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1、3810IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.11,NO.10,OCTOBER2012OptimalityofDualMethodsforDiscreteMultiuserMulticarrierResourceAllocationProblemsSimonG¨ortzenandAnkeSchmeink,Member,IEEEAbstractDualmethodsbasedonLagrangianrelaxationareusuallyinadditiontotheconcavecase.In[13],adiscretethest
2、ateofthearttosolvemultiusermulticarrierresourceRAPisanalysedwithaMinimumCostNetworkFlowmodelallocationproblems.Thisappliestoconcaveutilityfunctionsanddualmethodsareapplied.Thislistisnotintendedtobeaswellastopracticalsystemsemployingadaptivemodulation,exhaustivebutshowcasesthepopularityofdualme
3、thodsininwhichusersdataratescanbedescribedbystepfunctions.Weshowthatthisdiscreteresourceallocationproblemcanthisfield.Todate,dualmethodsareappliedasapowerfultoolbeformulatedasanintegerlinearprogrambelongingtothetosolvearbitraryRAPs,whichareoftendiscreteduetotheirclassofmultiple-choiceknapsackpr
4、oblems.Asaknapsackapplicationinpracticalsystems.Theperformanceoftheseproblemwithadditionalconstraints,thisproblemisNP-hard,methodsisattributedtothefactthatthedualitygapvanishesbutfacilitatesapproximationalgorithmsbasedonLagrangianwhenthenumberofsubcarrierstendstoinfinity,whichcanrelaxation.Wesh
5、owthatthesedualmethodscanbedescribedasroundingmethods.Asanimmediateresult,weconcludethatbeinterpretedasallowingarbitrarytime-sharingwithinthepriorclaimsofoptimality,basedonavanishingdualitygap,aresystem.Inthispaper,weshowthatthevanishingdualitygapinsufficient.Toanswerthequestionofoptimalityofdu
6、almethodsofdiscreteRAPsaloneisnotsufficienttoguaranteenear-fordiscretemulticarrierresourceallocationproblems,wepresentoptimalperformanceofdualmethods.boundsontheabsoluteintegralitygapforthreeexemplarydownlinkresourceallocationproblemswithdifferentobjectivesTothebestoftheauthorsknowledge,thesedi
7、screteprob-whenemployingroundingmethods.Theobtainedboundsarelemformulationshaveneverbeenanalysedindetailfromtheasymptoticallyoptimalinthesensethattherelativeperformanceperspectiveoflinearandintegerlinearoptimization.Inthislossvanishesas
