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A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization

A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization

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时间：2019-07-03

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1、AGENERALFRAMEWORKFORACLASSOFFIRSTORDERPRIMAL-DUALALGORITHMSFORCONVEXOPTIMIZATIONINIMAGINGSCIENCEERNIEESSERXIAOQUNZHANGTONYCHANAbstract.Wegeneralizetheprimal-dualhybridgradient(PDHG)algorithmproposedbyZhuandChanin[M.Zhu,andT.F.Chan,AnEfficientPrimal-DualHybrid

2、GradientAlgorithmforTotalVariationImageRestoration,UCLACAMReport[08-34],May2008],explainconnectionstosimilarmethodsanddiscussconvergenceofseveralspecialcasesandmodifications.Inparticular,wepointoutaconvergenceresultforamodifiedversionofPDHGthathasasimilarlygo

3、odempiricalconvergenceratefortotalvariation(TV)minimizationproblems.Itsconvergencefollowsfromin-terpretingitasaninexactUzawamethod.WealsoproveaconvergenceresultforPDHGappliedtoTVdenoisingwithsomerestrictionsonthePDHGstepsizeparameters.Itisshownhowtointerpre

4、tthisspecialcaseasaprojectedaveragedgradientmethodappliedtothedualfunctional.WediscusstherangeofparametersforwhichtheinexactUzawamethodandtheprojectedaveragedgradientmethodcanbeshowntoconverge.WealsopresentsomenumericalcomparisonsofthesealgorithmsappliedtoT

5、Vdenoising,TVdeblurringandconstrainedl1minimizationproblems.Keywords.convexoptimization,totalvariationminimization,primal-dualmethods,operatorsplitting,l1basispursuitAMSsubjectclassifications.90C25,90C06,49K35,49N45,65K101.Introduction.Totalvariationminimiza

6、tionproblemsariseinmanyimageprocessingapplicationsforregularizinginverseproblemswhereoneexpectsthere-coveredimageorsignaltobepiecewiseconstantorhavesparsegradient.However,alackofdifferentiabilitymakesminimizingTVregularizedfunctionalscomputationallychallengi

7、ng,andsothereisconsiderableinterestinefficientalgorithms,especiallyforlargescaleproblems.Moregenerally,thereisinterestinpracticalmethodsforsolvingnon-differentiableconvexoptimizationproblems,TVminimizationbeinganimportantspecialcase.ThePDHGalgorithm[57]inagene

8、ralsettingisamethodforsolvingproblemsoftheformminJ(Au)+H(u),u∈RmwhereJandHareclosedproperconvexfunctionsandA∈Rn×m.Usually,J(Au)willcorrespondtoaregularizingtermoftheformkAuk,inwhichcasethePDHGmethodwor

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