机器学习题库

1、机器学习题库极大似然1、 MLestimationofexponentialmodel(10)AGaussiandistributionisoftenusedtomodeldataontherealline,butissometimesinappropriatewhenthedataareoftenclosetozerobutconstrainedtobenonnegative.Insuchcasesonecanfitanexponentialdistribution,whoseprobabilitydensityfunctionisgivenby1展pxe节bGivenNobservatio

2、nsxidrawnfromsuchadistribution:(a) Writedownthelikelihoodasafunctionofthescaleparameterb.(b) Writedownthederivativeoftheloglikelihood.(c)GiveasimpleexpressionfortheMLestimateforb.期(LL)IX:6=TT-d二人t工3工(b)-Alog(i)-TkHI=I髀 T 十由i=ldistributionp|withhyperparameters 丫，suchthattheposteriordistributionp|X,pX

3、|p|p|与先验的分布族相同(a)Supposethatthelikelihoodisgivenbytheexponentialdistributionwithrateparameter 人:x2、换成 Poisson 分布：px|e,y0,1,2x!，Nllogpx|i1NxilogNi1Nxilogi1Nlogxii1logx!贝叶斯1、贝叶斯公式应用假设在考试的多项选择中，考生知道正确答案的概率为 p,猜测答案的概率为 1-p,并且假设考生知道正确答案答对题的概率为 1,猜中正确答案的概率为 1/-m,其中 m 为多选项的数目。那么已知考生答对题目，求他知道正确答案的概率。：pknown

4、|correctpknown,correctppknown11p1p一m2、ConjugatepriorsGivenalikelihoodpx|foraclassmodelswithparameters0,aconjugatepriorisapx|ShowthatthegammadistributionGamma|,isaconjugatepriorfortheexponential.Derivetheparameterupdategivenobservationsx1,，xNandthepredictiondistributionpxN1|4，xN.(n)Ey|ioiiontial；mdGm

5、ninnTheiktilinclisF(XA)=口3%cxp(Mrandtlirpriori/J(A|八，J)=中打门后壮，|nr：”i-LtKth。L山山LJiepowtcriuiinP(A|KXAexp(-A)-fr,r1+exp%(；+产国)Tlircforothepiraiiwtrrupclatesarcasfollows:FortbepredictiondistributionwecomputetheIblljriugintegral；XcxM事州|tr+A.d+(rg+wr(Or+.V)(3+HJV+.nnirTlirn=it|fl)=(1-即ifandpriorisabft+1b

6、+k-1FVrUBptHdlctioiidlstributtnncomputethsfallowingkrtcgr近jp(工工s件Xi=kM阳/(L-例八%曰，怙W+L匕+4-1%出口“+/*+*-)r(A+jrg+r曲r(fj+i)r(&+ji-jr(+A+A-+-i)r(fj+匕+Q/(七+/+/-2)a+1r(4+A:1)r(ci+14-A+f_Ijcj+fe+Zk+f-1r(nb+k)r(&+A+f-2)=1)一十十+mwluretlwlipinill：imntestepHw时打口4加。1fnmuUan/依+cnwricd?)(j)(Forextracredit)A,tatisticT

7、(1)isjsaidtuheforsipajEigriet.erorinotherwords,itisiiidcpciHlcntof/ShowthatforarandomvariallcXtirnwiifromancxpoDcutialfhiuilydeikiityij),inasiifficieutstatisticfor牛(Shuwthatafactorization川 北 廿 (/ ： ) ； *)=/( 止 ) 七 (1” : 叶 )hiHciryrindsiiffi.ci.eiLtfor打(看)to1KUastatisticMrtf).(k) (Forextracredit)Supp

8、oseX一XnaredrawniidfromanexponentiaJfamilydensity吓What如nowthesufficientstatisticT(2r14.u_j-ft)tbrif?三、判断题(1)给定 n 个数据点，如果其中一半用于训练，另一半用于测试，则训练误差和测试误差之间的差别会随着 n的增加而减小。(2)极大似然估计是无偏估计且在所有的无偏估计中方差最小，所以极大似然估计的风险最小。(3)回归函数 A 和 B,如果 A 比 B 更简单，则 A 几乎一定会比 B 在测试集上表现更好。(4)全局线性回归需要利用全部样本点来预测新输入的对应输出值，而局部线性回归只需利用查询

