Separable　nonlinear　models　are　very　common　in　various　research　fields,　such　as　machine　learning　and　system　identification.　The　variable　projection　(VP)　approach　is　efficient　for　the　optimization　of　such　models.　In　this　paper,　we　study　various　VP　algorithms　based　on　different　matrix　decompositions.　Compared　with　the　previous　method,　we　use　the　analytical　expression　of　the　Jacobian　matrix　instead　of　finite　differences.　This　improves　the　efficiency　of　the　VP　algorithms.　In　particular,　based　on　the　modified　Gram-Schmidt　(MGS)　method,　a　more　robust　implementation　of　the　VP　algorithm　is　introduced　for　separable　nonlinear　least-squares　problems.　In　numerical　experiments,　we　compare　the　performance　of　five　different　implementations　of　the　VP　algorithm.　Numerical　results　show　the　efficiency　and　robustness　of　the　proposed　MGS　method-based　VP　algorithm.