[分析] convex optimization — boyd & vandenberghe

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1、ConvexOptimization—Boyd&Vandenberghe1.Introduction•mathematicaloptimization•least-squaresandlinearprogramming•convexoptimization•example•coursegoalsandtopics•nonlinearoptimization•briefhistoryofconvexoptimization1–1Mathematicaloptimization(mathematical)optimiz

2、ationproblemminimizef0(x)subjecttofi(x)≤bi,i=1,...,m•x=(x1,...,xn):optimizationvariablesn•f0:R→R:objectivefunctionn•fi:R→R,i=1,...,m:constraintfunctionsoptimalsolutionx⋆hassmallestvalueoffamongallvectorsthat0satisfytheconstraintsIntroduction1–2Examplesportfoli

3、ooptimization•variables:amountsinvestedindifferentassets•constraints:budget,max./min.investmentperasset,minimumreturn•objective:overallriskorreturnvariancedevicesizinginelectroniccircuits•variables:devicewidthsandlengths•constraints:manufacturinglimits,timingre

4、quirements,maximumarea•objective:powerconsumptiondatafitting•variables:modelparameters•constraints:priorinformation,parameterlimits•objective:measureofmisfitorpredictionerrorIntroduction1–3Solvingoptimizationproblemsgeneraloptimizationproblem•verydifficulttosolve•

5、methodsinvolvesomecompromise,e.g.,verylongcomputationtime,ornotalwaysfindingthesolutionexceptions:certainproblemclassescanbesolvedefficientlyandreliably•least-squaresproblems•linearprogrammingproblems•convexoptimizationproblemsIntroduction1–4Least-squaresminimize

6、kAx−bk22solvingleast-squaresproblems•analyticalsolution:x⋆=(ATA)−1ATb•reliableandefficientalgorithmsandsoftware•computationtimeproportionalton2k(A∈Rk×n);lessifstructured•amaturetechnologyusingleast-squares•least-squaresproblemsareeasytorecognize•afewstandardtech

7、niquesincreaseflexibility(e.g.,includingweights,addingregularizationterms)Introduction1–5LinearprogrammingminimizecTxsubjecttoaTx≤b,i=1,...,miisolvinglinearprograms•noanalyticalformulaforsolution•reliableandefficientalgorithmsandsoftware•computationtimeproportion

8、alton2mifm≥n;lesswithstructure•amaturetechnologyusinglinearprogramming•notaseasytorecognizeasleast-squaresproblems•afewstandardtricksusedtoconvertproblemsintolinearprograms(e.g.,pr