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一个修正的Hooke-Jeeves方法(英文)
AModifiedHooke-JeevesMethodforNonlinearOptimization
TheHooke-Jeevesmethodisapopularoptimizationalgorithmthatbelongstotheclassofderivative-freeoptimizationmethods.Themethodisoftenusedtosolvenonlinearoptimizationproblems,wheretheobjectivefunctionisblack-boxorexpensivetoevaluate.Inthispaper,wepresentamodifiedHooke-Jeevesmethodthatincorporatessomeimportantmodificationstoimproveitsconvergenceproperties.
TheHooke-Jeevesmethodstartswithaninitialguessoftheoptimumsolutionandgeneratesasequenceoftrialpointsbyexploringthesearchspaceinasystematicway.Themethodusesthedirectionofthelastmoveandthestepsizetogeneratethenexttrialpoint.Iftheobjectivefunctionvalueatthetrialpointissmallerthanthecurrentbestvalue,thecurrentpointisupdated.Thisprocesscontinuesuntilastoppingcriterionismet.
However,theHooke-Jeevesmethodsuffersfromslowconvergencewhentheobjectivefunctionhasasmallcurvatureorthesearchspaceisverylarge.Toaddresstheseissues,weproposesomemodificationstotheoriginalHooke-Jeevesmethod.
Thefirstmodificationistheuseofavariablestepsize.Insteadofusingafixedstepsize,weproposetouseastepsizeschedulethatadaptstothecurvatureoftheobjectivefunction.Thisensuresthatthemethodtakeslargerstepsinregionswithalargecurvatureandsmallerstepsinregionswithasmallcurvature.Thismodificationhelpstoimprovetheconvergencerateofthealgorithm.
Thesecondmodificationistheuseofatrustregionapproach.Insteadofexploringtheentiresearchspace,weproposetolimitthesearchwithinaboundedregion.Thisboundedregionisdefinedbyaradiusaroundthecurrentpointandisupdatedateachiterationbasedontheprogressmadebythealgorithm.Thismodificationhelpstospeeduptheconvergencebyreducingthenumberoffunctionevaluationsandimprovingthequalityofthetrialpoints.
Thethirdmodificationistheuseofarestartstrategy.Whenthealgorithmgetsstuckinalocalminimum,weproposetorestartthealgorithmfromadifferentinitialpoint.Thishelpstoescapefromthelocalminimumandsearchforabetteroptimum.Therestartstrategycanbecombinedwithothermodificationstofurtherimprovetheperfor