9、点附近的样本来预测输出值。所以全局线性回归比局部线性回归计算代价更高。(5)Boosting 和 Bagging 都是组合多个分类器投票的方法，二者都是根据单个分类器的正确率决定其权重。(6)Intheboostingiterations,thetrainingerrorofeachnewdecisionstumpandthetrainingerrorofthecombinedclassifiervaryroughlyinconcert(F)Whilethetrainingerrorofthecombinedclassifiertypicallydecreasesasafunctionofbo

10、ostingiterations,theerroroftheindividualdecisionstumpstypicallyincreasessincetheexampleweightsbecomeconcentratedatthemostdifficultexamples.(7)OneadvantageofBoostingisthatitdoesnotoverfit.(F)(8)Supportvectormachinesareresistanttooutliers,i.e.,verynoisyexamplesdrawnfromadifferentdistribution.(F)(9)在回归

11、分析中，最佳子集选择可以做特征选择，当特征数目较多时计算量大；岭回归和 Lasso模型计算量小，且 Lasso 也可以实现特征选择。(10)当训练数据较少时更容易发生过拟合。(12)在核回归中，最影响回归的过拟合性和欠拟合之间平衡的参数为核函数的宽度。(13)IntheAdaBoostalgorithm,theweightsonallthemisclassifiedpointswillgoupbythesamemultiplicativefactor.(T)=exp(d)7.2pointstrue/falsenAdaBoost,weightedtrainingerrorqofthetif,we

12、akclassifierontrainingdatdwitliweightsDttendstoiiKTeaseasafunctionoff.SOLUTION：True.Inthecourseofboostingiterationstheweakclassifiersareforcedtotrytoclarifymoredifficultexamples.Theweightswillincreaseforexamplesthatarerepeatedlymisclassifiedbytheweakclassifiers.Theweightedtrainingerrorctoftheitfweak

13、classifieronthetrainingdat日thereforetendstoincrease.9.2pointsConsiderapointthatiscorrectlyclassifiedanddistantfromthedecisionboundary.WhywouldSrMsdecisionboundarybeunaffectedbyihispoint,buttheoneIrrtTiHtlyloginlicrrgriss.ioiibrHlfFd(HI?SOLUJION:Thehingelo5susedbySUM5giveszeroweighttothesepointswhile

14、thelog-lossusedbylogisticregressiongivesalittlebitofweighttothesepoints.(14) True/False:Inaleast-squareslinearregressionproblem,addinganL2regularizationpenaltycannotdecreasetheL2errorofthesolutionw?onthetrainingdata.(F)(15) True/False:Inaleast-squareslinearregressionproblem,addinganL2regularizationp

15、enaltyalwaysdecreasestheexpectedL2errorofthesolutionw?onunseentestdata(F).(16)除了 EM 算法，梯度下降也可求混合高斯模型的参数。(T)(20)Anydecisionboundarythatwegetfromagenerativemodelwithclass-conditionalGaussiandistributionscouldinprinciplebereproducedwithanSVMandapolynomialkernel.True!Infact,sinceclass-conditionalGaussia

16、nsalwaysyieldquadraticdecisionboundaries,theycanbereproducedwithanSVMwithkernelofdegreelessthanorequaltotwo.(21)AdaBoostwilleventuallyreachzerotrainingerror,regardlessofthetypeofweakclassifierituses,providedenoughweakclassifiershavebeencombined.False!Ifthedataisnotseparablebyalinearcombinationofthew

17、eakclassifiers,AdaBoostcantachievezerotrainingerror.(22)TheL2penaltyinaridgeregressionisequivalenttoaLaplacepriorontheweights.(F)(23)Thelog-likelihoodofthedatawillalwaysincreasethroughsuccessiveiterationsoftheexpectationmaximationalgorithm.(F)(24) Intrainingalogisticregressionmodelbymaximizingthelik

18、elihoodofthelabelsgiventheinputswehavemultiplelocallyoptimalsolutions.(F)四、回归1、考虑回归一个正则化回归问题。在下图中给出了惩罚函数为二次正则函数，当正则化参数 C取不同值时，在训练集和测试集上的 10g 似然（meanlog-probability）。（10 分）（1）说法“随着 C 的增加，图 2 中训练集上的 10g 似然永远不会增加”是否正确，并说明理由。（2）解释当 C 取较大值时，图 2 中测试集上的 10g 似然下降的原因。22、考虑线性回归模型：yNWoW1X,，训练数据如下图所不。（10 分）（1）用

