MATLAB优化工具.docx

MATLAB优化工具.docx

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MATLAB优化工具

Findminimumofconstrainednonlinearmultivariablefunction

EquationFindstheminimumofaproblemspecifiedby

bandbeqarevectors,AandAeqarematrices,c（x）andceq（x）arefunctionsthatreturnvectors,andf（x）isafunctionthatreturnsascalar.f（x）,c（x）,andceq（x）canbenonlinearfunctions

.x,lb,andubcanbepassedasvectorsormatrices;seeMatrixArguments.

Syntax

x=fmincon（fun,x0,A,b）

x=fmincon（fun,x0,A,b,Aeq,beq）

x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub）

x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub,nonlcon）

x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options）

x=fmincon（problem）

[x,fval]=fmincon（...）

[x,fval,exitflag]=fmincon（...）

[x,fval,exitflag,output]=fmincon（...）

[x,fval,exitflag,output,lambda]=fmincon（...）

[x,fval,exitflag,output,lambda,grad]=fmincon（...）

[x,fval,exitflag,output,lambda,grad,hessian]=fmincon（...）

Descriptionfminconattemptstofindaconstrainedminimumofascalarfunctionofseveralvariablesstartingataninitialestimate.Thisisgenerallyreferredtoasconstrainednonlinearoptimizationornonlinearprogramming.

Note:

PassingExtraParametersexplainshowtopassextraparameterstotheobjectivefunctionandnonlinearconstraintfunctions,ifnecessary.

x=fmincon（fun,x0,A,b）startsatx0andattemptstofindaminimizerxofthefunctiondescribedinfunsubjecttothelinear

inequalitiesA*x ≤ b.x0can

beascalar,vector,ormatrix.x=fmincon（fun,x0,A,b,Aeq,beq）minimizesfunsubject

tothelinearequalitiesAeq*x = beqandA*x ≤ b.Ifnoinequalities

exist,setA=[]andb=[].x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub）defines

asetoflowerandupperboundsonthedesignvariablesinx,

sothatthesolutionisalwaysintherangelb ≤ x ≤ ub.

Ifnoequalitiesexist,setAeq=[]andbeq

=[].Ifx（i）isunboundedbelow,setlb（i）

=-Inf,andifx（i）isunboundedabove,

setub（i）=Inf.Note:

Ifthespecifiedinputboundsforaproblemareinconsistent,

theoutputxisx0andtheoutputfvalis[].Componentsofx0thatviolatetheboundslb ≤ x ≤ ubareresettotheinterioroftheboxdefined

bythebounds.Componentsthatrespecttheboundsarenotchanged.SeeIterationsCanViolateConstraints.x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub,nonlcon）subjects

theminimizationtothenonlinearinequalitiesc（x）or

equalitiesceq（x）definedinnonlcon.fminconoptimizes

suchthatc（x） ≤ 0andceq（x） = 0.Ifnoboundsexist,setlb

=[]and/orub=[].x=fmincon（fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options）minimizes

withtheoptimizationoptionsspecifiedinoptions.

Useoptimoptionstosetthese

options.Iftherearenononlinearinequalityorequalityconstraints,

setnonlcon=[].x=fmincon（problem）findstheminimum

forproblem,whereproblemis

astructuredescribedinInputArguments.CreatetheproblemstructurebyexportingaproblemfromOptimizationapp,asdescribedinExportingYourWork.

[x,fval]=fmincon（...）returnsthevalueoftheobjectivefunctionfunatthesolutionx.

[x,fval,exitflag]=fmincon（...）returnsavalueexitflagthatdescribestheexitconditionoffmincon.

[x,fval,exitflag,output]=fmincon（...）returnsastructureoutputwithinformationabouttheoptimization.

[x,fval,exitflag,output,lambda]=fmincon（...）returnsastructurelambdawhosefieldscontaintheLagrangemultipliersatthesolutionx.

[x,fval,exitflag,output,lambda,grad]=fmincon（...）returnsthevalueofthegradientoffunatthesolutionx.

[x,fval,exitflag,output,lambda,grad,hessian]=fmincon（...）returnsthevalueoftheHessianatthesolutionx.SeefminconHessian.

InputArguments

FunctionArgumentsdescribestheargumentspassedtofmincon.Optionsprovidesthefunction-specificdetailsfortheoptionsvalues.Thissectionprovidesfunction-specificdetailsforfun,nonlcon,andproblem.

Fun

Thefunctiontobeminimized.funisafunctionthatacceptsavectorxandreturnsascalarf,theobjectivefunctionevaluatedatx.funcanbespecifiedasafunctionhandleforafile:

x=fmincon（@myfun,x0,A,b）wheremyfunisaMATLAB®functionsuchas

functionf=myfun（x）

f=...%Computefunctionvalueatx

funcanalsobeafunctionhandleforananonymous

function:

x=fmincon（@（x）norm（x）^2,x0,A,b）;

IfthegradientoffuncanalsobecomputedandtheGradObjoptionis'on',assetby

options=optimoptions（'fmincon','GradObj','on'）

thenfunmustreturnthegradientvectorg（x）inthesecondoutputargument.IftheHessianmatrixcanalsobecomputedandtheHessianoptionis'on'viaoptions = optimoptions（'fmincon','Hessian','user-supplied'）andtheAlgorithmoptionistrust-region-reflective,funmustreturntheHessianvalueH（x）,asymmetricmatrix,inathirdoutputargument.funcangiveasparseHessian.SeeWritingObjectiveFunctionsfordetails.

