Optimization modeling is fundamental to decision-making in fields such as supply chain management, logistics, and financial engineering, but its complexity presents a major barrier to adoption. Automating model creation from natural language is key to improving efficiency and access. However, while Large Language Models (LLMs) are a promising tool for this, they often produce flawed or infeasible results due to errors and hallucinations. To address this issue, we propose Solver-Informed Reinforcement Learning (SIRL), a framework that uses Reinforcement Learning with Verifiable Reward to improve LLMs’ ability to generate accurate and executable optimization models. Specifically, SIRL automatically assesses the executable code and the instance-level mathematical model represented by the associated .lp files. This process yields precise feedback on syntactic validity, feasibility, and solution quality, which serves as a direct reward signal to guide the reinforcement learning process. Furthermore, this verification mechanism also supports our instance-enhanced self-consistency method for creating high-quality training data. Extensive experiments on diverse public benchmarks demonstrate that models trained with our SIRL framework achieve state-of-the-art performance, substantially outperforming existing methods in generating accurate and executable optimization models. Specifically, our SIRL-32B model surpasses DeepSeek-V3 and OpenAI-o3 on the majority of these benchmarks. Our code is publicly available at https://github.com/Cardinal-Operations/SIRL.

We consider decentralized gradient-free optimization of minimizing Lipschitz continuous functions that satisfy neither smoothness nor convexity assumption. We propose two novel gradient-free algorithms, the Decentralized Gradient-Free Method (DGFM) and its variant, the Decentralized Gradient-Free Method+ (DGFM+). Based on the techniques of randomized smoothing and gradient tracking, DGFM requires the computation of the zeroth-order oracle of a single sample in each iteration, making it less demanding in terms of computational resources for individual computing nodes. Theoretically, DGFM achieves a complexity of O(d^(3/2)δ^(-1)ε^(-4)) for obtaining a (δ,ε)-Goldstein stationary point. DGFM+, an advanced version of DGFM, incorporates variance reduction to further improve the convergence behavior. It samples a mini-batch at each iteration and periodically draws a larger batch of data, which improves the complexity to O(d^(3/2)δ^(-1)ε^(-3)). Moreover, experimental results underscore the empirical advantages of our proposed algorithms when applied to real-world datasets.