{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:57:59Z","timestamp":1780361879979,"version":"3.54.1"},"reference-count":183,"publisher":"Emerald","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2011,7,26]]},"abstract":"<jats:p>Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas\u2013Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for \u21131 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.<\/jats:p>","DOI":"10.1561\/2200000016","type":"journal-article","created":{"date-parts":[[2011,8,11]],"date-time":"2011-08-11T07:21:18Z","timestamp":1313047278000},"page":"1-122","source":"Crossref","is-referenced-by-count":14297,"title":["Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers"],"prefix":"10.1108","volume":"3","author":[{"given":"Stephen","family":"Boyd","sequence":"first","affiliation":[{"name":"Electrical Engineering Department, Stanford University , Stanford, 94305,","place":["CA, USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Neal","family":"Parikh","sequence":"additional","affiliation":[{"name":"Computer Science Department, Stanford University , Stanford, 94305,","place":["CA, USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"Chu","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Stanford University , Stanford, 94305,","place":["CA, USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Borja","family":"Peleato","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Stanford University , Stanford, 94305,","place":["CA, USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan","family":"Eckstein","sequence":"additional","affiliation":[{"name":"Management Science and Information Systems Department and RUTCOR, Rutgers University , Piscataway, 08854,","place":["NJ, USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"140","published-online":{"date-parts":[[2011,7,26]]},"reference":[{"issue":"9","key":"2026040910314629300_ref001","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1109\/TIP.2010.2047910","article-title":"\u201cFast image recovery using variable splitting and constrained optimization\u201d","volume":"19","author":"Afonso","year":"2010","journal-title":"IEEE Transactions on Image Processing"},{"key":"2026040910314629300_ref002","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1109\/TIP.2010.2076294","article-title":"\u201cAn augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems\u201d","volume":"20","author":"Afonso","year":"2011","journal-title":"IEEE Transactions on Image Processing"},{"key":"2026040910314629300_ref003","volume-title":"LAPACK: A portable linear algebra library for high-performance computers","author":"Anderson","year":"1990"},{"issue":"3","key":"2026040910314629300_ref004","doi-asserted-by":"crossref","first-page":"265","DOI":"10.2307\/1907353","article-title":"\u201cExistence of an equilibrium for a competitive economy\u201d","volume":"22","author":"Arrow","year":"1954","journal-title":"Econometrica"},{"key":"2026040910314629300_ref005","volume-title":"Studies in Linear and Nonlinear Programming","author":"Arrow","year":"1958"},{"key":"2026040910314629300_ref006","volume-title":"Studies in Linear and Nonlinear Programming","author":"Arrow","year":"1958"},{"key":"2026040910314629300_ref007","first-page":"485","article-title":"\u201cModel selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data\u201d","volume":"9","author":"Banerjee","year":"2008","journal-title":"Journal of Machine Learning Research"},{"issue":"473","key":"2026040910314629300_ref008","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1198\/016214505000000907","article-title":"\u201cConvexity, classification, and risk bounds\u201d","volume":"101","author":"Bartlett","year":"2006","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2026040910314629300_ref009","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1006\/jath.1994.1136","article-title":"\u201cDykstra\u2019s alternating projection algorithm for two sets\u201d","volume":"79","author":"Bauschke","year":"1994","journal-title":"Journal of Approximation Theory"},{"issue":"3","key":"2026040910314629300_ref010","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1137\/S0036144593251710","article-title":"\u201cOn projection algorithms for solving convex feasibility problems\u201d","volume":"38","author":"Bauschke","year":"1996","journal-title":"SIAM Review"},{"issue":"1","key":"2026040910314629300_ref011","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"\u201cA fast iterative shrinkage-thresholding algorithm for linear inverse problems\u201d","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026040910314629300_ref012","article-title":"\u201cNESTA: A fast and accurate first-order method for sparse recovery\u201d","volume-title":"Available online","author":"Becker","year":"2009"},{"key":"2026040910314629300_ref013","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1007\/BF01386316","article-title":"\u201cPartitioning procedures for solving mixed-variables programming problems\u201d","volume":"4","author":"Benders","year":"1962","journal-title":"Numerische Mathematik"},{"key":"2026040910314629300_ref014","first-page":"133","article-title":"\u201cSur les m\u00e9thodes de d\u00e9composition, de d\u00e9centralisation et de coordination et applications\u201d","author":"Bensoussan","year":"1976","journal-title":"M\u00e9thodes Math\u00e9matiques de l\u2019Informatique"},{"key":"2026040910314629300_ref015","volume-title":"Constrained Optimization and Lagrange Multiplier Methods","author":"Bertsekas","year":"1982"},{"key":"2026040910314629300_ref016","volume-title":"Nonlinear Programming","author":"Bertsekas","year":"1999","edition":"2nd"},{"key":"2026040910314629300_ref017","volume-title":"Parallel and Distributed Computation: Numerical Methods","author":"Bertsekas","year":"1989"},{"key":"2026040910314629300_ref018","article-title":"\u201cAlternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing\u201d","author":"Bioucas-Dias","year":"2010","journal-title":"arXiv"},{"key":"2026040910314629300_ref019","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-9859-3","volume-title":"Convex Analysis and Nonlinear