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一种解无约束优化问题的新的非单调自适应信赖域方法(英文).docx

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一种解无约束优化问题的新的非单调自适应信赖域方法(英文) NewNonmonotonicAdaptiveTrustRegionMethodsforSolvingUnconstrainedOptimizationProblems Introduction Optimizationproblemsareubiquitousinengineering,economics,andsciences,amongotherfields.Manypracticalapplicationsrequirefindinganoptimalsolutionforanobjectivefunctionwhilesatisfyingcertainconstraints.Insomecases,however,theproblemisunconstrained,meaningthattheoptimizationprocesscanrangeovertheentirespace.Forsuchcases,severalmethodshavebeenproposed,includinggradientdescent,conjugategradient,quasi-Newton,andtrustregionmethods.Inthispaper,weproposeanewfamilyofnonmonotonicadaptivetrustregionmethodsforsolvingunconstrainedoptimizationproblems. Background Trustregionmethodsareapopularchoiceforsolvingunconstrainedoptimizationproblemsbecausetheycombinethebenefitsofbothlocalandglobaloptimization.Ateachiteration,trustregionmethodsbuildaquadraticmodeloftheobjectivefunctionandsolveasub-problemtodetermineacandidatesolution.Thecandidatesolutionisacceptedifitisdeemedgoodenough,andthetrustregionradiusisupdatedtocontrolthestepsizeinthenextiteration.Alternatively,ifthecandidatesolutionisnotsatisfactory,thetrustregionradiusisreduced,andtheprocessrepeats. Onelimitationoftraditionaltrustregionmethodsisthattheyassumeamonotonicdecreaseintheobjectivefunctionateachiteration,whichmaynotholdtruefornon-smoothornon-convexfunctions.Nonmonotonictrustregionmethodsattempttoovercomethislimitationbyrelaxingthemonotonicityassumptionandacceptingsolutionsthatincreasetheobjectivefunctionundercertainconditions. However,nonmonotonictrustregionmethodsareoftendesignedapriori,meaningthattheparametersaresetbeforetheoptimizationprocessstarts.Thiscanleadtopoorperformanceiftheproblempropertieschangeduringoptimization.Adaptivetrustregionmethodsattempttoaddressthisissuebytuningtheparametersdynamicallyduringtheoptimizationprocess. NewNonmonotonicAdaptiveTrustRegionMethods Weproposeanewfamilyofnonmonotonicadaptivetrustregionmethodsthatupdatetheirparametersbasedontheobservedproblemproperties.Them