e EE364A Chance Constrained Optimization brPage 7br Portfolio optimization example gives portfolio allocation is fractional position in asset must satisfy 1 8712 C convex portfolio constraint set portfolio return say in percent is where 8764 N p ID: 32129
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Chanceconstrainedoptimizationchanceconstraintsandpercentileoptimizationchanceconstraintsforlog-concavedistributionsconvexapproximationofchanceconstraintssources:Rockafellar&Uryasev,Nemirovsky&ShapiroEE364A|ChanceConstrainedOptimization1 Chanceconstraintsandpercentileoptimization`chanceconstraints'(is`con dencelevel'):Prob(fi(x;!)0){convexinsomecases(later){generallyinterestedin=0:9;0:95;0:99{=0:999meaningless(unlessyou'resureaboutthedistributiontails)percentileoptimization(\ris`-percentile'):minimize\rsubjecttoProb(f0(x;!)\r){convexorquasi-convexinsomecases(later)EE364A|ChanceConstrainedOptimization2 Value-at-riskandconditionalvalue-at-riskvalue-at-riskofrandomvariablez,atlevel:VaR(z;)=inff\rjProb(z\r)g{chanceconstraintProb(fi(x;!)0)sameasVaR(fi(x;!);)0conditionalvalue-at-risk:CVaR(z;)=inf ( +1=(1 )E(z )+){CVaR(z;)VaR(z;)(moreonthislater)EE364A|ChanceConstrainedOptimization3 CVaRinterpretation(forcontinuousdistributions)inCVaRde nition, =VaR(z;):0=d d ( +1=(1 )E(z )+)=1 1=(1 )Prob(z )soProb(z )=1 conditionaltailexpectation(orexpectedshortfall)E(zjz )=E( +(z )jz )= +E((z )+)=Prob(z )=CVaR(z;)EE364A|ChanceConstrainedOptimization4 Chanceconstraintsforlog-concavedistributionssuppose{!haslog-concavedensityp(!){C=f(x;!)jf(x;!)0gisconvexin(x;!)thenProb(f(x;!)0)=Z1((x;!)2C)p(!)d!islog-concave,sinceintegrandissochanceconstraintProb(f(x;!)0)canbeexpressedasconvexconstraintlogProb(f(x;!)0)logEE364A|ChanceConstrainedOptimization5 LinearinequalitywithnormallydistributedparameterconsideraTxb,withaN(a;)thenaTx bN(aTx b;xTx)henceProb(aTxb)=b aTx p xTxandsoProb(aTxb)()b aTx 1()k12xk2asecond-orderconeconstraintfor0:5(i.e., 1()0)EE364A|ChanceConstrainedOptimization6 Portfoliooptimizationexamplex2Rngivesportfolioallocation;xiis(fractional)positioninassetixmustsatisfy1Tx=1,x2C(convexportfolioconstraintset)portfolioreturn(say,inpercent)ispTx,wherepN(p;)(amorerealisticmodelisplog-normal)maximizeexpectedreturnsubjecttolimitonprobabilityoflossEE364A|ChanceConstrainedOptimization7 problemismaximizeEpTxsubjecttoProb(pTx0) 1Tx=1;x2Ccanbeexpressedasconvexproblem(provided 1=2)maximizepTxsubjecttopTx 1(1 )k12xk21Tx=1;x2C(anSOCPwhenCispolyhedron)EE364A|ChanceConstrainedOptimization8 Examplen=10assets, =0:05,C=fxjx 0:1gcompare{optimalportfolio{optimalportfoliow/olossriskconstraint{uniformportfolio(1=n)1portfolio EpTx Prob(pTx0) optimal 7.51 5.0%w/olossconstraint 10.66 20.3%uniform 3.41 18.9%EE364A|ChanceConstrainedOptimization9 returndistributions: -20 -15 -10 -5 0 5 10 15 20 25 30 -20 -15 -10 -5 0 5 10 15 20 25 30 -20 -15 -10 -5 0 5 10 15 20 25 30 optimalw/olossconstraintuniformEE364A|ChanceConstrainedOptimization10 Convexapproximationofchanceconstraintboundassumefi(x;!)isconvexinxsuppose :R!Risnonnegativeconvexnondecreasing,with (0)=1forany i�0, (z= i)1(z�0)forallz,soE (fi(x;!)= i)Prob(fi(x;!)�0)hence(convex)constraintE (fi(x;!)