A Hybrid Three-Term Conjugate Gradient Method For Symmetric Nonlinear Equations

Wit h the deve lopment of sc ient ific techno logy and the widely applicat ion of the electronic computers,more and more proble ms about solving nonlinear equatio ns have been raised,and draw widespread attent ion.Since the ear ly 1970 s,a lot o f researches have been conducted on both theoret ical and numer ical solut ions.However,the prob lem of solving nonlinear equations is not as effect ive as linear equat ions in theory or solut ion.Therefore,there are many proble ms in solving t he existence o f solut io ns and find ing e ffective numer ical met hods to nonlinear equations,which need further research and explorat ion.Numer ical techniques for large-scale no nlinear constra ined optimizat ion proble ms are another important direct ion for fut ure constrained optimizat ion research,for examp le,constrained optimizat ion a lgor it hm for finite storage,or for sub-proble m inexact solut io n,etc.Conjugate grad ient met hod is one of the most commo nly used met hods in optimizat ion.It has the advanta ges of s imple algorithm and sma ll storage require ments,whic h is very suitable for large-scale optimizat ion proble ms.Conjugate gradient method is often used to so lve lar ge-scale optimizat ion proble ms in o il explorat ion,atmospher ic simulat ion,aerospace and o ther fie lds.The conjugate gradient met hod was first proposed by Hestenes and Stie fle.Then,in 1964,Fletcher and Reeves first proposed a conj ugate grad ient met hod for solving nonlinear optimizat ion proble ms.In the past decades,the conj ugate grad ient me thod has been greatly improved,and the study in the global converge nce has made great progress.The well-know conj ugate grad ient methods inc lude the Fletcher-Reeves(FR)met hod,Polak-Rib iere-Polyak(PRP)met hod,Hestenes-Stie fel(HS)met hod,Dai-Yua n(DY)method,Conjugate Descent(CD)met hod,etc.This paper focuses on the proble m of symmetr ic nonlinear equat ions by us ing the hybr id HS conjugate grad ient correction method and the appropriate line search met hod,to reach a hybr id three-term conjugate gradient met hod for solving symmetr ic nonlinear equat ions.And the convergence of t he hybr id three-ter m conjugate gradie nt met hod has been proved.Under certa in condit io ns,we can prove that the met hod has global converge nce and local linear convergence properties.The effect ive ness of the algor it hm has been illustrated by numer ical exper iments.And the numer ical exa mp les in the paper can also illustrate the effect iveness of t he a lgor it hm,for high-dimens iona l symmetr ic nonlinear equat ions,this method is a lso ap plicable.Fina lly,we hope to extend the non-derivat ive algorit hm to non-smooth problems.