Optimization: Theory and Examples","author":"Borwein","year":"2000"},{"key":"2026040910314629300_ref020","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex Optimization","author":"Boyd","year":"2004"},{"issue":"3","key":"2026040910314629300_ref021","first-page":"487","article-title":"\u201cFinding the common point of convex sets by the method of successive projections\u201d","volume":"162","author":"Bregman","year":"1965","journal-title":"Proceedings of the USSR Academy of Sciences"},{"issue":"3","key":"2026040910314629300_ref022","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","article-title":"\u201cThe relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming\u201d","volume":"7","author":"Bregman","year":"1967","journal-title":"USSR Computational Mathematics and Mathematical Physics"},{"key":"2026040910314629300_ref023","volume-title":"Op\u00e9rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert","author":"Br\u00e9zis","year":"1973"},{"issue":"1","key":"2026040910314629300_ref024","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1137\/060657704","article-title":"\u201cFrom sparse solutions of systems of equations to sparse modeling of signals and images\u201d","volume":"51","author":"Bruckstein","year":"2009","journal-title":"SIAM Review"},{"key":"2026040910314629300_ref025","article-title":"\u201cHaLoop: Efficient iterative data processing on large clusters\u201d","author":"Bu","year":"2010","journal-title":"Proceedings of the 36th International Conference on Very Large Databases"},{"issue":"5","key":"2026040910314629300_ref026","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.1137\/0916069","article-title":"\u201cA limited memory algorithm for bound constrained optimization\u201d","volume":"16","author":"Byrd","year":"1995","journal-title":"SIAM Journal on Scientific and Statistical Computing"},{"issue":"5A","key":"2026040910314629300_ref027","doi-asserted-by":"crossref","first-page":"2145","DOI":"10.1214\/08-AOS653","article-title":"\u201cNear-ideal model selection by \u21131 minimization\u201d","volume":"37","author":"Cand\u00e8s","year":"2009","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026040910314629300_ref028","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"\u201cRobust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information\u201d","volume":"52","author":"Cand\u00e8s","year":"2006","journal-title":"IEEE Transactions on Information Theory"},{"issue":"12","key":"2026040910314629300_ref029","doi-asserted-by":"crossref","first-page":"5406","DOI":"10.1109\/TIT.2006.885507","article-title":"\u201cNear-optimal signal recovery from random projections: Universal encoding strategies\u201d","volume":"52","author":"Cand\u00e8s","year":"2006","journal-title":"IEEE Transactions on Information Theory"},{"issue":"3","key":"2026040910314629300_ref030","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/BF00940051","article-title":"\u201cProximal minimization algorithm with D-functions\u201d","volume":"73","author":"Censor","year":"1992","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref031","volume-title":"Parallel Optimization: Theory, Algorithms, and Applications","author":"Censor","year":"1997"},{"issue":"2","key":"2026040910314629300_ref032","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1365815.1365816","article-title":"\u201cBigTable: A distributed storage system for structured data\u201d","volume":"26","author":"Chang","year":"2008","journal-title":"ACM Transactions on Computer Systems"},{"key":"2026040910314629300_ref033","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF01582566","article-title":"\u201cA proximal-based decomposition method for convex minimization problems\u201d","volume":"64","author":"Chen","year":"1994","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref034","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1137\/S003614450037906X","article-title":"\u201cAtomic decomposition by basis pursuit\u201d","volume":"43","author":"Chen","year":"2001","journal-title":"SIAM Review"},{"issue":"3","key":"2026040910314629300_ref035","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1145\/1391989.1391995","article-title":"\u201cAlgorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update\/downdate\u201d","volume":"35","author":"Chen","year":"2008","journal-title":"ACM Transactions on Mathematical Software"},{"issue":"3","key":"2026040910314629300_ref036","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1090\/S0002-9939-1959-0105008-8","article-title":"\u201cProximity maps for convex sets\u201d","volume":"10","author":"Cheney","year":"1959","journal-title":"Proceedings of the American Mathematical Society"},{"key":"2026040910314629300_ref037","article-title":"\u201cMapReduce for machine learning on multicore\u201d","author":"Chu","year":"2007","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026040910314629300_ref038","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1190\/1.1440378","article-title":"\u201cRobust modeling with erratic data\u201d","volume":"38","author":"Claerbout","year":"1973","journal-title":"Geophysics"},{"key":"2026040910314629300_ref039","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S1076-5670(08)70157-5","article-title":"\u201cThe convex feasibility problem in image recovery\u201d","volume":"95","author":"Combettes","year":"1996","journal-title":"Advances in Imaging and Electron Physics"},{"issue":"4","key":"2026040910314629300_ref040","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1109\/JSTSP.2007.910264","article-title":"\u201cA Douglas\u2013Rachford splitting approach to nonsmooth convex variational signal recovery\u201d","volume":"1","author":"Combettes","year":"2007","journal-title":"IEEE Journal on Selected Topics in Signal Processing"},{"key":"2026040910314629300_ref041","article-title":"\u201cProximal splitting methods in signal processing\u201d","author":"Combettes","year":"2009","journal-title":"arXiv"},{"issue":"4","key":"2026040910314629300_ref042","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/050626090","article-title":"\u201cSignal recovery by proximal forward-backward