Many　inverse　problems　in　machine　learning,　system　identification,　and　image　processing　include　nuisance　parameters,　which　are　important　for　the　recovering　of　other　parameters.　Separable　nonlinear　optimization　problems　fall　into　this　category.　The　special　separable　structure　in　these　problems　has　inspired　several　efficient　optimization　strategies.　A　well-known　method　is　the　variable　projection　(VP)　that　projects　out　a　subset　of　the　estimated　parameters,　resulting　in　a　reduced　problem　that　includes　fewer　parameters.　The　expectation　maximization　(EM)　is　another　separated　method　that　provides　a　powerful　framework　for　the　estimation　of　nuisance　parameters.　The　relationships　between　EM　and　VP　were　ignored　in　previous　studies,　though　they　deal　with　a　part　of　parameters　in　a　similar　way.　In　this　article,　we　explore　the　internal　relationships　and　differences　between　VP　and　EM.　Unlike　the　algorithms　that　separate　the　parameters　directly,　the　hierarchical　identification　algorithm　decomposes　a　complex　model　into　several　linked　submodels　and　identifies　the　corresponding　parameters.　Therefore,　this　article　also　studies　the　difference　and　connection　between　the　hierarchical　algorithm　and　the　parameter-separated　algorithms　like　VP　and　EM.　In　the　numerical　simulation　part,　Monte　Carlo　experiments　are　performed　to　further　compare　the　performance　of　different　algorithms.　The　results　show　that　the　VP　algorithm　usually　converges　faster　than　the　other　two　algorithms　and　is　more　robust　to　the　initial　point　of　the　parameters.