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1、AnnOperRes(2009)166:7390DOI10.1007/s10479-008-0420-4AmodifiedPRPconjugategradientmethodGonglinYuan·XiwenLuPublishedonline:28August2008©SpringerScience+BusinessMedia,LLC2008AbstractThispapergivesamodifiedPRPmethodwhichpossessestheglobalconvergenceofnonconvexfunctionandtheR-linearconvergencerateofunifor
2、mlyconvexfunction.Fur-thermore,thepresentedmethodhassufficientlydescentpropertyandcharacteristicofauto-maticallybeinginatrustregionwithoutcarryingoutanylinesearchtechnique.Numericalresultsindicatethatthenewmethodisinterestingforthegiventestproblems.KeywordsLinesearch·Unconstrainedoptimization·Conjuga
3、tegradientmethod·Globalconvergence·R-linearconvergence1IntroductionConsidertheunconstrainedoptimizationproblemminf(x),(1.1)x∈nwheref:n→iscontinuouslydifferentiable.Thenonlinearconjugategradient(CG)methodusuallytakesthefollowingiterativeformulaxk+1=xk+αkdk,k=0,1,2,...,(1.2)ThisworkissupportedbyGua
4、ngxiUniversitySFgrandsX061041andChinaNSFgrands10761001.G.Yuan()·X.LuCollegeofMathematicsandInformationScience,GuangxiUniversity,Nanning,Guangxi,530004,PeoplesRepublicofChinae-mail:glyuan@tom.comG.YuanSchoolofScience,EastChinaUniversityofScienceandTechnology,Shanghai,200237,PeoplesRepublicofChina74A
5、nnOperRes(2009)166:7390forsolving(1.1),wherexkisthecurrentiteratepoint,αk>0isasteplength,anddkisasearchdirectiondefinedby−gk+βkdk−1,ifk≥1,dk=(1.3)−gk,ifk=0,wheregkisthegradient∇f(xk)off(x)atxk,βk∈isascalarwhichdeterminesthedifferentconjugategradientmethods.(1.2)and(1.3)arecalledtheconjugategra-dien
6、tmethods.Well-knownformulasforβkarecalledtheDaiYuan(DY)(DaiandYuan2000a),Conjugate-Descent(CD)(Fletcher1997),Fletcher-Reeves(FR)(FletcherandReeves1964),HestensesStiefel(HS)(HestenesandStiefel1952),LiuStorrey(LS)(LiuandStorey1992),andPolakRibièrePolyak(PRP)(PolakandRibière1969;Polyak1969),andsomemodi
7、fiedformulas(Dai2002;DaiandYuan2001,1995;Qietal.1996;Weietal.2006b).Theconvergencebehaviorofthedifferentconjugategradientmethodswithsomelinesearchconditions(Armijo1966;Al-baali1985;Dai2001;Daietal.1999