splitting\u201d","volume":"4","author":"Combettes","year":"2006","journal-title":"Multiscale Modeling and Simulation"},{"key":"2026040910314629300_ref043","volume-title":"Linear Programming and Extensions","author":"Dantzig","year":"1963"},{"key":"2026040910314629300_ref044","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1287\/opre.8.1.101","article-title":"\u201cDecomposition principle for linear programs\u201d","volume":"8","author":"Dantzig","year":"1960","journal-title":"Operations Research"},{"key":"2026040910314629300_ref045","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1002\/cpa.20042","article-title":"\u201cAn iterative thresholding algorithm for linear inverse problems with a sparsity constraint\u201d","volume":"57","author":"Daubechies","year":"2004","journal-title":"Communications on Pure and Applied Mathematics"},{"issue":"1","key":"2026040910314629300_ref046","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1145\/1327452.1327492","article-title":"\u201cMapReduce: Simplified data processing on large clusters\u201d","volume":"51","author":"Dean","year":"2008","journal-title":"Communications of the ACM"},{"key":"2026040910314629300_ref047","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971446","volume-title":"Applied Numerical Linear Algebra","author":"Demmel","year":"1997"},{"issue":"1","key":"2026040910314629300_ref048","doi-asserted-by":"crossref","first-page":"157","DOI":"10.2307\/2528966","article-title":"\u201cCovariance selection\u201d","volume":"28","author":"Dempster","year":"1972","journal-title":"Biometrics"},{"key":"2026040910314629300_ref049","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/18.382009","article-title":"\u201cDe-noising by soft-thresholding\u201d","volume":"41","author":"Donoho","year":"1995","journal-title":"IEEE Transactions on Information Theory"},{"issue":"4","key":"2026040910314629300_ref050","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"\u201cCompressed sensing\u201d","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Transactions on Information Theory"},{"issue":"45","key":"2026040910314629300_ref051","doi-asserted-by":"crossref","first-page":"18914","DOI":"10.1073\/pnas.0909892106","article-title":"\u201cMessage-passing algorithms for compressed sensing\u201d","volume":"106","author":"Donoho","year":"2009","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2026040910314629300_ref052","volume-title":"Tech. Rep.","author":"Donoho","year":"2006"},{"key":"2026040910314629300_ref053","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","article-title":"\u201cOn the numerical solution of heat conduction problems in two and three space variables\u201d","volume":"82","author":"Douglas","year":"1956","journal-title":"Transactions of the American Mathematical Society"},{"key":"2026040910314629300_ref054","article-title":"\u201cDistributed Dual Averaging in Networks\u201d","author":"Duchi","year":"2010","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026040910314629300_ref055","article-title":"\u201cProjected subgradient methods for learning sparse Gaussians\u201d","author":"Duchi","year":"2008","journal-title":"Proceedings of the Conference on Uncertainty in Artificial Intelligence"},{"key":"2026040910314629300_ref056","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1080\/01621459.1983.10477029","article-title":"\u201cAn algorithm for restricted least squares regression\u201d","volume":"78","author":"Dykstra","year":"1983","journal-title":"Journal of the American Statistical Association"},{"key":"2026040910314629300_ref057","volume-title":"Splitting methods for monotone operators with applications to parallel optimization","author":"Eckstein","year":"1989"},{"key":"2026040910314629300_ref058","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1287\/moor.18.1.202","article-title":"\u201cNonlinear proximal point algorithms using Bregman functions, with applications to convex programming\u201d","author":"Eckstein","year":"1993","journal-title":"Mathematics of Operations Research"},{"issue":"1","key":"2026040910314629300_ref059","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF02196592","article-title":"\u201cParallel alternating direction multiplier decomposition of convex programs\u201d","volume":"80","author":"Eckstein","year":"1994","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"1","key":"2026040910314629300_ref060","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1080\/10556789408805578","article-title":"\u201cSome saddle-function splitting methods for convex programming\u201d","volume":"4","author":"Eckstein","year":"1994","journal-title":"Optimization Methods and Software"},{"issue":"1","key":"2026040910314629300_ref061","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10107-003-0374-x","article-title":"\u201cA practical general approximation criterion for methods of multipliers based on Bregman distances\u201d","volume":"96","author":"Eckstein","year":"2003","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref062","volume-title":"Tech. Rep.","author":"Eckstein","year":"1990"},{"key":"2026040910314629300_ref063","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","article-title":"\u201cOn the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators\u201d","volume":"55","author":"Eckstein","year":"1992","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref064","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1287\/ijoc.10.2.218","article-title":"\u201cOperator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control\u201d","volume":"10","author":"Eckstein","year":"1998","journal-title":"INFORMS Journal on Computing"},{"key":"2026040910314629300_ref065","first-page":"119","article-title":"\u201cSome reformulations and applications of the alternating direction method of multipliers\u201d","author":"Eckstein","year":"1993","journal-title":"Large Scale Optimization: State of the Art"},{"issue":"1-2","key":"2026040910314629300_ref066","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10107-006-0070-8","article-title":"\u201cA family of projective splitting methods for the sum of two maximal monotone