This　paper　delves　into　an　in-depth　exploration　of　the　Variable　Projection　(VP)　algorithm,　a　powerful　tool　for　solving　separable　nonlinear　optimization　problems　across　multiple　domains,　including　system　identification,　image　processing,　and　machine　learning.　We　first　establish　a　theoretical　framework　to　examine　the　effect　of　the　approximate　treatment　of　the　coupling　relationship　among　parameters　on　the　local　convergence　of　the　VP　algorithm　and　theoretically　prove　that　the　Kaufman＇s　VP　algorithm　can　achieve　a　similar　convergence　rate　as　the　Golub　&　Pereyra＇s　form.　These　studies　fill　the　gap　in　the　existing　convergence　theory　analysis,　and　provide　a　solid　foundation　for　understanding　the　mechanism　of　VP　algorithm　and　broadening　its　application　horizons.　Furthermore,　inspired　by　these　theoretical　insights,　we　design　a　refined　VP　algorithm,　termed　VPLR,　to　address　separable　nonlinear　optimization　problems　with　large　residual.　This　algorithm　enhances　convergence　performance　by　addressing　the　coupling　relationship　between　parameters　in　separable　models　and　continually　refining　the　approximated　Hessian　matrix　to　counteract　the　influence　of　large　residual.　The　effectiveness　of　this　refined　algorithm　is　corroborated　through　numerical　experiments.　©　2025　Elsevier　Ltd