系统优化与调度读书报告.docx
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系统优化与调度读书报告.docx
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系统优化与调度读书报告
BookReportofSystem
Optimizationand
Scheduling
Theconjugategradientmethodanditsapplicationinsolving
optimizationproblems
1.Introductionofproblem'background
Optimizationtheoryandmethodsisaveryactiveyoungdiscipline,itdiscussesthecharacteristicsofthebestchoicedecidingproblems.Structthecalculationsofseekingoptimalsolution,studythetheoreticalpropertiesandtheactualcomputingperformaneeofthesecalculations.Withtherapiddevelopmentofhigh-tech,computerandinformationtechnology,optimizationtheoryandmethodsbecomemoreandmoreimportant,ithasbeenwidelyusedinvariousaspectsofthenaturalsciencesandtheengineeringdesign.Conjugategradientmethodisoneofthemostcommonlyusedoptimizationmethods.Inalloptimizationsneedtocalculatederivative,thesteepestdescentmethodisthemostsimple,butitistooslowconvergenee.Quasi-Newtonmethodconvergesquickly,iswidelyregardedasthemosteffectivemethodfornonlinearprogramming,butthequasi-Newtonmethodrequiresthestoragematrixandbysolvinglinearequationstocalculatethesearchdirection,whichisalmostimpossibletosolvelarge-scaleproblems.
Conjugategradientmethodcantransformann-dimensionaloptimizationproblemintonequivalentone-dimensionalproblems,thealgorithmissimple,smallstoragerequirements,theconvergeneeratesurpassesfaststeepestdescentmethod,andisparticularlysuitableforsolvinglarge-scaleproblems.Suchaselectricitydistribution,oilexploration,atmosphericmodeling,aerospaceandotherproposedoptimizationproblems.
ConjugategradientmethodwasfirstproposedbyHestenesandStieflecamein1952,forthesolutionoflinearequationsofdefinitecoefficientmatrix.Thefamousarticletheycooperate-“Methodofconjugategradientsforsolvinglinearsystems⑴isconsideredtobethefounderofthearticlesaboutconjugategradientmethod.Thisarticlediscussesindetailthenatureoftheconjugategradientmethodforsolvinglinearequationsanditsrelationshipwithothermethods.Onthisbasis,FletcherandReevesin1964firstproposedtheconjugategradientmethodtosolveanonlinearoptimizationproblem,makingitanimportantoptimizationmethod.Subsequently、BealeFletcher、Powellandotherscholarsbein-depthstudy,givenearlyresultsofnonlinearconjugategradientmethodsomeconvergeneeanalysis.Sincetheconjugategradientmethoddoesnotrequirematrixstorage,andhasafasterconvergencerateandsecondarytermination,etc.,andnowtheconjugategradientmethodhasbeenwidelyusedinpracticalproblems.
2.Mathematicaldescriptionoftheproblem
x
WeproceedfromthepointXqR,inturnone-dimensionalsearchalongthe
groupofconjugatedirectiontosolvingunconstrainedoptimizationproblemsisknownasconjugatedirectionmethod.Conjugategradientmethodusesconjugatedirectionasakindofsearchdirection.Itisatypicalmethodofconjugatedirection,eachofsearchdirectionsaremutuallyconjugate,andthesesearchdirectionsareonlyacombinationofthenegativegradientdirectionwiththedirectionofthepreviousiterationofthesearch,Therefore,storeless,calculateconveniently.Meanwhile,theconjugategradientmethodisamethodbetweenbetweensteepestdescentmethodandNewton'smethod,itusesonlythefirstorderderivativeinformation,butovercomesthedisadvantageofslowconvergeneeofthesteepestdescentmethod,butalsoavoidstheneedtostoreandcalculateHessematrixinversionofNewton'smethod.
Thebasicideaofconjugategradientmethodistocombineconjugationandthesteepestdescentmethod,usingtheknownpoint'gradienttostructagroupofconjugatedirections,andsearchelementsalongthedirectionofthisgroup,tofindtheminimumpointoftheobjectivefunction.Accordingtothebasicnatureoftheconjugatedirection,thismethodhasthesecondtermination.Intheconjugatedirection,iftaketheinitialsearchdirectiondQ-f(Xq),thefollowingconjugatedirectiondkisdeterminedbythenegativegradient」f(xjfromiterationofktimesandlinearcombinationwhoseconjugatedirectiondk4hasbeenobtainednegativek-thiterationoftheconjugategradientdirectionhasbeenobtainedbylinearcombination,whichconstructaspecificconjugatedirectionmethod:
(2-1)
dk-f(xj:
k」dk」,k=1,2,...,n-1
Becauseeachconjugatedirectionisdependentonthenegativegradientofiterationpoint,soitiscalledtheconjugategradientmethod.
3.Thealgorithm
(1)Linearconjugategradientmethod
Linearconjugategradientmethod^isproposedindependentlybyHesetnesand
SteleflinsolvinglinearequationsAx=b,xRn.(3-1)
WhenAisasymmetricpositivedefinitematrix,linearequationsisequivalenttothequadraticoptimizationproblemstosolvethefollowingformula:
1
T",Tn
xAx「bx,xR
2
Therefore,Hestneesandsteiefl'sapproachcanbeviewedastheconjugategradientmethodtoevaluattheminimumvalueofthequadraticfunction.
Linearconjugategradientmethodsstepsareasfollows:
a.