operators\u201d","volume":"111","author":"Eckstein","year":"2008","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref067","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1137\/070698816","article-title":"\u201cGeneral projective splitting methods for sums of maximal monotone operators\u201d","volume":"48","author":"Eckstein","year":"2009","journal-title":"SIAM Journal on Control and Optimization"},{"key":"2026040910314629300_ref068","first-page":"31","article-title":"\u201cApplications of Lagrangian-based alternating direction methods and connections to split Bregman\u201d","volume":"9","author":"Esser","year":"2009","journal-title":"CAM report"},{"issue":"3","key":"2026040910314629300_ref069","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1287\/opre.11.3.399","article-title":"\u201cGeneralized Lagrange multiplier method for solving problems of optimum allocation of resources\u201d","volume":"11","author":"Everett","year":"1963","journal-title":"Operations Research"},{"key":"2026040910314629300_ref070","article-title":"\u201cMonotone operator splitting for optimization problems in sparse recovery\u201d","author":"Fadili","year":"2009","journal-title":"IEEE ICIP"},{"key":"2026040910314629300_ref071","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971316","volume-title":"Nonlinear Programming: Sequential Unconstrained Minimization Techniques","author":"Fiacco","year":"1990"},{"key":"2026040910314629300_ref072","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.1109\/TIP.2010.2053941","article-title":"\u201cRestoration of Poissonian Images Using Alternating Direction Optimization\u201d","volume":"19","author":"Figueiredo","year":"2010","journal-title":"IEEE Transactions on Image Processing"},{"issue":"4","key":"2026040910314629300_ref073","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1109\/JSTSP.2007.910281","article-title":"\u201cGradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems\u201d","volume":"1","author":"Figueiredo","year":"2007","journal-title":"IEEE Journal on Selected Topics in Signal Processing"},{"key":"2026040910314629300_ref074","first-page":"1663","article-title":"\u201cConsensus-based distributed support vector machines\u201d","volume":"11","author":"Forero","year":"2010","journal-title":"Journal of Machine Learning Research"},{"key":"2026040910314629300_ref075","volume-title":"Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems","author":"Fortin","year":"1983"},{"key":"2026040910314629300_ref076","volume-title":"Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems","author":"Fortin","year":"1983"},{"key":"2026040910314629300_ref077","volume-title":"MPI: A Message-Passing Interface Standard, version 2.2","author":"Forum","year":"2009"},{"key":"2026040910314629300_ref078","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/3-540-59119-2_166","volume-title":"Computational Learning Theory","author":"Freund","year":"1995"},{"issue":"3","key":"2026040910314629300_ref079","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1093\/biostatistics\/kxm045","article-title":"\u201cSparse inverse covariance estimation with the graphical lasso\u201d","volume":"9","author":"Friedman","year":"2008","journal-title":"Biostatistics"},{"key":"2026040910314629300_ref080","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00247655","article-title":"\u201cApplication of the alternating direction method of multipliers to separable convex programming problems\u201d","volume":"1","author":"Fukushima","year":"1992","journal-title":"Computational Optimization and Applications"},{"key":"2026040910314629300_ref081","volume-title":"Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems","author":"Gabay","year":"1983"},{"key":"2026040910314629300_ref082","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0898-1221(76)90003-1","article-title":"\u201cA dual algorithm for the solution of nonlinear variational problems via finite element approximations\u201d","volume":"2","author":"Gabay","year":"1976","journal-title":"Computers and Mathematics with Applications"},{"key":"2026040910314629300_ref083","volume-title":"GNU Scientific Library Reference Manual","author":"Galassi","year":"2002","edition":"3rd ed."},{"issue":"4","key":"2026040910314629300_ref084","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/BF00934810","article-title":"\u201cGeneralized Benders decomposition\u201d","volume":"10","author":"Geoffrion","year":"1972","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"5","key":"2026040910314629300_ref085","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1145\/1165389.945450","article-title":"\u201cThe Google file system\u201d","volume":"37","author":"Ghemawat","year":"2003","journal-title":"ACM SIGOPS Operating Systems Review"},{"key":"2026040910314629300_ref086","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1051\/m2an\/197509R200411","article-title":"\u201cSur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9, d\u2019une classe de probl\u00e8mes de Dirichlet non lin\u00e9aires\u201d","volume":"9","author":"Glowinski","year":"1975","journal-title":"Revue Fran\u00e7aise d\u2019Automatique, Informatique et Recherche Op\u00e9rationnelle"},{"key":"2026040910314629300_ref087","article-title":"\u201cAugmented Lagrangian methods for the solution of variational problems\u201d","author":"Glowinski","year":"1987","journal-title":"University of Wisconsin\u2013Madison"},{"issue":"2","key":"2026040910314629300_ref088","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1137\/080725891","article-title":"\u201cThe split Bregman method for \u21131 regularized problems\u201d","volume":"2","author":"Goldstein","year":"2009","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026040910314629300_ref089","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/BFb0120845","article-title":"\u201cModified Lagrangians in convex programming and their generalizations\u201d","author":"Gol\u2019shtein","year":"1979","journal-title":"Point-to-Set Maps and Mathematical Programming"},{"key":"2026040910314629300_ref090","volume-title":"Matrix Computations","author":"Golub","year":"1996","edition":"3rd ed."