In　this　paper,　we　investigate　the　$l-{0}$　quasi-norm　constrained　optimization　problem　in　the　Lagrange　dual　framework　and　show　that　the　strong　duality　property　holds.　Motivated　by　the　property,　we　propose　a　Lagrange　dual　method　for　the　sparsity　constrained　optimization　problem.　The　method　adopts　the　bisection　search　technique　to　maximize　the　Lagrange　dual　function.　For　each　Lagrange　multiplier,　we　adopt　the　iterative　hard　thresholding　method　to　minimize　the　Lagrange　function.　We　show　that　the　proposed　method　converges　to　an　$L$　-stationary　point　of　the　primal　problem.　Computational　experiments　and　comparisons　on　a　number　of　test　instances　(including　random　compressed　sensing　instances　and　random　and　real　sparse　logistic　regression　instances)　demonstrate　the　effectiveness　of　the　proposed　method　in　generating　sparse　solution　accurately.　©　2013　IEEE.