19、极大似然估计参数，并在图（a）中画出模型。（3 分）（2）用正则化的极大似然估计参数，即在 10g 似然目标函数中加入正则惩罚函数并在图（b）中画出当参数 C 取很大值时的模型。（3 分）（3）在正则化后，高斯分布的方差2-是变大了、变小了还是不变？（4 分）3,1,我们用 1-10 阶多项式特征，采用线性回归模型来学习 x 与 y 之间的关系（高阶特征模型包含所有低阶特征）（1）现在n20个样本上，训练 1 阶、2 阶、8 阶和 10 阶特征的模型，然后在一个大规模的独立的测试集上测试，则在下 3 列中选择合适的模型（可能有多个选项），并解释第 3 列中你选择的模型为什么测试误差小。（10

20、分）训练误差最小训练误差最大测试误差最小1 阶特征的线性模型X2 阶特征的线性模型X8 阶特征的线性模型X10 阶特征的线性模型X现在n106个样本上，训 I 练 1 阶、2 阶、8 阶和 10 阶特征的模型，然后在一个大规模的独3.考虑二维输入空间点XTX1,X2上的回归问题，其中Xj1,1,j1,2在单位正方形内训练样本和测试样本在单位正方形中均匀分布输出模型为yNX3X510 x1x27x125x2,损失函数取平方误差损失。立的测试集上测试，则在下 3 列中选择合适的模型（可能有多个选项），并解释第 3 列中你选择的模型为什么测试误差小。（10 分）训练误差最小训练误差最大测试误差最小1

21、 阶特征的线性模型X2 阶特征的线性模型8 阶特征的线性模型XX10 阶特征的线性模型X(3)Theapproximationerrorofapolynomialregressionmodeldependsonthenumberoftrainingpoints.(T)(4)Thestructuralerrorofapolynomialregressionmodeldependsonthenumberoftrainingpoints.(F)4、Wearetryingtolearnregressionparametersforadatasetwhichweknowwasgeneratedfroma

22、polynomialofacertaindegree,butwedonotknowwhatthisdegreeis.Assumethedatawasactuallygeneratedfromapolynomialofdegree5withsomeaddedGaussiannoise(thatis2345yw)wxw?xwxw4xwxFortrainingwehave100 x,ypairsandfortestingweareusinganadditionalsetof100 x,ypairs.Sincewedonotknowthedegreeofthepolynomialwelearntwom

23、odelsfromthedata.ModelAlearnsparametersforapolynomialofdegree4andmodelBlearnsparametersforapolynomialofdegree6.Whichofthesetwomodelsislikelytofitthetestdatabetter?Answer:Degree6polynomial.Sincethemodelisadegree5polynomialandwehaveenoughtrainingdata,themodelwelearnforasixdegreepolynomialwilllikelyfit

24、averysmallcoefficientforx6.Thus,eventhoughitisasixdegreepolynomialitwillactuallybehaveinaverysimilarwaytoafifthdegreepolynomialwhichisthecorrectmodelleadingtobetterfittothedata.5、Input-dependentnoiseinregression八,0,1Ordinaryleast-squaresregressionisequivalenttoassumingthateachdatapointisgeneratedacc

25、ordingtoalinearfunctionoftheinputpluszero-mean,constant-varianceGaussiannoise.Inmanysystems,however,thenoisevarianceisitselfapositivelinearfunctionoftheinput(whichisassumedtobenon-negative,i.e.,x=0).a)Whichofthefollowingfamiliesofprobabilitymodelscorrectlydescribesthissituationintheunivariatecase?(H

26、int:onlyoneofthemdoes.)(iii)iscorrect.InaGaussiandistributionovery,thevarianceisdeterminedbythecoefficientofy2;so2.2byreplacingbyx,wegetavariancethatincreaseslinearlywithx.(Notealsothechangetothenormalization“constant.)(i)hasquadraticdepenXe 的 edoesnotchangethevarianceatall,itjustrenamesw1.b)Circlet

27、heplotsinFigure1thatcouldplausiblyhavebeengeneratedbysomeinstanceofthemodelfamily(ies)youchose.(11)and(iii).(Notethat(iii)worksfor20.)(i)exhibitsalargevarianceatx=0,andthevarianceappearsindependentofx.c)True/False:Regressionwithinput-dependentnoisegivesthesamesolutionasordinaryregressionforaninfinit