},{"key":"2026040910314629300_ref091","article-title":"\u201cThe Parallel BGL: A generic library for distributed graph computations\u201d","author":"Gregor","year":"2005","journal-title":"Parallel Object-Oriented Scientific Computing"},{"issue":"2","key":"2026040910314629300_ref092","doi-asserted-by":"crossref","DOI":"10.1109\/MIS.2009.36","article-title":"\u201cThe unreasonable effectiveness of data\u201d","volume":"24","author":"Halevy","year":"2009","journal-title":"IEEE Intelligent Systems"},{"key":"2026040910314629300_ref093","article-title":"\u201cMapReduce\/BigTable for distributed optimization\u201d","author":"Hall","year":"2010","journal-title":"Neural Information Processing Systems Workshop on Learning on Cores, Clusters, and Clouds"},{"key":"2026040910314629300_ref094","volume-title":"Generalized Additive Models","author":"Hastie","year":"1990"},{"key":"2026040910314629300_ref095","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-84858-7","volume-title":"The Elements of Statistical Learning: Data Mining, Inference and Prediction","author":"Hastie","year":"2009","edition":"2nd ed."},{"issue":"2","key":"2026040910314629300_ref096","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1023\/A:1004603514434","article-title":"\u201cAlternating direction method with self-adaptive penalty parameters for monotone variational inequalities\u201d","volume":"106","author":"He","year":"2000","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref097","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1007\/BF00927673","article-title":"\u201cMultiplier and gradient methods\u201d","volume":"4","author":"Hestenes","year":"1969","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref098","volume-title":"Computing Methods in Optimization Problems","author":"Hestenes","year":"1969"},{"key":"2026040910314629300_ref099","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-56468-0","volume-title":"Fundamentals of Convex Analysis","author":"Hiriart-Urruty","year":"2001"},{"key":"2026040910314629300_ref100","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1214\/aoms\/1177703732","article-title":"\u201cRobust estimation of a location parameter\u201d","volume":"35","author":"Huber","year":"1964","journal-title":"Annals of Mathematical Statistics"},{"issue":"2","key":"2026040910314629300_ref101","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1137\/070690274","article-title":"\u201c\u21131 trend filtering\u201d","volume":"51","author":"Kim","year":"2009","journal-title":"SIAM Review"},{"issue":"4","key":"2026040910314629300_ref102","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1109\/JSTSP.2007.910971","article-title":"\u201cAn interior-point method for large-scale \u21131-regularized least squares\u201d","volume":"1","author":"Kim","year":"2007","journal-title":"IEEE Journal on Selected Topics in Signal Processing"},{"key":"2026040910314629300_ref103","first-page":"1519","article-title":"\u201cAn interior-point method for large-scale \u21131-regularized logistic regression\u201d","volume":"8","author":"Koh","year":"2007","journal-title":"Journal of Machine Learning Research"},{"key":"2026040910314629300_ref104","volume-title":"Probabilistic Graphical Models: Principles and Techniques","author":"Koller","year":"2009"},{"key":"2026040910314629300_ref105","unstructured":"Kontogiorgis, S. A.\n           (1994). Alternating directions methods for the parallel solution of large-scale block-structured optimization problems. University of Wisconsin\u2013Madison."},{"key":"2026040910314629300_ref106","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF02680549","article-title":"\u201cA variable-penalty alternating directions method for convex optimization\u201d","volume":"83","author":"Kontogiorgis","year":"1998","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref107","volume-title":"Optimization Theory for Large Systems","author":"Lasdon","year":"1970"},{"issue":"3","key":"2026040910314629300_ref108","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1112\/plms\/s3-55.3.605","article-title":"\u201cOn fixed points of non-expansive piecewise isometric mappings\u201d","volume":"3","author":"Lawrence","year":"1987","journal-title":"Proceedings of the London Mathematical Society"},{"issue":"3","key":"2026040910314629300_ref109","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1145\/355841.355847","article-title":"\u201cBasic linear algebra subprograms for Fortran usage\u201d","volume":"5","author":"Lawson","year":"1979","journal-title":"ACM Transactions on Mathematical Software"},{"key":"2026040910314629300_ref110","article-title":"\u201cAlgorithms for non-negative matrix factorization\u201d","volume":"13","author":"Lee","year":"2001","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026040910314629300_ref111","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1145\/1830252.1830263","article-title":"\u201cDesign patterns for efficient graph algorithms in MapReduce\u201d","author":"Lin","year":"2010","journal-title":"Proceedings of the Eighth Workshop on Mining and Learning with Graphs"},{"key":"2026040910314629300_ref112","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","article-title":"\u201cSplitting algorithms for the sum of two nonlinear operators\u201d","volume":"16","author":"Lions","year":"1979","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"3","key":"2026040910314629300_ref113","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/BF01589116","article-title":"\u201cOn the limited memory method for large-scale optimization\u201d","volume":"45","author":"Liu","year":"1989","journal-title":"Mathematical Programming"},{"key":"2026040910314629300_ref114","article-title":"\u201cGraphLab: A new parallel framework for machine learning\u201d","author":"Low","year":"2010","journal-title":"Conference on Uncertainty in Artificial Intelligence"},{"issue":"4","key":"2026040910314629300_ref115","doi-asserted-by":"crossref","first-page":"1807","DOI":"10.1137\/070695915","article-title":"\u201cSmooth optimization approach for sparse covariance selection\u201d","volume":"19","author":"Lu","year":"2009","journal-title":"SIAM