凸优化2017作业及答案4

SI251-ConvexOptimization,Spring2017Homework4Dueon08:00a.m.,April6,2017,beforeclassNote:PleasecompressyourcodesintooneleandsentittoTAs,andprintyourguresorresultsandanswerthequestionsonA4paper.FinishyoursimulationwithCVXpackage(MATLAB/Python/


).Andinitializeyourprogramwithcom-mandstoxyourrandomizedresultsandmakesurethatyourresultsarerepeatable.Forexample,ifyouareusingMATLAB,youmayaddrng('default');rng(1);inthepreamble.AndyoumayneedtoreprogramthegivenMATLABcodesegmentstootherprogramminglanguagesthatyou'dliketochoose.1.Feasibility1)(Multiusertransmitbeamforming.)PowerminimizationprobleminwirelesscommunicationP:minimizew1;


;wKKXk=1kwkk2subjecttoSINRkk;k=1;


;K;(1)wherew1;


;wK2Cnarethebeamformingvectorsforreceiverk=1;


;K.Signal-to-interference-plus-noise-ratioforthek-thuserSINRkisgivenbySINRk=jhHkwkj2Pi6=kjhHkwij2+2;(2)wherehk2Cnisthechannelcoecientvectorbetweenthetransmitterandthek-threceiverand2isnoisepower.Inthesimulation,considerthecomplexGaussianchannel,i.e.hkCN(0;s2I)inwhichs=1=pK.Andthenoisepower2canbesetas1withoutlossofgenerality.EachtargetSINRk0andit'softenrepresentedwithdB,whichisdenedas10logk.(a)ConsidertherelationshipbetweentargetSINRandthefeasibilityofP.Pleasedrawthephasetransition1gurewhereX-axisistargetSINRindB(1=


=K=),andY-axisistheratiowhentheproblemisfeasibleovermultiplerealizationsofchannel,i.e.R=#fPisfeasibleg#oftests(channelrealizations):(3)AssumeK=50;n=3:Youneedtorun20timesandtakeaverage.(5points)(b)PleasedrawthephasetransitiongureabouttherelationshipbetweenthenumberofusersKandthefeasibilityofP.Assumen=3;=15dB:Youneedtorun20timesandtakeaverage.(5points)(c)PleasedrawthephasetransitiongureabouttherelationshipbetweenthenumberofantennasnandthefeasibilityofP.AssumeK=100;=10dB:Youneedtorun20timesandtakeaverage.(5points)2)(Second-orderconeoptimizationproblem.)RandomlygeneratestandardSOCPPSOCP:minimizex2RnfTxsubjecttokAix+bikcTix+di;i=1;


;K(4)whereeachentryofAi2Rmn;bi2Rm;ci2Rn;di2Risalldrawofi.i.d.standardGaussiandistributionN(0;1).PleasedrawthephasetransitiongureabouttherelationshipbetweenthenumberofconstriantsKandthefeasibilityofPSOCP.Assumem=20;n=100.Youneedtorun20timesandtakeaverage.(10points)1Formoreaboutphasetransition,refertoDennisAmelunxenetal.:Livingontheedge:Phasetransitionsinconvexprogramswithrandomdata,in:InformationandInference2014,iau0051凸优化2017作业及答案2.Optimizationproblems.(a)(LASSO.)Wewishtorecoverasparsevectorx2Rnfrommeasurementsy2Rm.Ourmeasurementmodeltellsusthaty=Ax+v;whereA2Rmnisaknownmatrixandv2Rmisunknownmeasurementerror.TheentriesofvaredrawnIIDfromthedistributionN(0;2).WecanrsttrytorecoverxbysolvingtheoptimizationproblemminxkAxyk22+jjxjj22:(5)Thisproblemiscalledridgeregression.AmoresuccessfulapproachistosolvetheLASSOproblemminxkAxyk22+jjxjj1:(6)Pleaseusethecodebelowtodenen,m,A,x,andy.1n=200;2m=100;3truex=sparse(fix(rand(20,1)∗100),ones(20,1),100∗rand(20,1),n,1);4A=randn(m,n);5sigma=1;6v=normrnd(0,sigma,m,1);7y=A∗truex+v;(a)UseCVXtoestimatexfromyusingridgeregressionandLASSOproblem,respectively.(15points)(b)Plotyourresulttocomparetheestimatedxwiththetruex.(5points)(c)Howmanymeasurementsmareneededtondanaccuratexwithridgeregression?HowaboutwiththeLASSO?(5points)(b)(PortfolioOptimization.)Findminimum-riskportfolioswiththesameexpectedreturnastheuniformportfolio(w=(1=n)1),withriskmeasuredbyportfolioreturnvariance,andthefollowingportfolioconstraints(inadditionto1Tw=1):No(additional)constraints.Long-only:w0.Limitontotalshortposition:1Tw0:5,where(w)i=maxfwi;0g.(a)UseCVXtocomparetheoptimalriskintheseportfolioswitheachotherandtheuniformportfolio.(10points)(b)Plottheoptimalrisk-returntrade-ocurvesforthelong-onlyportfolio,andfortotalshortpositionlimitedto0.5,inthesamegure.Commentontherelationshipbetweenthetwotrade-ocurves.(10points)(c)(EnergyStorageTrade-os.)Weconsidertheuseofastoragedevice(say,abattery)toreducethetotalcostofelectricityconsumedoveroneday.WedividethedayintoTtimeperiods,andletptdenotethe(positive,time-varying)electricityprice,andutdenotethe(nonnegative)usageorconsumption,inperiodt,fort=1;:::;T.Withouttheuseofabattery,thetotalcostispTu.Letqtdenotethe(nonnegative)energystoredinthebatteryinperiodt.Forsimplicity,weneglectenergyloss(althoughthisiseasilyhandledaswell),sowehaveqt+1=qt+ct,t=1;:::;T1,wherectisthechargingofthebatteryinperiodt;ct0meansthebatteryisdischarged.Wewillrequirethatq1=qT+cT,i.e.,wenishwiththesamebatterychargethatwestartwith.Withthebatteryoperating,thenetconsumptioninperiodtisut+ct;werequirethistobenonnegative(i.e.,wedonotpumppowerbackintothegrid).ThetotalcostisthenpT(u+c).Thebatteryischaracterizedbythreeparameters:ThecapacityQ,whereqtQ;themaximumchargerateC,wherectC;andthemaximumdischargerateD,wherectD.(TheparametersQ,C,andDarenonnegative.)(a)Explainhowtondthechargingprolec2RT(andassociatedstoredenergyproleq2RT)thatminimizesthetotalcost,subjecttotheconstraints.(5points)minq;cpT(u+c)s.tqt+1=qt+ct;t=1;:::;T1q1=qT+cT0qtQ;t=1;:::;TDctC;t=1;:::;T0ut+ct;t=1;:::;T2(b)UseCVXtosolvetheproblemabovewithQ=35,C=