28、edatasetgeneratedaccordingtothecorrespondingmodel.True.Inbothcasesthealgorithmwillrecoverthetrueunderlyingmodel.d)Forthemodelyouchoseinpart(a),writedownthederivativeofthenegativeloglikelihoodwithrespecttowi.Theuegativeluglikclilujod今(Wi-(+可13)2工卬2Hiidtheileiivatiivew.r.t认NotethatforlinestlLioiiglith

29、e-orisin(w=0),theoptiiidHolmtionliasthepartknlarhsimpleform南=w工LtisputssHilctoLikeI1ULLdeiiviitiM?dfdielugwirlioutiwjticiiiUiutIUJJ;exp(j)=/：weumyIUKlikcliliuixlatorHgoorlrcfion!Phi，theyiuipli0rlivh：nKlliiigofniidriplcdatap“in,IICCHHSPtlicproductofpr口bM/Hticshcouucsa即川ioflogprobhliiitios.II111.五、分类1

30、 .产生式模型vs,判别式模型(a) Yourbillionairefriendneedsyourhelp.Sheneedstoclassifyjobapplicationsintogood/badcategories,andalsotodetectjobapplicantswholieintheirapplicationsusingdensityestimationtodetectoutliers.Tomeettheseneeds,doyourecommendusingadiscriminativeorgenerativeclassifier?Why?产生式模型因为要估计密度px|y(b)

31、Yourbillionairefriendalsowantstoclassifysoftwareapplicationstodetectbug-proneapplicationsusingfeaturesofthesourcecode.Thispilotprojectonlyhasafewapplicationstobeusedastrainingdata,though.Tocreatethemostaccurateclassifier,doyourecommendusingadiscriminativeorgenerativeclassifier?Why?判别式模型样本数较少，通常用判别式模

32、型直接分类效果会好些(d)Finally,yourbillionairefriendalsowantstoclassifycompaniestodecidewhichonetoacquire.Thisprojecthaslotsoftrainingdatabasedonseveraldecadesofresearch.Tocreatethemostaccurateclassifier,doyourecommendusingadiscriminativeorgenerativeclassifier?Why?产生式模型样本数很多时，可以学习到正确的产生式模型2、logstic 回归averageI

33、og-probabilityoftestlabels00.511522.533.54regularizationparameterCFigure2:Log-probabilityoflabelsasafunctionofregularizationparameterCHereweusealogisticregressionmodeltosolveaclassificationproblem.InFigure2,wehaveplottedthemeanlog-probabilityoflabelsinthetrainingandtestsetsafterhavingtrainedtheclass

34、ifierwithquadraticregularizationpenaltyanddifferentvaluesoftheregularizationparameterC.1、 Intrainingalogisticregressionmodelbymaximizingthelikelihoodofthelabelsgiventheinputswehavemultiplelocallyoptimalsolutions.(F)Answer:Thelog-probabilityoflabelsgivenexamplesimpliedbythelogisticregressionmodelisac

35、oncave(convexdown)functionwithrespecttotheweights.The(only)locallyoptimalsolutionisalsogloballyoptimal2、 Astochasticgradientalgorithmfortraininglogisticregressionmodelswithafixedlearningratewillfindtheoptimalsettingoftheweightsexactly.(F)Answer:Afixedlearningratemeansthatwearealwaystakingafinitestep

36、towardsimprovingthelog-probabilityofanysingletrainingexampleintheupdateequation.Unlesstheexamplesaresomehow“aligned”,wecoilitinuejumpingfromsidetosideoftheoptimalsolution,andwillnotbeabletogetarbitrarilyclosetoit.Thelearningratehastoapproachtozerointhecourseoftheupdatesfortheweightstoconverge.3、 The

37、averagelog-probabilityoftraininglabelsasinFigure2canneverincreaseasweincreaseC.(T)Strongerregularizationmeansmoreconstraintsonthesolutionandthusthe(average)log-probabilityofthetrainingexamplescanonlygetworse.4、 ExplainwhyinFigure2thetestlog-probabilityoflabelsdecreasesforlargevaluesofC.AsCincreases,

38、wegivemoreweighttoconstrainingthepredictor,andthusgivelessflexibilitytofittingthetrainingset.Theincreasedregularizationguaranteesthatthetestperformancegetsclosertoaveragelog-prcoftraininglabels4-O.thetrainingperformance,butasweover-constrainourallowedpredictors,wearenotabletofitthetrainingsetatall,a

