/BF02739237 – reference: LuoZ-QTsengPOn the Linear convergence of descent methods for convex essentially smooth minimizationSIAM J. Control Optim.199230408425114907610.1137/03300250756.90084 – reference: PangJ-SOn the Convergence of Dual Ascent Methods for Large Scale Linearly Constrained Optimization Problems1984DallasThe University of Texas, School of Management – reference: HeBSYuanXMOn the O(1/n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/n)$$\end{document} convergence rate of the Douglas–Rachford alternating direction methodSIAM J. Numer. Anal.201250700709291428210.1137/1108369361245.90084 – reference: Deng, W., Yin, W.: On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers. J. Sci. Comput. 66(3), 889–916 (2012) – reference: Ventura, J.A., Hearn, D.W.: Computational Development of a Lagrangian Dual Approach for Quadratic Networks. 21(4), 469–485 (1991) – reference: BertsekasDPGafniEProjection methods for variational inequalities with application to the traffic assignment problemMath. Prog. Study19821713915965469710.1007/BFb01209650478.90071 – reference: BertsekasDPTsitsiklisJNParallel and Distributed Computation: Numerical Methods1989Englewood CliffsPrentice-Hall0743.65107 – reference: TaoMYuanXMRecovering low-rank and sparse components of matrices from incomplete and noisy observationsSIAM J. Optim.2011215781276548910.1137/1007818941218.90115 – reference: DouglasJRachfordHHOn the numerical solution of the heat conduction problem in 2 and 3 space variablesTrans. Am. Math. Soc.1956824214398419410.1090/S0002-9947-1956-0084194-40070.35401 – reference: EcksteinJSvaiterBFGeneral projective splitting methods for sums of maximal monotone operatorsSIAM J. Control Optim.201048787811248609410.1137/0706988161194.49038 – reference: De PierroARIusemANOn the convergence properties of Hildreth’s quadratic programming algorithmMath. Prog.1990473751105484010.1007/BF015808510712.90054 – reference: LionsPLMercierBSplitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. Anal.19791696497955131910.1137/07160710426.65050 – reference: YangJFZhangYAlternating direction algorithms for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_1$$\end{document}-problems in compressive sensingSIAM J. Sci. Comput.201133250278278319410.1137/0907777611256.65060 – reference: Eckstein, J.: Splitting Methods for Monotone Operators with Applications to Parallel Optimization. Ph.D. Thesis, Operations Research Center, MIT (1989) – reference: GlowinskiRNumerical Methods for Nonlinear Variational Problems1984New YorkSpringer10.1007/978-3-662-12613-40536.65054 – reference: BoydSParikhNChuEPeleatoBEcksteinJDistributed optimization and statistical learning via the alternating direction method of multipliersFound. Trends Mach. Learn.20113112210.1561/22000000161229.90122(Michael Jordan, Editor in Chief) – reference: GabayDMercierBA dual algorithm for the solution of nonlinear variational problems via finite-element approximationsComput. Math. Appl.19762174010.1016/0898-1221(76)90003-10352.65034 – reference: Zhang, H., Jiang, J.J., Luo, Z.-Q.: On the linear convergence of a proximal gradient method for a class of nonsmooth convex minimization problems. J. Oper. Res. Soc. Chin. 1(2), 163–186 (2013) – reference: TsengPBertsekasDPRelaxation methods for problems with strictly convex separable costs and linear constraintsMath. Prog.19873830332190376910.1007/BF025920170636.90072 – reference: Ma, S.: Alternating proximal gradient method for convex minimization. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0150-0 – reference: Goldfarb, D., Ma, S.: Fast multiple splitting algorithms for convex optimization. SIAM J. Optim. 