Journal on Optimization"},{"key":"2026040910314629300_ref116","article-title":"\u201cAn alternating direction method for finding Dantzig selectors\u201d","author":"Lu","year":"2010","journal-title":"arXiv:1011.4604"},{"key":"2026040910314629300_ref117","volume-title":"Introduction to Linear and Nonlinear Programming","author":"Luenberger","year":"1973"},{"key":"2026040910314629300_ref118","article-title":"\u201cNetwork flow algorithms for structured sparsity\u201d","volume":"24","author":"Mairal","year":"2010","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026040910314629300_ref119","first-page":"135","article-title":"\u201cPregel: A system for large-scale graph processing\u201d","author":"Malewicz","year":"2010","journal-title":"Proceedings of the 2010 International Conference on Management of Data"},{"key":"2026040910314629300_ref120","article-title":"\u201cAn augmented Lagrangian approach to constrained MAP inference\u201d","author":"Martins","year":"2011","journal-title":"International Conference on Machine Learning"},{"key":"2026040910314629300_ref121","doi-asserted-by":"crossref","first-page":"5262","DOI":"10.1109\/TSP.2010.2055862","article-title":"\u201cDistributed sparse linear regression\u201d","volume":"58","author":"Mateos","year":"2010","journal-title":"IEEE Transactions on Signal Processing"},{"key":"2026040910314629300_ref122","volume-title":"Generalized Linear Models","author":"McCullagh","year":"1991"},{"issue":"3","key":"2026040910314629300_ref123","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1214\/009053606000000281","article-title":"\u201cHigh-dimensional graphs and variable selection with the lasso\u201d","volume":"34","author":"Meinshausen","year":"2006","journal-title":"Annals of Statistics"},{"key":"2026040910314629300_ref124","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF00928472","article-title":"\u201cUse of the augmented penalty function in mathematical programming problems, part 1\u201d","volume":"8","author":"Miele","year":"1971","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref125","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/BF00928473","article-title":"\u201cUse of the augmented penalty function in mathematical programming problems, part 2\u201d","volume":"8","author":"Miele","year":"1971","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref126","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00934960","article-title":"\u201cOn the method of multipliers for mathematical programming problems\u201d","volume":"10","author":"Miele","year":"1972","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref127","first-page":"2897","article-title":"\u201cFonctions convexes duales et points proximaux dans un espace Hilbertien\u201d","volume":"255","author":"Moreau","year":"1962","journal-title":"Reports of the Paris Academy of Sciences, Series A"},{"key":"2026040910314629300_ref128","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-75142-7_41","article-title":"\u201cFully distributed algorithms for convex optimization problems\u201d","author":"Mosk-Aoyama","year":"2007"},{"issue":"11","key":"2026040910314629300_ref129","doi-asserted-by":"crossref","first-page":"2674","DOI":"10.1109\/TAC.2008.2007159","article-title":"\u201cApplication of a smoothing technique to decomposition in convex optimization\u201d","volume":"53","author":"Necoara","year":"2008","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"1","key":"2026040910314629300_ref130","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/TAC.2008.2009515","article-title":"\u201cDistributed subgradient methods for multi-agent optimization\u201d","volume":"54","author":"Nedi\u0107","year":"2009","journal-title":"IEEE Transactions on Automatic Control"},{"key":"2026040910314629300_ref131","volume-title":"Convex Optimization in Signal Processing and Communications","author":"Nedi\u0107","year":"2010"},{"issue":"2","key":"2026040910314629300_ref132","first-page":"372","article-title":"\u201cA method of solving a convex programming problem with convergence rate O(1\/k2)\u201d","volume":"27","author":"Nesterov","year":"1983","journal-title":"Soviet Mathematics Doklady"},{"key":"2026040910314629300_ref133","unstructured":"Nesterov, Y.\n           (2007). \u201cGradient methods for minimizing composite objective function\u201d. CORE Discussion Paper, Catholic University of Louvain."},{"key":"2026040910314629300_ref134","doi-asserted-by":"crossref","unstructured":"Ng, M., Weiss, P., & Yuang, X. (2009). \u201cSolving constrained total-variation image restoration and reconstruction problems via alternating direction methods\u201d. ICM Research Report.","DOI":"10.1137\/090774823"},{"key":"2026040910314629300_ref135","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"Nocedal","year":"1999"},{"key":"2026040910314629300_ref136","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1016\/j.automatica.2010.03.013","article-title":"\u201cSegmentation of ARX-models using sum-of-norms regularization\u201d","volume":"46","author":"Ohlsson","year":"2010","journal-title":"Automatica"},{"key":"2026040910314629300_ref137","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1137\/0103003","article-title":"\u201cThe numerical solution of parabolic and elliptic differential equations\u201d","volume":"3","author":"Peaceman","year":"1955","journal-title":"Journal of the Society for Industrial and Applied Mathematics"},{"key":"2026040910314629300_ref138","volume-title":"Optimization","author":"Powell","year":"1969"},{"key":"2026040910314629300_ref139","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1109\/MCS.2009.935569","article-title":"\u201cKalman filtering in wireless sensor networks\u2014Incorporating communication cost in state estimation problems\u201d","volume":"30","author":"Ribeiro","year":"2010","journal-title":"IEEE Control Systems Magazine"},{"key":"2026040910314629300_ref140","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"Rockafellar","year":"1970"},{"key":"2026040910314629300_ref141","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1287\/moor.1.2.97","article-title":"\u201cAugmented Lagrangians and