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1、NORTHWESTERNUNIVERSITYDepartmentofElectricalEngineeringandComputerScienceALIMITEDMEMORYALGORITHMFORBOUNDCONSTRAINEDOPTIMIZATIONby1222RichardH.Byrd,PeihuangLu,JorgeNocedalandCiyouZhuTechnicalReportNAM-08Revised:May19941ComputerScienceDepartment,UniversityofColoradoatBoulder,BoulderColorado80309.T
2、hisauthorwassupportedbyNSFgrantCCR-9101795,AROgrantDAAL03-91-G-0151,andAFOSRgrantAFOSR-90-0109.2DepartmentofElectricalEngineeringandComputerScience,NorthwesternUniversity,EvanstonIl60208
nocedal@eecs.nwu.edu.TheseauthorsweresupportedbyNationalScienceFoundationGrantsCCR-9101359andASC-9213149,andb
3、yDepartmentofEnergyGrantDE-FG02-87ER25047-A004.1ALIMITEDMEMORYALGORITHMFORBOUNDCONSTRAINEDOPTIMIZATIONbyRichardH.Byrd,PeihuangLu,JorgeNocedalandCiyouZhuABSTRACTAnalgorithmforsolvinglargenonlinearoptimizationproblemswithsimpleboundsisde-scribed.Itisbasedonthegradientprojectionmethodandusesalimite
4、dmemoryBFGSmatrixtoapproximatetheHessianoftheobjectivefunction.Itisshownhowtotakeadvan-tageoftheformofthelimitedmemoryapproximationtoimplementthealgorithmeciently.Theresultsofnumericaltestsonasetoflargeproblemsarereported.Keywords:boundconstrainedoptimization,limitedmemorymethod,nonlinearoptimi
5、zation,quasi-Newtonmethod,large-scaleoptimization.Abbreviatedtitle:ALimitedMemoryMethod1.Introduction.Inthispaperwedescribealimitedmemoryquasi-Newtonalgorithmforsolvinglargenonlinearoptimizationproblemswithsimpleboundsonthevariables.Wewritethisproblemasminf(x)(1:1)subjecttolxu(1:2)nwheref:
6、isanonlinearfunctionwhosegradientgisavailable,thevectorslandurepresentlowerandupperboundsonthevariables,andthenumberofvariablesnisassumedtobelarge.Thealgorithmdoesnotrequiresecondderivativesorknowledgeofthestructureoftheobjectivefunction,andcanthereforebeappliedwhentheHessianmatrixisnotpractical
7、tocompute.Alimitedmemoryquasi-NewtonupdateisusedtoapproximatetheHessianmatrixinsuchawaythatthestoragerequiredislinearinn.1ThealgorithmdescribedinthispaperissimilartothealgorithmsproposedbyConn,GouldandToint[9]andMoreandTora