T
一猛.Orderthenext
Selecttheinitialpointx^Rn,setr°二Axq-b,d°--r0,k=0.
b.Ifrk_;,thenstop,orcalculationstepfactork=
iterationpoint:
Xk1=兀:
“dk,R1=1-b.
Anotablefeatureoflinearconjugategradientmethodisaboutthedirectiondk,k=0,1...generatedbythealgorithm,aboutAconjugate,sowithlimitedtermination.
(2)Nonlinearconjugategradientmethod
Hypothesisf:
Rn—Riscontinuousanddifferentiable,g(x)isthegradientoffatpointx.Thegeneralformatofnonlinearconjugategradientmethodforsolvingunconstrainedminimizationproblem:
minf(x),xRnisasfollows:
(3-3)
kd,kk=0,1,...
Amongthem,:
kcanbegettedbysomelinearsearch,thesearchdirectiondkisdefinedbythefollowingequation:
(3-4)
-gk,k=0
k一©:
kdk」,k0
kisparameter.Thedifferenttypeofkcorrespondstothedifferentnonlinear
conjugategradientmethod.Thefollowingsgivessomeknownmethocfsparameter:
k:
Amongthem,yk」二gk74二g(xj・
Accordingtothecharacteristicsofthegradient,intheexperimentalpointsx(k),selectnegativegradientdirectionasthesearchdirection,thefastestdecreaseinthefunctionvalue,whichcanbeshownthat,foranon-negativedefinitequadraticfunctionatn-dimensionalEuclideanspace,canbeachievedtheminimumvalueofthepointwithoutexceedingnsearchtimes.Nowusethepositivedefinitequadratic
1Tt
functionf(x)xAxbxc(Aisnnsymmetricpositivedefinitematrix;cisstandardconstants;x=(%,x2,,xn)T;bT=(b,b2,,bh))asanexampletoexplain,
alsocanbecalledFRconjugategradientmethod.Ifyoutakethesteplengthparameterh,thentheiterativeequationsofnextstepc(kd):
(3-6)
Set&=h(x(k))|asastepofone-dimensionalsearchfromx(k)alongthe
searchdirection,wecanget:
Ifanegativegradientangleofrotation,sothatthesearchdirectionbecomes
conjugatedirection,namely
(k1)
■kisconjugatecoefficient,ifHessianmatrixisA,multiply(d(k))TAatboth
sidesoftheequation,whichontheleftandguarantee(k)andd(k刊aboutAconjugate,thenwehave:
(d(k))TAd(k1丿-(dk())TAgk(7(dk)(T)A£k(0(3-9)
(3-10)
(d(k))TAg(k1)
(d(k))TAd(k)
InordertoavoidthetroubleofcalculatingtheHessianmatrixA,nowtrytoeliminateAfromtheaboveformula,multiplyAsimultaneouslybybothsidesontheleftandthenplusb(bT=(b1,b2/,bh))ofone-dimensionalsearchingiterative
formulax(k十)=x(k)+?
^d(k),weget:
(3-11)
Ax(k1)-b=Ax(k)bkAd(k)
£f
Accordingtothequationf(x)=bAx,theformulacanbechangedto;
ex
\f(x(k»f(x(k))kAd(k)org(k°-g(k)「kAd(k)(3-12)
(d(k))TAg(f
Substitutingthetopquationintok二(护,weget:
(3-13)
Theaboveistheconjugategradientmethod,themethodhassimplestructure,onlyneedtostorethreevariables,occupylessstorageunits,andfacilitatetoiterativelycalculateforcomputer.
Iterativestepsofconjugategradientmethodasfollow®:
a.Setthenumberofiterationk=0,setthetoleranceerror;andtheinitialpoint
(k)
X
b.Calculatethegradientvectorg(k)-'■f(x(k))offunctionf(x)atx(k),testingitifmeets]g(k)兰名,ifitmeetsthenthex(k)wegetistheapproximatelyoptimalsolutionx;orletd(k)--g(k).
c.Determinethebestofstepk(fromx(k),alongthedirectiond(k)),toconductone-dimensionalsearch,ifitmeetsf(x(k):
:
;;kdk)=minf(x(k):
:
;;,kdk),let
k
x(k1)=x(k)」dk.
d.Calculateg(k4F)=Vf(x(k*)),andtestitifmeet^g(k+l^^,ifitmeetsthenletx(k+)=x*,orturntoe.
e.
Judgeit,ifk=n,itshowsthetimesofiterationisn,whichhasrunnedoutofallconjugatedirections,letx(0)=x(k卩,k=0,turntob;orturntof.
g.Letk=k1,turntoc.
Thesearetheunconstrainedoptimalsolutionalalgorithmwhichagainstthetarget
1tt
functionf(x)xAxbxc,wecangeneralizethemtothegeneraldifferentiablefunction,beginfromanypointx(0),respectively,conductone-dimensionalsearchalongeachaxisdirection,conductover(totalofntimeslinearsearch)later,theoptimalsolutionoff(x)willbeabletoget.Forthequadraicdifferentiablefunctionswhosetagetfunctionsaren-dimension,usingconjugategradientmethodcantheoreticallyreachuptotheminimumpointaslongasneedingntimesiterations,butintheactualcalculation,duetoroundingerrors,alwayscarrymoretimescanachievesatisfactoryresults,whilefornon-quadraticfunctionsthenumberofiteratio
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