= i)1 ensureschanceconstraintProb(fi(x;!)0)holdsthisholdsforany i�0;wenowshowhowtooptimizeover iEE364A|ChanceConstrainedOptimization11 writeconstraintasE i (fi(x;!)= i) i(1 ){(perspectivefunction)v (u=v)isconvexin(u;v)forv�0,nondecreasinginu{socomposition i (fi(x;!)= i)isconvexin(x; i)for i�0{henceconstraintaboveisconvexinxand i{sowecanoptimizeoverxand i�0viaconvexoptimizationyieldsaconvexstochasticoptimizationproblemthatisaconservativeapproximationofthechance-constrainedproblemwe'lllookatsomespecialcasesEE364A|ChanceConstrainedOptimization12 Markovchanceconstraintboundtaking (u)=(u+1)+givesMarkovbound:forany i�0,Prob(fi(x;!)�0)E(fi(x;!)= i+1)+convexapproximationconstraintE i(fi(x;!)= i+1)+ i(1 )canbewrittenasE(fi(x;!)+ i)+ i(1 )wecanoptimizeoverxand i0EE364A|ChanceConstrainedOptimization13 Interpretationviaconditionalvalue-at-riskwriteconservativeapproximationasE(fi(x;!)+ i)+ 1 i0LHSisconvexin(x; i),sominimumover i,inf i�0E(fi(x;!)+ i)+ 1 iisconvexinxthisisCVaR(fi(x;!);)(canshow i�0canbedropped)soconvexapproximationreplacesVaR(fi(x;!);)0withCVaR(fi(x;!);)0whichisconvexinxEE364A|ChanceConstrainedOptimization14 Chebyshevchanceconstraintboundtaking (u)=(u+1)2+yieldsChebyshevbound:forany i�0,Prob(fi(x;!)�0)E(fi(x;!)= i+1)2+convexapproximationconstraintE i(fi(x;!)= i+1)2+ i(1 )canbewrittenasE(fi(x;!)+ i)2+= i i(1 )EE364A|ChanceConstrainedOptimization15 TraditionalChebyshevbounddroppingsubscript+wegetmoreconservativeconstraintE i(fi(x;!)= i+1)2 i(1 )whichwecanwriteas2Efi(x;!)+(1= i)Efi(x;!)2+ i0minimizingover igives i= Efi(x;!)2=12;yieldsconstraintEfi(x;!)+ Efi(x;!)2120whichdependsonlyon rstandsecondmomentsoffiEE364A|ChanceConstrainedOptimization16 Examplefi(x)=aTx b,whereaisrandomwithEa=a,EaaT=traditionalChebyshevapproximationofchanceconstraintisaTx b+12 xTx 2baTx+b2120canwriteassecond-orderconeconstraintaTx b+12k(z;y)k20withz=12x b 12a,y=b 1 aT 1a12caninterpretascertainty-equivalentconstraint,withnormtermas`extramargin'EE364A|ChanceConstrainedOptimization17 Cherno chanceconstraintboundtaking (u)=expuyieldsCherno bound:forany i�0,Prob(fi(x;!)�0)Eexp(fi(x;!)= i)convexapproximationconstraintE iexp(fi(x;!)= i) i(1 )canbewrittenaslogEexp(fi(x;!)= i)log(1 )(LHSiscumulantgeneratingfunctionoffi(x;!),evaluatedat1= i)EE364A|ChanceConstrainedOptimization18 Examplemaximizealinearrevenuefunction(say)subjecttorandomlinearconstraintsholdingwithprobability:maximizecTxsubjecttoProb(max(Ax b)0)withvariablex2Rn;A2Rmn,b2Rmrandom(Gaussian)Markov/CVaRapproximation:maximizecTxsubjecttoE(max(Ax b)+ )+ (1 )withvariablesx2Rn, 2REE364A|ChanceConstrainedOptimization19 Chebyshevapproximation:maximizecTxsubjecttoE(max(Ax b)+ )2+= (1 )withvariablesx2Rn, 2RoptimalvaluesoftheseapproximateproblemsarelowerboundsfororiginalproblemEE364A|ChanceConstrainedOptimization20 instancewithn=5,m=10,=0:9solveapproximationswithsamplingmethodwithN=1000trainingsamples,validatewith=10000samplescomparetosolutionofdeterministicproblemmaximizecTxsubjecttoEAxEbestimatesofProb(max(Ax b)0)ontraining/validationdata cTx train validate Markov 3:60 0:97 0:96Chebyshev 3:43 0:97 0:96deterministic 7:98 0:04 0:03EE364A|ChanceConstrainedOptimization21 PDFofmax(Ax b)forMarkovapproximationsolution -25 -20 -15 -10 -5 0 5 10 15 EE364A|ChanceConstrainedOptimization22