22(2), 533–556 (2012) – reference: Zhou, Z., Li, X., Wright, J., Candes, E.J., Ma, Y.: Stable principal component pursuit. In: Proceedings of 2010 IEEE International Symposium on Information Theory (2010) – reference: WangXFYuanXMThe linearized alternating direction method of multipliers for dantzig selectorSIAM J. Sci. Comput.20123427922811302372610.1137/1108335431263.90061 – reference: KontogiorgisSMeyerRRA variable-penalty alternating directions method for convex optimizationMath. Program.199883295316439630920.90118 – reference: Goldfarb, D., Ma, S., Scheinberg, K.: Fast alternating linearization methods for minimizing the sum of two convex functions. Math. Prog. A. 141(1,2), 349–382 (2013) – reference: TsengPBertsekasDPRelaxation methods for problems with strictly convex costs and linear constraintsMath. Oper. Res.199116462481112046410.1287/moor.16.3.4620755.90067 – reference: EcksteinJBertsekasDPOn the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operatorsMath. Program.199255293318116818310.1007/BF015812040765.90073 – reference: MangasarianOLShiauT-HLipschitz continuity of solutions of linear inequalities, programs and complementarity problemsSIAM J. Control Optim.19872558359588518710.1137/03250330613.90066 – reference: YuanMLinYModel selection and estimation in regression with grouped variablesJ. R. Stat. Soc. Ser. B (Statistical Methodology)2006684967221257410.1111/j.1467-9868.2005.00532.x1141.62030 – reference: Chen, C., He, B. , Yuan, X., Ye, Y.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Prog. 155(1), 57–79 (2013) – reference: Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Prog. 125(2), 263–295 (2010) – reference: Levitin, E.S., Poljak, B.T.: Constrained minimization methods. Z. Vycisl. Mat. i Mat. Fiz. 6, 787–823 (1965). English translation in USSR Comput. Math. Phys. 6, 1–50 (1965) – reference: BregmanLMThe relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programmingUSSR Comput. Math. Math. Phys.1967720021721561710.1016/0041-5553(67)90040-70186.23807 – reference: OhuchiAKajiILagrangian dual coordinatewise maximization algorithm for network transportation problems with quadratic costsNetworks19841451553010.1002/net.32301404040585.90062 – reference: MonteiroRSvaiterBIteration-complexity of block-decomposition algorithms and the alternating direction method of multipliersSIAM J. Optim.201323475507303311610.1137/1108494681267.90181 – reference: RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/97814008731730193.18401 – reference: LuoZ-QTsengPOn the convergence rate of dual ascent methods for strictly convex minimizationMath. Oper. Res.199318846867125168310.1287/moor.18.4.8460804.90103 – reference: PangJ-SA posteriori error bounds for the linearly-constrained variational inequality problemMath. Oper. Res.19871247448490641910.1287/moor.12.3.474 – reference: LinYYPangJ-SIterative methods for large convex quadratic programs: a surveySIAM J. Control Optim.19871838341187706910.1137/03250230624.90083 – reference: IusemANOn dual convergence and the rate of primal convergence of bregman’s convex programming methodSIAM J. Control Optim.19911401423111252710.1137/08010250753.90051 – reference: GlowinskiRLe TallecPAugmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics1989PhiladelphiaSIAM Studies in Applied Mathematics10.1137/1.97816119708380698.73001 – reference: HeBSTaoMYuanXMAlternating direction method with gaussian back substitution for separable convex programmingSIAM J. Optim.201222313340296885610.1137/1108223471273.90152 – reference: Goldstein, T., O’Donoghue, B., Setzer, S.: Fast alternating direction optimization methods. SIAM J. Imaging Sci, 7(3), 1588–1623 (2014) – reference: LuoZ-QTsengPOn the convergence of the coordinate descent method for convex differentiable minimizationJ. Optim. 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SubjectTerms	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Convergence Convex analysis Convexity Decomposition Errors Feasibility Full Length Paper Inequality Lagrange multiplier Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Methods Multipliers Numerical Analysis Optimization Studies Texts Theoretical
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