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无约束优化问题的修正PRP共轭梯度法(英文).docx

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无约束优化问题的修正PRP共轭梯度法(英文) Unconstrainedoptimizationproblemsarecommonlyencounteredinvariousfieldssuchasmachinelearning,engineeringandfinance.Thegoaloftheseproblemsistofindtheminimumofagivenobjectivefunctionintheabsenceofanyconstraintsonthedecisionvariables.Theproblemcanbeapproachedusingvariousmethodssuchasgradientdescent,Newton'smethod,andconjugategradientmethods.Inthispaper,wefocusontheconjugategradientmethod,specificallythecorrectedPRP(Polak-Ribiére-Polyak)conjugategradientmethod,whichisaniterativealgorithmforsolvingunconstrainedoptimizationproblems. Theconjugategradientmethodisapopularoptimizationtechniqueknownforitsefficiencyinsolvinglinearsystemsandquadraticoptimizationproblems.Thismethodhasbeenextendedtononlinearoptimizationproblems,wherethedirectionofdescentisdeterminedbyaconjugategradientsearchdirection.ThecorrectedPRPconjugategradientmethodisanextensionofthePRPconjugategradientmethod,whichimprovesitsconvergenceperformance.ThePRPmethod(Polak-Ribiére)isawidelyusedconjugategradientmethod,whichusesthepreviousgradientinformationtocomputetheconjugategradientsearchdirection.However,thePRPmethodmayconvergeslowlyorevendivergeforcertaintypesofobjectivefunctions.ThecorrectedPRPmethodaddressesthisissuebyaddingascalingfactortothesearchdirection,leadingtofasterconvergencerates. ThecorrectedPRPconjugategradientmethodcanbedescribedasfollows:givenaninitialpointx₀,computethegradientg₀oftheobjectivefunctionf(x)atx₀.Thensetd₀=-g₀andα=1.Ateachiterationk,computethestepsizeαkusingalinesearchalgorithm,whichminimizesf(x)alongthesearchdirectiondk.Thenupdatexk,gk,anddkasfollows: αk=argminα≥0f(xk+αdk) xk+1=xk+αkdk gk+1=∇f(xk+1) βk+1=(gk+1-gk)·gk+1/||gk||² dk+1=-gk+1+βk+1(dk-||gk+1||²/||gk||²dk) where·denotesthedotproductand||.||denotestheEuclideannorm. ThecorrectedPRPmethodusestheparameterβk+1toscalethepreviousconjugatesearchdirection(dk)toimprovetheconvergencerate.Whenβk+1isnegative,themethodfallsbacktotheclassicalPRPmethodforwhichβk+1issettozero.ItcanbeshownthatthecorrectedPRPmethodhasa