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非线性比式和问题的全局优化算法(英文).docx

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非线性比式和问题的全局优化算法(英文) Introduction Nonlinearinequalitiesandequationsfrequentlyoccurinmathematicalmodelsfromseveralfieldssuchasengineering,physics,economics,andfinance.Additionally,nonlinearoptimizationproblemsareimportantforfindingglobaloptimainvariousapplications.Moreover,nonlinearinequalitiesandequationsarechallengingtosolveandrequireadvancedtechniquesandalgorithms.Inthispaper,wepresentabriefoverviewofglobaloptimizationalgorithmsfornonlinearinequalitiesandequations. NonlinearInequalities Anonlinearinequalityisexpressedasafunctionthatshouldsatisfyaspecificcondition,suchasless-thanorgreater-than.Oneofthemostcommonlyknownalgorithmsforsolvingnonlinearinequalitiesisthebranchandboundalgorithm.Thebranchandboundalgorithmworksbypartitioningthefeasibleregionintosmallerregionsandsolvingeachsub-regioninarecursivemanner.Itisagloballyconvergentalgorithmbutcanbecomputationallyexpensiveifthenumberofvariablesishigh. Anotherglobaloptimizationalgorithmfornonlinearinequalitiesisthesimulatedannealingalgorithm.Simulatedannealingisaprobabilisticalgorithmthatgraduallyreducesthetemperatureofthesystem,allowingittoannealintotheglobalminimum.Itisaheuristicalgorithmthatcanworkwellforproblemswithalargenumberofvariables. NonlinearEquations Anonlinearequationisamathematicalexpressionwhereoneormorevariablesappearinanonlinearform.Manydifferentalgorithmsexistforsolvingnonlinearequations,includingtheNewton-Raphsonmethod,theSecantmethod,andtheBroydenmethod.TheNewton-Raphsonmethodisawidelyusedalgorithmforsolvingnonlinearequations.Itworksbystartingwithaninitialguessandtheniterativelyupdatingtheguessuntilitconvergestothedesiredsolution. However,theNewton-Raphsonmethoddoesnotalwaysconvergeandmayfailiftheinitialguessisfarfromthesolution.TheSecantmethodisamodifiedversionoftheNewton-Raphsonmethodandrequirestwoinitialguesses.TheBroydenmethod,alsoknownasthequasi-Newtonmethod,isanotheralternativethatsometimesworkswhentheNewton-Raphsonmethodfails. GlobalOptimization Aglobaloptimizationalgorithmfindstheglobalminim