Nonlinear dynamics sparse identification methods based on system sparsity priors (hereafter referred to as sparse modeling) represent a quintessential technique in data-driven dynamic modeling. By reformulating dynamic modeling into sparse optimization problems, this method effectively achieves a balance between model interpretability and practical performance. However, traditional sparse optimization methods based on specific loss functions and regularizers often struggle to balance the accuracy, sparsity, robustness, and computational efficiency of the models. To address this, focusing on the core optimization problem of sparse modeling, this paper proposes a unified iterative solution framework based on the Alternating Direction Method of Multipliers (ADMM) to flexibly accommodate sparse optimization problems with diverse objectives. Within this framework, the loss functions encompass squared loss, absolute deviation loss, and Huber loss, while the regularization terms include the conventional one-norm and its variants, alongside the one-half quasi-norm non-convex regularizer. By comprehensively integrating non-convex regularizers, robust loss functions, and accelerated iterative techniques, the proposed method systematically enhances the overall performance of the model across the aforementioned multidimensional metrics. Finally, the effectiveness of the proposed method is validated using the six-dimensional Lorenz 96 chaotic system. The results indicate that: (1) the proposed framework can uniformly solve sparse optimization problems containing various combinations of loss functions and regularizers, encompassing non-smooth/non-convex optimization problems; (2) the introduction of non-convex regularizers significantly enhances the ability to characterize the sparse features of the system; (3) the adoption of an accelerated iterative strategy effectively improves the efficiency of solving for sparse solutions.