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1、Chanceconstrainedoptimization•chanceconstraintsandpercentileoptimization•chanceconstraintsforlog-concavedistributions•convexapproximationofchanceconstraintssources:Rockafellar&Uryasev,Nemirovsky&ShapiroEE364A—ChanceConstrainedOptimization1Chanceconstraintsandpercentileoptimization•‘chan
2、ceconstraints’(ηis‘confidencelevel’):Prob(fi(x,ω)≤0)≥η–convexinsomecases(later)–generallyinterestedinη=0.9,0.95,0.99–η=0.999meaningless(unlessyou’resureaboutthedistributiontails)•percentileoptimization(γis‘η-percentile’):minimizeγsubjecttoProb(f0(x,ω)≤γ)≥η–convexorquasi-convexinsomecases
3、(later)EE364A—ChanceConstrainedOptimization2Value-at-riskandconditionalvalue-at-risk•value-at-riskofrandomvariablez,atlevelη:VaR(z;η)=inf{γ
4、Prob(z≤γ)≥η}–chanceconstraintProb(fi(x,ω)≤0)≥ηsameasVaR(fi(x,ω);η)≤0•conditionalvalue-at-risk:CVaR(z;η)=inf(β+1/(1−η)E(z−β)+)β–CVaR(z;η)≥VaR(z;η)(m
5、oreonthislater)EE364A—ChanceConstrainedOptimization3CVaRinterpretation(forcontinuousdistributions)•inCVaRdefinition,β⋆=VaR(z;η):d0=(β+1/(1−η)E(z−β)+)=1−1/(1−η)Prob(z≥β)dβsoProb(z≥β⋆)=1−η•conditionaltailexpectation(orexpectedshortfall)⋆⋆⋆⋆E(z
6、z≥β)=E(β+(z−β)
7、z≥β)⋆⋆⋆=β+E((z−β)+)/Prob(z≥β)=C
8、VaR(z;η)EE364A—ChanceConstrainedOptimization4Chanceconstraintsforlog-concavedistributions•suppose–ωhaslog-concavedensityp(ω)–C={(x,ω)
9、f(x,ω)≤0}isconvexin(x,ω)•thenZProb(f(x,ω)≤0)=1((x,ω)∈C)p(ω)dωislog-concave,sinceintegrandis•sochanceconstraintProb(f(x,ω)≤0)≥ηcanbeexpressedasconvexconst
10、raintlogProb(f(x,ω)≤0)≥logηEE364A—ChanceConstrainedOptimization5Linearinequalitywithnormallydistributedparameter•consideraTx≤b,witha∼N(¯a,Σ)•thenaTx−b∼N(¯aTx−b,xTΣx)•henceb−a¯TxT√Prob(ax≤b)=ΦxTΣx•andsoTT−11/2Prob(ax≤b)≥η⇐⇒b−a¯x≥Φ(η)kΣxk2asecond-orderconeconstraintforη≥0.5(i.e.,Φ−1(η)≥
11、0)EE364A—ChanceConstrainedOptimization6Portfoliooptimizationexamplen•x∈Rgivesportfolioallocation;xiis(fractional)positioninasseti•xmustsatisfy1Tx=1,x∈C(convexportfolioconstraintset)•portfolioreturn(say,inpercent)ispTx,wherep∼N(¯p,Σ)(amorerealisticmodelisplog-normal)•maximizeexp