39、ndalthoughthetestperformanceisnowveryclosetothetrainingperformance,botharelow.5、 Thelog-probabilityoflabelsinthetestsetwoulddecreaseforlargevaluesofCevenifwehadalargenumberoftrainingexamples.(T)Theaboveargumentstillholds,butthevalueofCforwhichwewillobservesuchadecreasewillscaleupwiththenumberofexamp

40、les.6、 Addingaquadraticregularizationpenaltyfortheparameterswhenestimatingalogisticregressionmodelensuresthatsomeoftheparameters(weightsassociatedwiththecomponentsoftheinputvectors)vanish.Aregularizationpenaltyforfeatureselectionmusthavenon-zeroderivativeatzero.Otherwise,theregularizationhasnoeffect

41、atzero,andweightwilltendtobeslightlynon-zero,evenwhenthisdoesnotimprovethelog-probabilitiesbymuch.3、正则化的 Logstic 回归ThisproblemwewillrefertothebinaryclassificationtaskdepictedinFigure1(a),whichweattempttosolvewiththesimplelinearlogisticregressionmodel户=L|x,叫，wa)=g(皿/-n吧工幻=；;(forsimplicitywedonotuseth

42、ebiasparameterw0).Thetrainingdatacanbeseparatedwithzerotrainingerror-seelineL1inFigure1(b)forinstance.(a)The2-dimensionaldatasetusedinProblem2Consideraregularizationapproachwherewetrytomaximize加耳以此|如 tik 心)-uj：=i-forlargeC.Notethatonlyw2ispenalized.WedliketoknowwhichofthefiiussinFigure1(b)couldarise

43、asaresultofsuchregularization.ForeachpotentiallineL2,L3orL4determinewhetheritcanresultfromregularizingw2.Ifnot,explainverybrieflywhynot.L2:No.Whenweregularizew2,theresultingboundarycanrelylessonthevalueofx2andthereforebecomesmorevertical.L2hereseemstobemorehorizontalthantheunregularizedsolutionsoitc

44、annotcomeasaresultofpenalizingw2L3:Yes.Herew2A2issmallrelativetow1A2(asevidencedbyhighslope),andeventhoughitwouldassignaratherlowlog-probabil(b)ThepointscanbeseparatedbyL1(solidline).PossibleotherdecisionboundariesareshownbyL2;L3；L4.itytotheobservedlabels,itcouldbeforcedbyalargeregularizationparamet

45、erC.L4:No.ForverylargeC,wegetaboundarythatisentirelyvertical(linex1=0orthex2axis).L4hereisreflectedacrossthex2axisandrepresentsapoorersolutionthanitscounterpartontheotherside.Formoderateregularizationwehavetogetthebestsolutionthatwecanconstructwhilekeepingw2small.L4isnotthebestandthuscannotcomeasare

46、sultofregularizingw2.(2)Ifwechangetheformofregularizationtoone-norm(absolutevalue)andalsoregularizew1wegetthefollowingpenalizedlog-likelihoodlog一洲X;，叫,帆)-(IWi|4-w2)-i=lConsideragaintheprobleminFigure1(a)andthesamelinearlogisticregressionmodel.AsweincreasetheregularizationparameterCwhichofthefollowin

47、gscenariosdoyouexpecttoobserve(chooseonlyone):(x)Firstw1willbecome0,thenw2.()w1andw2willbecomezerosimultaneously()Firstw2willbecome0,thenw1.()Noneoftheweightswillbecomeexactlyzero,onlysmallerasCincreasesThedatacanbeclassifiedwithzerotrainingerrorandthereforealsowithhighlog-probabilitybylookingatthev

48、alueofx2alone,i.e.makingw1=0.Initiallywemightprefertohaveanon-zerovalueforw1butitwillgotozeroratherquicklyasweincreaseregularization.Notethatwepayaregularizationpenaltyforanon-zerovalueofw1andifitdoesnhelp1classificationwhywouldwepaythepenalty?Theabsolutevalueregularizationensuresthatw1willindeedgot

49、oexactlyzero.AsCincreasesfurther,evenw2willeventuallybecomezero.Wepayhigherandhighercostforsettingw2toanon-zerovalue.Eventuallythiscostoverwhelmsthegainfromthelog-probabilityoflabelsthatwecanachievewithanon-zerow2.Notethatwhenw1=w2=0,thelog-probabilityoflabelsisafinitevaluenlog(0:5).1、SVMFigure4:Tra

50、iningset,maximummarginlinearseparator,andthesupportvectors(inbold).(1)Whatistheleave-one-outcross-validationerrorestimateformaximummarginseparationinfigure4?(weareaskingforanumber)(0)Basedonthefigurewecanseethatremovinganysinglepointwouldnotchancetheresultingmaximummarginseparator.Sinceallthepointsa

