.　Non-convex　optimization　regularized　with　a　sparsity　function　has　many　applications　in　machine　learning　and　other　fields.　Non-convex　relaxation　of　the　sparsity　function　often　leads　to　more　effective　sparse　optimization　algorithms.　In　this　paper,　we　propose　a　new　weighted　regularizer　to　approximate　the　sparsity　function,　and　derive　a　weighted　thresholding　operator　for　the　sparse　optimization　problem　with　the　regularizer.　Then,　iterative　weighted　thresholding　algorithms　are　designed,　followed　with　an　acceleration　by　using　Nesterov＇s　acceleration　method　and　non-monotone　line　search.　Under　the　Kurdyka-　Lojasiewicz　(KL)　property,　the　smoothness　and　the　appropriate　convexity　assumptions,　we　prove　that　the　two　algorithms　are　convergent　and　the　convergence　rates　are　O(1/k)　and　O(1/k2)　respectively,　where　k　is　the　iteration　counter.　Moreover,　we　develop　convergent　practical　homotopy　algorithms　by　invoking　the　two　iterative　weighted　thresholding　algorithms　as　subroutines　respectively.　A　series　of　numerical　experiments　demonstrate　that　our　algorithms　are　superior　in　both　average　recovery　rate　and　average　running　time　for　the　sparse　recovery　problem,　and　are　competitive　in　solution　quality　and　average　running　time　for　the　logistic　regression　problem,　compared　to　state-of-the-art　algorithms.