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1、RFoundationsandTrendsinMachineLearningVol.3,No.1(2010)1–122c2011S.Boyd,N.Parikh,E.Chu,B.PeleatoandJ.EcksteinDOI:10.1561/2200000016DistributedOptimizationandStatisticalLearningviatheAlternatingDirectionMethodofMultipliersStephenBoyd1,NealParikh2,EricChu3BorjaPeleato4andJonathanEcks
2、tein51ElectricalEngineeringDepartment,StanfordUniversity,Stanford,CA94305,USA,boyd@stanford.edu2ComputerScienceDepartment,StanfordUniversity,Stanford,CA94305,USA,npparikh@cs.stanford.edu3ElectricalEngineeringDepartment,StanfordUniversity,Stanford,CA94305,USA,echu508@stanford.edu4Ele
3、ctricalEngineeringDepartment,StanfordUniversity,Stanford,CA94305,USA,peleato@stanford.edu5ManagementScienceandInformationSystemsDepartmentandRUTCOR,RutgersUniversity,Piscataway,NJ08854,USA,jeckstei@rci.rutgers.eduContents1Introduction32Precursors72.1DualAscent72.2DualDecomposition92
4、.3AugmentedLagrangiansandtheMethodofMultipliers103AlternatingDirectionMethodofMultipliers133.1Algorithm133.2Convergence153.3OptimalityConditionsandStoppingCriterion183.4ExtensionsandVariations203.5NotesandReferences234GeneralPatterns254.1ProximityOperator254.2QuadraticObjectiveTerms
5、264.3SmoothObjectiveTerms304.4Decomposition315ConstrainedConvexOptimization335.1ConvexFeasibility345.2LinearandQuadraticProgramming3661-NormProblems386.1LeastAbsoluteDeviations396.2BasisPursuit416.3General1RegularizedLossMinimization426.4Lasso436.5SparseInverseCovarianceSelection4
6、57ConsensusandSharing487.1GlobalVariableConsensusOptimization487.2GeneralFormConsensusOptimization537.3Sharing568DistributedModelFitting618.1Examples628.2SplittingacrossExamples648.3SplittingacrossFeatures669NonconvexProblems739.1NonconvexConstraints739.2Bi-convexProblems7610Impleme
7、ntation7810.1AbstractImplementation7810.2MPI8010.3GraphComputingFrameworks8110.4MapReduce8211NumericalExamples8711.1SmallDenseLasso8811.2Distributed1RegularizedLogisticRegression9211.3GroupLassowithFeatureSplitting9511.4DistributedLarge-ScaleLassowithMPI9711.5RegressorSelection1001
8、2Conclusions103Ackn
