Separable　nonlinear　models　are　widely　used　in　various　fields　such　as　time　series　analysis,　system　modeling,　and　machine　learning,　due　to　their　flexible　structures　and　ability　to　capture　nonlinear　behavior　of　data.　However,　identifying　the　parameters　of　these　models　is　challenging,　especially　when　sparse　models　with　better　interpretability　are　desired　by　practitioners.　Previous　theoretical　and　practical　studies　have　shown　that　variable　projection　(VP)　is　an　efficient　method　for　identifying　separable　nonlinear　models,　but　these　are　based　on　L2　penalty　of　model　parameters,　which　cannot　be　directly　extended　to　deal　with　sparse　constraint.　Based　on　the　exploration　of　the　structural　characteristics　of　separable　models,　this　paper　proposes　gradient-based　and　trust-region-based　variable　projection　algorithms,　which　mainly　solve　two　key　problems:　how　to　eliminate　linear　parameters　under　sparse　constraint;　and　how　to　deal　with　the　coupling　relationship　between　linear　and　nonlinear　parameters　in　the　model.　Finally,　numerical　experiments　on　synthetic　data　and　real　time　series　data　are　conducted　to　verify　the　effectiveness　of　the　proposed　algorithms.　©　The　Author(s),　under　exclusive　licence　to　South　China　University　of　Technology　and　Academy　of　Mathematics　and　Systems　Science,　Chinese　Academy　of　Sciences　2024.