IftheHessianmatrixcanbecomputedandtheAlgorithmoptionisinterior-point,thereareseveralwaystopasstheHessiantofmincon.Formoreinformation,seeHessian.

A,b,Aeq,beq

LinearconstraintmatricesAandAeq,andtheircorrespondingvectorsbandbeq,canbesparseordense.Thetrust-region-reflectiveandinterior-pointalgorithmsusesparselinearalgebra.IfAorAeqislarge,withrelativelyfewnonzeroentries,saverunningtimeandmemoryinthetrust-region-reflectiveorinterior-pointalgorithmsbyusingsparsematrices.

Nonlcon

Thefunctionthatcomputesthenonlinearinequalityconstraintsc（x）≤0andthenonlinearequalityconstraintsceq（x）=0.nonlconacceptsavectorxandreturnsthetwovectorscandceq.cisavectorthatcontainsthenonlinearinequalitiesevaluatedatx,andceqisavectorthatcontainsthenonlinearequalitiesevaluatedatx.nonlconshouldbespecifiedasafunctionhandletoafileortoananonymousfunction,suchasmycon:

x=fmincon（@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon）

wheremyconisaMATLABfunctionsuchas

function[c,ceq]=mycon（x）

c=...%Computenonlinearinequalitiesatx.

ceq=...%Computenonlinearequalitiesatx.

IfthegradientsoftheconstraintscanalsobecomputedandtheGradConstroptionis'on',assetbyoptions=optimoptions（'fmincon','GradConstr','on'）thennonlconmustalsoreturn,inthethirdandfourthoutputarguments,GC,thegradientofc（x）,andGCeq,thegradientofceq（x）.GCandGCeqcanbesparseordense.IfGCorGCeqislarge,withrelativelyfewnonzeroentries,saverunningtimeandmemoryintheinterior-pointalgorithmbyrepresentingthemassparsematrices.Formoreinformation,seeNonlinearConstraints.

Note  BecauseOptimizationToolbox™functionsonlyacceptinputsoftypedouble,user-suppliedobjectiveandnonlinearconstraintfunctionsmustreturnoutputsoftypedouble. 

Problem

objectiveObjectivefunction 

x0Initialpointforx 

AineqMatrixforlinearinequalityconstraints 

bineqVectorforlinearinequalityconstraints 

AeqMatrixforlinearequalityconstraints 

beqVectorforlinearequalityconstraints 

lbVectoroflowerbounds 

ubVectorofupperbounds 

nonlconNonlinearconstraintfunction 

solver'fmincon' 

optionsOptionscreatedwithoptimoptions 

OutputArgumentsFunctionArgumentsdescribesargumentsreturnedbyfmincon.Thissectionprovidesfunction-specificdetailsforexitflag,lambda,andoutput:

Exitflag

Integeridentifyingthereasonthealgorithmterminated.Thefollowingliststhevaluesofexitflagandthecorrespondingreasonsthealgorithmterminated.

AllAlgorithms:

1First-orderoptimalitymeasurewaslessthanoptions.TolFun,andmaximumconstraintviolationwaslessthanoptions.TolCon.

0Numberofiterationsexceededoptions.MaxIterornumberoffunctionevaluationsexceededoptions.MaxFunEvals.

-1Stoppedbyanoutputfunctionorplotfunction.

-2Nofeasiblepointwasfound.

trust-region-reflective,interior-point,andsqpalgorithms:

2Changeinxwaslessthanoptions.TolXandmaximumconstraintviolationwaslessthanoptions.TolCon.

trust-region-reflectivealgorithmonly:

3Changeintheobjectivefunctionvaluewaslessthanoptions.TolFunandmaximumconstraintviolationwaslessthanoptions.TolCon.

active-setalgorithmonly:

4Magnitudeofthesearchdirectionwaslessthan2*options.TolXandmaximumconstraintviolationwaslessthanoptions.TolCon.

5Magnitudeofdirectionalderivativeinsearchdirectionwaslessthan2*options.TolFunandmaximumconstraintviolationwaslessthanoptions.TolCon.

interior-pointandsqpalgorithms:

-3Objectivefunctionatcurrentiterationwentbelowoptions.ObjectiveLimitandmaximumconstraintviolationwaslessthanoptions.TolCon.

GradGradientatx 

HessianHessianatx 

LambdaStructurecontainingtheLagrangemultipliersatthesolutionx（separatedbyconstrainttype）.Thefieldsofthestructureare:

LowerLowerboundslb

UpperUpperboundsub

IneqlinLinearinequalities

eqlinLinearequalities

ineqnonlinNonlinearinequalities

eqnonlinNonlinearequalities

outputStructurecontaininginformationabouttheoptimization.Thefieldsofthestructureare:

iterationsNumberofiterationstaken

funcCountNumberoffunctionevaluations

lssteplengthSizeoflinesearchsteprelativetosearchdirection（active-setalgorithmonly）

constrviolationMaximumofconstraintfunctions

stepsizeLengthoflastdisplacementinx（active-setandinterior-pointalgorithms）

algorithmOptimizationalgorithmused

cgiterationsTotalnumberofPCGiterations（trust-region-reflectiveandinterior-pointalgorithms）

firstorderoptMeasureoffirst-orderoptimality

messageExitmessage

Hessian

fminconusesaHessianasanoptionalinput.ThisHessianisthesecondderivativesoftheLagrangian（seeEquation 3-1）,namely,

ThevariousfminconalgorithmshandleinputHessiansdifferently:

Theactive-setandsqpalgorithmsdonotacceptauser-suppliedHessian.Theycomputeaquasi-Newtonap