applications of the proximal point algorithm in convex programming\u201d","volume":"1","author":"Rockafellar","year":"1976","journal-title":"Mathematics of Operations Research"},{"key":"2026040910314629300_ref142","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","article-title":"\u201cMonotone operators and the proximal point algorithm\u201d","volume":"14","author":"Rockafellar","year":"1976","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"1","key":"2026040910314629300_ref143","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1287\/moor.16.1.119","article-title":"\u201cScenarios and policy aggregation in optimization under uncertainty\u201d","volume":"16","author":"Rockafellar","year":"1991","journal-title":"Mathematics of Operations Research"},{"key":"2026040910314629300_ref144","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"Rockafellar","year":"1998"},{"key":"2026040910314629300_ref145","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","article-title":"\u201cNonlinear total variation based noise removal algorithms\u201d","volume":"60","author":"Rudin","year":"1992","journal-title":"Physica D"},{"issue":"5","key":"2026040910314629300_ref146","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0167-6377(89)90055-2","article-title":"\u201cAn augmented Lagrangian decomposition method for block diagonal linear programming problems\u201d","volume":"8","author":"Ruszczy\u0144ski","year":"1989","journal-title":"Operations Research Letters"},{"issue":"3","key":"2026040910314629300_ref147","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1287\/moor.20.3.634","article-title":"\u201cOn convergence of an augmented Lagrangian decomposition method for sparse convex optimization\u201d","volume":"20","author":"Ruszczy\u0144ski","year":"1995","journal-title":"Mathematics of Operations Research"},{"key":"2026040910314629300_ref148","article-title":"\u201cSparse inverse covariance selection via alternating linearization methods\u201d","volume-title":"Advances in Neural Information Processing Systems","author":"Scheinberg","year":"2010"},{"key":"2026040910314629300_ref149","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1109\/TSP.2007.908943","article-title":"\u201cConsensus in ad hoc WSNs with noisy links\u2014Part II: Distributed estimation and smoothing of random signals\u201d","volume":"56","author":"Schizas","year":"2008","journal-title":"IEEE Transactions on Signal Processing"},{"key":"2026040910314629300_ref150","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1109\/TSP.2007.906734","article-title":"\u201cConsensus in ad hoc WSNs with noisy links\u2014Part I: Distributed estimation of deterministic signals\u201d","volume":"56","author":"Schizas","year":"2008","journal-title":"IEEE Transactions on Signal Processing"},{"key":"2026040910314629300_ref151","volume-title":"Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond","author":"Sch\u00f6lkopf","year":"2002"},{"key":"2026040910314629300_ref152","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-82118-9","volume-title":"Minimization Methods for Non-Differentiable Functions","author":"Shor","year":"1985"},{"key":"2026040910314629300_ref153","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/BF01586091","article-title":"\u201cApplications of the method of partial inverses to convex programming: decomposition\u201d","volume":"32","author":"Spingarn","year":"1985","journal-title":"Mathematical Programming"},{"issue":"2","key":"2026040910314629300_ref154","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1007\/s10851-009-0179-5","article-title":"\u201cRemoving multiplicative noise by Douglas-Rachford splitting methods\u201d","volume":"36","author":"Steidl","year":"2010","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2026040910314629300_ref155","first-page":"311","article-title":"\u201cBundle methods for regularized risk minimization\u201d","volume":"11","author":"Teo","year":"2010","journal-title":"Journal of Machine Learning Research"},{"key":"2026040910314629300_ref156","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"\u201cRegression shrinkage and selection via the lasso\u201d","volume":"58","author":"Tibshirani","year":"1996","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"2026040910314629300_ref157","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1137\/0329006","article-title":"\u201cApplications of a splitting algorithm to decomposition in convex programming and variational inequalities\u201d","volume":"29","author":"Tseng","year":"1991","journal-title":"SIAM Journal on Control and Optimization"},{"key":"2026040910314629300_ref158","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1137\/S1052623495279797","article-title":"\u201cAlternating projection-proximal methods for convex programming and variational inequalities\u201d","volume":"7","author":"Tseng","year":"1997","journal-title":"SIAM Journal on Optimization"},{"key":"2026040910314629300_ref159","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","article-title":"\u201cA modified forward-backward splitting method for maximal monotone mappings\u201d","volume":"38","author":"Tseng","year":"2000","journal-title":"SIAM Journal on Control and Optimization"},{"key":"2026040910314629300_ref160","unstructured":"Tsitsiklis, J. N.\n           (1984). Problems in decentralized decision making and computation. Massachusetts Institute of Technology."