一种基于Wolfe搜索下的新共轭梯度法(英文)-豆柴文库

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一种基于Wolfe搜索下的新共轭梯度法(英文)
Title:ANovelConjugateGradientMethodbasedonWolfeLineSearch
Abstract:
Conjugategradientmethodsarepopularoptimizationalgorithmsusedtosolvesystemsoflinearequationsandtominimizeunconstrainedoptimizationproblems.Onecrucialcomponentofthesemethodsisthelinesearchprocess,whichdeterminesthestepsizealongthesearchdirection.Inthispaper,weproposeanewvariantoftheconjugategradientmethodbasedontheWolfelinesearch,knownastheWolfe-CGmethod,whichcombinestheefficientconvergencepropertiesoftheconjugategradientmethodwiththestrongglobalconvergenceguaranteesoftheWolfelinesearch.
1.Introduction
Conjugategradientmethodshavelongbeenknownfortheirfastconvergenceinsolvingoptimizationproblems.However,thechoiceofstepsizealongthesearchdirectioncansignificantlyaffecttheperformanceandconvergencerate.TheWolfelinesearchisareliableapproachinfindinganappropriatestepsizethatsatisfiesboththesufficientdecreaseandcurvatureconditions.Inthispaper,weintroducetheWolfe-CGmethod,whichutilizestheWolfelinesearchduringtheoptimizationprocesstoensureconvergencetoalocalminimum.
2.Background
2.1ConjugateGradientMethod
Theconjugategradientmethodisaniterativealgorithmusedtominimizeaquadraticfunction.Itiswell-knownforitsabilitytoavoidunnecessaryoscillationsandzig-zagsduringtheoptimizationprocess.Thekeyideaistochooseconjugatedirectionsthatensureadescentdirectionwhilepreventingcyclicbehavior.
2.2WolfeLineSearch
TheWolfeconditionsareasetoflinesearchconditionsthatguaranteeconvergencetoadesiredstepsize.Theyconsistoftwoconditions:sufficientdecreaseconditionandcurvaturecondition.Thesufficientdecreaseconditionensuresthattheselectedstepsizeprovidesasufficientdecreaseintheobjectivefunctionvalue,whilethecurvatureconditionpreventsexcessivestepsizesthatmightleadtodivergence.
3.ProposedWolfe-CGMethod
3.1Algorithm
TheWolfe-CGmethodcombinestheefficiencyoftheconjugategradientmethodwiththeguaranteesprovidedbytheWolfelinesearch.Thealgorithmcanbesummarizedasfollows:
1.Initializex0,gradientg0,searchdirectiond0.
2.Whilenotco