newton′s method with a model trust region modification外语英文电子书

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1、SIAMJ.NUMER.ANAL.©1982SocietyforIndustrialandAppliedMathematicsVol.19,No.2,April19820036-1429/82/1902-0014$01.00/0NEWTON'SMETHODWITHAMODELTRUSTREGIONMODIFICATION*D.C.SORENSENtAbstract.AmodifiedNewtonmethodforunconstrainedminimizationispresentedandanalyzed.The

2、modificationisbaseduponthemodeltrustregionapproach.ThisreportcontainsathoroughanalysisofthelocallyconstrainedquadraticminimizationsthatariseassubproblemsinthemodifiedNewtoniteration.Severalpromisingalternativesarepresentedforsolvingthesesubproblemsinwaysthato

3、vercomecertaintheoreticaldifficultiesexposedbythisanalysis.Verystrongconvergenceresultsarepresentedconcerningtheminimizationalgorithm.Inparticular,theexplicituseofsecondorderinformationisjustifiedbydemonstratingthattheiteratesconvergetoapointwhichsatisfiesthe

4、secondordernecessaryconditionsforminimization.Withtheexceptionofverypathologicalcasesthisoccurswheneverthealgorithmisappliedtoproblemswithcontinuoussecondpartialderivatives.1.Introduction.Theproblemofminimizingareal-valuedfunctionfofseveralrealvariablesisgene

5、rallyattackedbysomevariantofNewton'smethodforfindingazeroofthegradientoff.ThetermvarianthereismeanttoincludeanymethodbaseduponmaintaininganapproximationtotheHessianmatrixofmixedsecondorderpartialderivativesoff.Whenthismatrixisactuallycomputable,thenNewton'sme

6、thodisprobablythemethodofchoicefortheminimizationproblem.Asweshallpointoutin§3,thereareseveralthingstoconsiderwhenattemptingtoprovideapracticalimplementationofNewton'smethodforgeneraluse.Nottheleastoftheseistheproblemofforcingconvergenceofthemethodwhenagoodin

7、itialguessatthesolutionisnotavailable.Themainpurposeofthisreportistodescribeandanalyzeatechniqueforthesolutionofthisproblem.Theapproachweshallpresentiswellknown.Itisappropriatelycalledamodeltrustregionapproachinthatthesteptoanewiterateisobtainedbyminimizingal

8、ocalquadraticmodeltotheobjectivefunctionoverarestrictedellipsoidalregioncenteredaboutthecurrentiterate.Thediameterofthisregionisexpandedandcontractedinacontrolledwaybaseduponhowwelltheloc