---
title: "Towards Hierarchical Multi-Step Reward Models for Enhanced"
description: "The authors propose the Hierarchical Reward Model (HRM). 30 references, cited by 10 papers here."
authors: ["Teng Wang", "Jiang Zhangyi", "Zhenqi He", "Hailei Gong", "Shenyang Tong", "Wenhan Yang", "Zeyu Li", "Yanan Zheng", "Zifan He", "Zewen Ye", "Shengjie Ma", "Jianping Zhang"]
year: 2026
venue: "Findings of the Association for Computational Linguistics: ACL 2026"
source: "https://aclanthology.org/2026.findings-acl.27/"
doi: "10.18653/v1/2026.findings-acl.27"
licence: "CC BY 4.0 (ACL Anthology)"
licence_url: "https://creativecommons.org/licenses/by/4.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/wang-2026-hierarchical-multi-step-reward-models/"
---
[Elamur Research](https://md.research.elamur.ai/free/en/) · [this page in HTML](https://research.elamur.ai/free/en/papers/wang-2026-hierarchical-multi-step-reward-models/)

[home](https://md.research.elamur.ai/free/en/) / [papers](https://md.research.elamur.ai/free/en/papers/) / [2026](https://md.research.elamur.ai/free/en/papers/2026/page/2/) / Towards Hierarchical Multi-Step Reward Models for Enhanced Reasoning in Large Language Models

# Towards Hierarchical Multi-Step Reward Models for Enhanced Reasoning in Large Language Models

**Authors:** [Teng Wang](https://md.research.elamur.ai/free/en/authors/teng-wang/), Jiang Zhangyi, Zhenqi He, Hailei Gong, Shenyang Tong, Wenhan Yang, [Zeyu Li](https://md.research.elamur.ai/free/en/authors/zeyu-li/), [Yanan Zheng](https://md.research.elamur.ai/free/en/authors/yanan-zheng/), Zifan He, Zewen Ye, [Shengjie Ma](https://md.research.elamur.ai/free/en/authors/shengjie-ma/) and Jianping Zhang · **Year:** 2026 · **Venue:** [Findings of the Association for Computational Linguistics: ACL 2026](https://md.research.elamur.ai/free/en/venues/findings/) ([Findings of the Association for Computational Linguistics 2026](https://md.research.elamur.ai/free/en/venues/findings/2026/)) · **Publisher:** [Association for Computational Linguistics](https://md.research.elamur.ai/free/en/publishers/association-for-computational-linguistics/) · **Licence:** [CC BY 4.0 (ACL Anthology)](https://creativecommons.org/licenses/by/4.0/)

**Links:** [Publisher's page](https://aclanthology.org/2026.findings-acl.27/) · [PDF](https://aclanthology.org/2026.findings-acl.27.pdf) · [DOI 10.18653/v1/2026.findings-acl.27](https://doi.org/10.18653/v1/2026.findings-acl.27) · [OpenAlex](https://openalex.org/W4414900408)

**Citations:** 0 citations (OpenAlex, 2026-10-14) · [10](https://md.research.elamur.ai/free/en/papers/wang-2026-hierarchical-multi-step-reward-models/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Process Reward Models (PRMs) score each step of an LLM's mathematical reasoning. They have two weaknesses: they can be gamed through reward hacking, and they need costly step-level annotation. They also penalize a wrong step even when a later step fixes it.

The authors propose the Hierarchical Reward Model (HRM). Its training set contains single steps plus pairs of adjacent steps merged together, so the model learns local correctness and how well consecutive steps fit together, including self-corrections. At inference it still scores one step at a time, like a PRM.

To make automatic labeling with MCTS cheaper, they add Hierarchical Node Compression (HNC). HNC merges consecutive tree nodes, which yields extra, slightly noisy training examples at almost no extra compute.

The models are trained on PRM800K and tested with Best-of-N selection. HRM gives steadier accuracy than PRM or ORM as N grows. It also transfers better to MATH500 and slightly better to GSM8K.

This matters because it offers a cheaper and more robust way to supervise multi-step reasoning.

## Key points
- **Manually annotated PRM800K data:** with Qwen2.5-72B-Math-Instruct as the policy, HRM accuracy levels off around 80% as N increases. PRM and ORM fluctuate and drop at larger N.
- **HRM training data:** a superset of PRM data. It adds merged pairs of adjacent steps (s_i + s_{i+1}). Only two steps are merged at a time, because larger merges make labeling rules ambiguous.
- **Out-of-domain results (Qwen2.5-7B-Math-Instruct, largest N tested):**
  - MATH500: HRM 0.736 vs PRM 0.688, with an HRM peak of 0.742.
  - GSM8K: HRM 0.926 vs PRM 0.918, a small gap since these problems need few steps.
- **Self-training:** Qwen2.5-7B-Math-Instruct was fine-tuned on MCTS trajectories with MC-score > 0.9, using a log-scaled KL term with λ = 0.5. The resulting Qwen-7B-HRM policy reaches about 0.789 under HRM scoring on PRM800K.
- **Setup and code:** MCTS answers were limited to at most 6 steps in a fixed "# Step X" template. This constraint partly explains the weaker 72B results in the MCTS-based setup. Code is available at github.com/tengwang0318/hierarchial_reward_model.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Tsinghua University](https://md.research.elamur.ai/free/en/universities/tsinghua-university/) — University, Beijing, China · 415 works · 94,028 citations
- [National University of Singapore](https://md.research.elamur.ai/free/en/universities/national-university-of-singapore/) — University, Singapore, Singapore · 207 works · 39,576 citations
- [Georgia Institute of Technology](https://md.research.elamur.ai/free/en/universities/georgia-institute-of-technology/) — University, Atlanta, United States · 152 works · 27,763 citations
- [Renmin University of China](https://md.research.elamur.ai/free/en/universities/renmin-university-of-china/) — University, Beijing, China · 139 works · 17,752 citations
- [Chinese University of Hong Kong](https://md.research.elamur.ai/free/en/universities/chinese-university-of-hong-kong/) — University, Hong Kong, China · 159 works · 16,154 citations
- [Zhejiang University](https://md.research.elamur.ai/free/en/universities/zhejiang-university/) — University, Hangzhou, China · 224 works · 13,587 citations
- [University of Hong Kong](https://md.research.elamur.ai/free/en/universities/university-of-hong-kong/) — University, Pok Fu Lam, China · 118 works · 13,496 citations