},{"issue":"9","key":"2026040910314629300_ref161","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1109\/TAC.1986.1104412","article-title":"\u201cDistributed asynchronous deterministic and stochastic gradient optimization algorithms\u201d","volume":"31","author":"Tsitsiklis","year":"1986","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"4","key":"2026040910314629300_ref162","doi-asserted-by":"crossref","first-page":"872","DOI":"10.2307\/1907569","article-title":"\u201cMarket mechanisms and mathematical programming\u201d","volume":"28","author":"Uzawa","year":"1960","journal-title":"Econometrica"},{"issue":"3","key":"2026040910314629300_ref163","doi-asserted-by":"crossref","first-page":"182","DOI":"10.2307\/2296080","article-title":"\u201cWalras\u2019 t\u00e2tonnement in the theory of exchange\u201d","volume":"27","author":"Uzawa","year":"1960","journal-title":"The Review of Economic Studies"},{"issue":"8","key":"2026040910314629300_ref164","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1145\/79173.79181","article-title":"\u201cA bridging model for parallel computation\u201d","volume":"33","author":"Valiant","year":"1990","journal-title":"Communications of the ACM"},{"key":"2026040910314629300_ref165","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3264-1","volume-title":"The Nature of Statistical Learning Theory","author":"Vapnik","year":"2000"},{"key":"2026040910314629300_ref166","doi-asserted-by":"crossref","DOI":"10.1515\/9781400881895","volume-title":"Functional Operators, Volume 2: The Geometry of Orthogonal Spaces","author":"von Neumann","year":"1950"},{"issue":"1-2","key":"2026040910314629300_ref167","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000001","article-title":"\u201cGraphical models, exponential families, and variational inference\u201d","volume":"1","author":"Wainwright","year":"2008","journal-title":"Foundations and Trends in Machine Learning"},{"key":"2026040910314629300_ref168","volume-title":"\u00c9l\u00e9ments d\u2019\u00e9conomie politique pure, ou, Th\u00e9orie de la richesse sociale","author":"Walras","year":"1896"},{"issue":"2","key":"2026040910314629300_ref169","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1023\/A:1017522623963","article-title":"\u201cDecomposition method with a variable parameter for a class of monotone variational inequality problems\u201d","volume":"109","author":"Wang","year":"2001","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026040910314629300_ref170","volume-title":"Hadoop: The Definitive Guide","author":"White","year":"2010","edition":"2"},{"key":"2026040910314629300_ref171","volume-title":"Introductory Econometrics: A Modern Approach","author":"Wooldridge","year":"2009","edition":"4"},{"issue":"1","key":"2026040910314629300_ref172","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.sysconle.2004.02.022","article-title":"\u201cFast linear iterations for distributed averaging\u201d","volume":"53","author":"Xiao","year":"2004","journal-title":"Systems & Control Letters"},{"key":"2026040910314629300_ref173","doi-asserted-by":"crossref","unstructured":"Yang, A. Y., Ganesh, A., Zhou, Z., Sastry, S. S., & Ma, Y. (2010). \u201cA review of fast l1-minimization algorithms for robust face recognition\u201d. arXiv.","DOI":"10.21236\/ADA525384"},{"key":"2026040910314629300_ref174","unstructured":"Yang, J. & Yuan, X. (2010). \u201cAn inexact alternating direction method for trace norm regularized least squares problem\u201d. Optimization Online."},{"key":"2026040910314629300_ref175","unstructured":"Yang, J. & Zhang, Y. (2009). \u201cAlternating direction algorithms for l1-problems in compressive sensing\u201d. Preprint."},{"issue":"1","key":"2026040910314629300_ref176","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1137\/070703983","article-title":"\u201cBregman iterative algorithms for l1-minimization with applications to compressed sensing\u201d","volume":"1","author":"Yin","year":"2008","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"1","key":"2026040910314629300_ref177","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","article-title":"\u201cModel selection and estimation in regression with grouped variables\u201d","volume":"68","author":"Yuan","year":"2006","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2026040910314629300_ref178","unstructured":"Yuan, X. M.\n           (2009). \u201cAlternating direction methods for sparse covariance selection\u201d. Optimization Online."},{"key":"2026040910314629300_ref179","article-title":"\u201cSpark: Cluster computing with working sets\u201d","volume-title":"Proceedings of the USENIX Conference on Hot Topics in Cloud Computing","author":"Zaharia","year":"2010"},{"issue":"1","key":"2026040910314629300_ref180","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1214\/aos\/1079120130","article-title":"\u201cStatistical behavior and consistency of classification methods based on convex risk minimization\u201d","volume":"32","author":"Zhang","year":"2004","journal-title":"Annals of Statistics"},{"issue":"6A","key":"2026040910314629300_ref181","doi-asserted-by":"crossref","first-page":"3468","DOI":"10.1214\/07-AOS584","article-title":"\u201cThe composite absolute penalties family for grouped and hierarchical variable selection\u201d","volume":"37","author":"Zhao","year":"2009","journal-title":"Annals of Statistics"},{"issue":"6","key":"2026040910314629300_ref182","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1109\/TWC.2010.06.090890","article-title":"\u201cDistributed consensus-based demodulation: algorithms and error analysis\u201d","volume":"9","author":"Zhu","year":"2010","journal-title":"IEEE Transactions on Wireless Communications"},{"issue":"10","key":"2026040910314629300_ref183","doi-asserted-by":"crossref","first-page":"3970","DOI":"10.1109\/TSP.2009.2023936","article-title":"\u201cDistributed in-network channel decoding\u201d","volume":"57","author":"Zhu","year":"2009","journal-title":"IEEE Transactions on Signal Processing"}],"container-title":["Foundations and Trends\u00ae in Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/ftmal\/article-pdf\/3\/1\/1\/11505007\/2200000016en.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/www.emerald.com\/ftmal\/article-pdf\/3\/1\/1\/11505007\/2200000016en.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:10:40Z","timestamp":1777486240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.emerald.com\/ftmal\/article\/3\/1\/1\/1331527\/Distributed-Optimization-and-Statistical-Learning"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,26]]},"references-count":183,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,7,26]]}},"URL":"https:\/\/doi.org\/10.1561\/2200000016","relation":{},"ISSN":["1935-8237","1935-8245"],"issn-type":[{"value":"1935-8237","type":"print"},{"value":"1935-8245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,7,26]]}}}