Automated Optimization Modeling via a Localizable Error-Driven Perspective

Automated optimization modeling via Large Language Models (LLMs) has emerged as a promising approach to assist complex human decision-making. While post-training has become a pivotal technique to enhance LLMs’ capabilities in this domain, its effectiveness is severely constrained by the scarcity and underutilization of high-quality training data. However, through a detailed profiling of error patterns across various problem-response pairs drawn from post-training, we identify two fundamental limitations of existing automated optimization modeling approaches: (L1) the sparsity of error-specific problems and (L2) the sparse rewards associated with difficult problems. We demonstrate that these limitations can result in suboptimal performance in domain-specific post-training for LLMs. To tackle the above two limitations, we propose a novel error-driven learning framework – namely, auto\textbf{m}ated opt\textbf{i}mization modeli\textbf{n}g via a localizable error-\textbf{d}riven perspective (MIND) – that customizes the whole model training framework from data synthesis to post-training. MIND is based on our key observation of the unique localizable patterns in error propagation of optimization modelings, that is, modeling errors may remain localized to specific semantic segments and do not propagate throughout the entire solution. Thus, in contrast to holistic reasoning tasks such as mathematical proofs, MIND leverages the construction of a focused, high-density training corpus and proposes \textbf{D}ynamic Supervised \textbf{F}ine-Tuning \textbf{P}olicy \textbf{O}ptimization (DFPO) to tackle difficult problems through localized refinement. Experiments on six benchmarks demonstrate that MIND consistently outperforms all the state-of-the-art automated optimization modeling approaches.

💡 Research Summary

The paper tackles the emerging task of automated optimization modeling, where large language models (LLMs) translate natural‑language problem statements into formal mathematical formulations and executable solver code. While recent post‑training techniques have improved LLM performance in this domain, two fundamental bottlenecks remain: (L1) the scarcity of error‑specific training instances and (L2) the sparsity of reward signals for difficult problems. By profiling a large set of question‑response pairs, the authors discover a “localizable error” phenomenon: errors tend to be confined to specific semantic components—variables, constraints, or objectives—rather than propagating throughout the entire solution. This observation motivates the design of MIND (Automated Optimization Modeling via a Localizable Error‑Driven Perspective), a two‑stage framework that customizes both data synthesis and model fine‑tuning.

In the first stage, an error‑driven reverse synthesis pipeline identifies frequent error patterns in existing data, then deliberately perturbs those components to generate a high‑density, error‑aware corpus called MIND‑Train. Because the average error ratio is only 0.33, the generated corpus is highly sample‑efficient, focusing training effort on the most informative failure modes. In the second stage, the authors introduce Dynamic Supervised Fine‑tuning Policy Optimization (DFPO). DFPO blends supervised fine‑tuning (SFT) with reinforcement learning (RL) by automatically correcting wrong model outputs using the reverse‑synthesized ground‑truth, while keeping the corrected responses close to the base model’s distribution through KL‑regularization and reward shaping. This dynamic correction supplies dense learning signals for hard problems, mitigating the sparse‑reward issue that plagues prior RL‑based approaches such as SIRL.

Extensive experiments on six public benchmarks and a newly released MIND‑Bench demonstrate consistent gains over state‑of‑the‑art methods (e.g., ORLM, Step‑Opt, Resocratic, OptMATH). MIND improves average accuracy by 4–7 percentage points and shows faster convergence, especially on the hardest problem subsets where DFPO’s enriched rewards are most beneficial. Ablation studies confirm that both the error‑focused data generation and the DFPO policy contribute substantially to the performance boost. The authors also release the MIND‑Train dataset and the benchmark suite to foster reproducibility and further research.

Key contributions are: (1) empirical evidence of error locality in automated optimization modeling, (2) a novel error‑driven data synthesis pipeline, (3) the DFPO algorithm that stabilizes and enriches RL‑based fine‑tuning, (4) state‑of‑the‑art results across multiple benchmarks, and (5) open‑source resources for the community. Limitations include the reliance on the locality assumption, which may not hold for all optimization domains, and the added complexity of tuning DFPO hyper‑parameters. Future work could explore broader error pattern catalogs, automated hyper‑parameter optimization, and application to other structured reasoning tasks.