51、reinitiallyclassifiedcorrectly,theleave-one-outerroriszero.(2) Wewouldexpectthesupportvectorstoremainthesameingeneralaswemovefromalinearkerneltohigherorderpolynomialkernels.(F)Therearenoguaranteesthatthesupportvectorsremainthesame.Thefeaturevectorscorrespondingtopolynomialkernelsarenon-linearfunctio

52、nsoftheoriginalinputvectorsandthusthesupportpointsformaximummarginseparationinthefeaturespacecanbequitedifferent.(3) Structuralriskminimizationisguaranteedtofindthemodel(amongthoseconsidered)withthelowestexpectedloss.(F)Weareguaranteedtofindonlythemodelwiththelowestupperboundontheexpectedloss.(4) Wh

53、atistheVC-dimensionofamixtureoftwoGaussiansmodelintheplanewithequalcovariancematrices?Why?AmixtureoftwoGaussianswithequalcovariancematriceshasalineardecisionboundary.LinearseparatorsintheplanehaveVC-dimexactly3.4、SVM对如下数据点进行分类:(a) Plotthesesixtrainingpoints.Aretheclasses+，-linearlyseparable?yes(b) C

54、onstructtheweightvectorofthemaximummarginhyperplanebyinspectionandidentifythesupportvectors.Themaximummarginhyperplaneshouldhaveaslopeof-1andshouldsatisfyx1=3/2,x2=0.Thereforeitsequatixn+s2=3/2,andtheweightvectoris(1,1)T.(c) Ifyouremoveoneofthesupportvectorsdoesthesizeoftheoptimalmargindecrease,stay

55、thesame,orincrease?Inthisspecificdatasettheoptimalmarginincreaseswhenweremovethesupportvectors(1,0)or(1,1)m孙andstaysthesamewhenweremovetheothertwo.(d) (ExtraCredit)Isyouranswerto(c)alsotrueforanydataset?Provideacounterexampleorgiveashortproof.Whenwedropsomeconstraintsinaconstrainedmaximizationproble

56、m,wegetanoptimalvaluewhichisatleastasgoodthepreviousone.Itisbecausethesetofcandidatessatisfyingtheoriginal(larger,stronger)setofcontraintsisasubsetofthecandidatessatisfyingthenew(smaller,weaker)setofconstraints.So,fortheweakerconstraints,theoldoptimalsolutionisstillavailableandtheremaybeadditionssol

57、tonsthatareevenbetter.Inmathematicalform:maxfix)1,?=1.2.3UsingthemethodofLagrangemultipliersshowthatthesolutionisV?0,0,2,b1andthemarginislinearlyseparable?x1,.2x,x2.Arethe,-NoForoptimizationproblemswithinequalityconstraintssuchastheabove,weshouldapplyKKTconditionswhichisageneralizationofLagrangemult

58、ipliers.Howeverthisproblemcanbesolvedeasierbynotingthatwehavethreevectorsinthe3-dimensionalspaceandallofthemaresupportvectors.Hencetheall3constraintsholdwithequality.ThereforewecanapplythemethodofLagrangemultipliersto,min|wI2s.t.曲曲.1i)=1,i=1,2.3Vehave3constraints,andshouldhavp3Lagrangemultipliers.We

59、firstformtheLagrangianfunctionZ(w.A： whpreA=(ALh23)隈follows1,=71Ml同+%(如(与)+的1)i=lanfldificmitifitcwithrespectr.ooptimizationvarirtblnswandbmidtuzejo.Hww+尢物。(S)1=1吧产=3uby1=1l.Ningthedatapoints妣卬),wegetthefollowingequationsfromtheabovelines.wj+Aj而一入3=0(10)6-vAa=0(11)-A2-As=1)(12)Ai 一 43-人 3=0(13)(14)I

60、.m 电(10)and(14)嗝工1grt 支=0,Tlienph 喝日 iu 裁 tliisCoequalityconstraiiithintheoptiinizatioHprobleiikweget%=1(15)V2W2+tf3-1-t=1(16)4/22+L=1(17)(16)and(17)implytliatu”=0.mid七乜=2.llierctorctheo)tiniapciglitsarew=(0.0：2)an=LAntitheluarginis1/2.(e) Showthatthesolutionremainsthesameiftheconstraintsarechangedt