We　propose　in　this　paper　a　novel　weighted　thresholding　method　for　the　sparsity-constrained　optimization　problem.　By　reformulating　the　problem　equivalently　as　a　mixed-integer　programming,　we　investigate　the　Lagrange　duality　with　respect　to　an　l1-norm　constraint　and　show　the　strong　duality　property.　Then　we　derive　a　weighted　thresholding　method　for　the　inner　Lagrangian　problem,　and　analyze　its　convergence.　In　addition,　we　give　an　error　bound　of　the　solution　under　some　assumptions.　Further,　based　on　the　proposed　method,　we　develop　a　homotopy　algorithm　with　varying　sparsity　level　and　Lagrange　multiplier,　and　prove　that　the　algorithm　converges　to　an　L-stationary　point　of　the　primal　problem　under　some　conditions.　Computational　experiments　show　that　the　proposed　algorithm　is　competitive　with　state-of-the-art　methods　for　the　sparsity-constrained　optimization　problem.　©　2020,　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.