Inthispaperweimprovethetwoversionsofthetwo-sidedprojectedquasi-Newtonmethod-onewasproposedbyNocedal&Overtonin[1]andtheotherwasdiscussedinourpreviouspaper,byintroducingthreedifferentmeritfunctionstomakeinexactone-dimensionalsearches.Itisshownthattheseimprovedquasi-Newtonalgorithmshavegainedglobalconvergencepropertywhichisnotpossessedbytheoriginaltwoalgorithms.