Pruning is a critical strategy for compressing trained large language models (LLMs), aiming at substantial memory conservation and computational acceleration without compromising performance. However, existing pruning methods typically necessitate inefficient retraining for billion-scale LLMs or rely on heuristically designed metrics to determine pruning masks, leading to performance degradation. This paper presents, for the first time, a LASSO-like convex optimization model crafted to induce sparsity in LLMs. By leveraging the FISTA, we introduce FISTAPruner, a novel method that includes a cumulative error elimination mechanism within decoder layers and supports parallel pruning for unstructured pruning. Additionally, we extend this method to 2:4 semi-structured pruning. We comprehensively evaluate FISTAPruner on models such as OPT and LLaMA variants with 125M to 70B parameters under unstructured and 2:4 semi-structured sparsity, showcasing superior performance over existing methods across various language benchmarks. Notably, it can remove 50% of the model parameters for LLaMA-3-70B while retaining 98.6% and 95.6% of the zero-shot task performance under these two sparsity patterns, respectively.