📚 Shelves (5): [🖥️ Computer Science](https://md.research.elamur.ai/free/en/shelves/computer-science/) · 18,614 works · [🤖 Artificial Intelligence](https://md.research.elamur.ai/free/en/shelves/computer-science/artificial-intelligence/) · 17,965 works · [🎭 Arts and Humanities](https://md.research.elamur.ai/free/en/shelves/arts-and-humanities/) · 17,450 works · [🗣️ Language and Linguistics](https://md.research.elamur.ai/free/en/shelves/arts-and-humanities/language-and-linguistics/) · 17,296 works · [👁️ Computer Vision and Pattern Recognition](https://md.research.elamur.ai/free/en/shelves/computer-science/computer-vision-and-pattern-recognition/) · 3,486 works

🧭 Topics (2): [🧠 Language models & pretraining](https://md.research.elamur.ai/free/en/topics/language-models/) · 3,894 works · [🧮 Reasoning](https://md.research.elamur.ai/free/en/topics/reasoning/) · 1,685 works

## Cite

Cite as: Teng Wang, Jiang Zhangyi, Zhenqi He, Hailei Gong, Shenyang Tong, Wenhan Yang, Zeyu Li, Yanan Zheng, Zifan He, Zewen Ye, Shengjie Ma and Jianping Zhang. 2026. Towards Hierarchical Multi-Step Reward Models for Enhanced Reasoning in Large Language Models. Findings of the Association for Computational Linguistics: ACL 2026.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{wang2026hierarchical,
  title = {{Towards Hierarchical Multi-Step Reward Models for Enhanced Reasoning in Large Language Models}},
  author = {Teng Wang, Jiang Zhangyi, Zhenqi He, Hailei Gong, Shenyang Tong, Wenhan Yang, Zeyu Li, Yanan Zheng, Zifan He, Zewen Ye, Shengjie Ma and Jianping Zhang},
  booktitle = {Findings of the Association for Computational Linguistics: ACL 2026},
  year = {2026},
  doi = {10.18653/v1/2026.findings-acl.27},
  url = {https://doi.org/10.18653/v1/2026.findings-acl.27}
}
```

- [← previous The Sparse Frontier: Sparse Attention Trade-offs in Transformer LLMs](https://md.research.elamur.ai/free/en/papers/nawrot-2026-sparse-frontier-sparse-attention-trade/)
- [next → AbjadMed: Arabic Medical Text Classification at AbjadNLP 2026](https://md.research.elamur.ai/free/en/papers/gupta-2026-abjadmed-arabic-medical-text-classification/)
- [↑ back to papers of 2026](https://md.research.elamur.ai/free/en/papers/2026/page/2/)

## References

30 references, 15 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

1. Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, and 1 others. 2023. [Gpt-4 technical report](https://md.research.elamur.ai/free/en/papers/openai-2023-gpt-4-technical-report/). In ArXiv e-prints.
2. Reza Yazdani Aminabadi, Samyam Rajbhandari, Ammar Ahmad Awan, Cheng Li, Du Li, Elton Zheng, Olatunji Ruwase, Shaden Smith, Minjia Zhang, Jeff Rasley, and 1 others. 2022. Deepspeed-inference: enabling efficient inference of transformer models at unprecedented scale. In SC22: International Conference… [find](https://scholar.google.com/scholar?q=Reza+Yazdani+Aminabadi%2C+Samyam+Rajbhandari%2C+Ammar+Ahmad+Awan%2C+Cheng+Li%2C+Du+Li%2C+Elton+Zheng%2C+Olatunji+Ruwase%2C+Shaden+Smith%2C+Minjia+Zhang%2C+Jeff+Rasley%2C+and+1+others%2E+2022%2E+Deepspeed%2Dinference%3A+enabling+)
3. Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané. 2016. [Concrete problems in ai safety](https://md.research.elamur.ai/free/en/papers/amodei-2016-concrete-problems-ai-safety/). In arXiv preprint arXiv:1606.06565.
4. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, and 1 others. 2021. [Training verifiers to solve math word problems](https://md.research.elamur.ai/free/en/papers/cobbe-2021-training-verifiers-solve-math-word/). In ArXiv e-prints.
5. Tri Dao. 2024. [FlashAttention-2: Faster attention with better parallelism and work partitioning](https://md.research.elamur.ai/free/en/papers/dao-2023-flashattention-2-faster-attention-better/). In International Conference on Learning Representations (ICLR).
6. Tri Dao, Daniel Y. Fu, Stefano Ermon, Atri Rudra, and Christopher Ré. 2022. FlashAttention: Fast and memory-efficient exact attention with IO-awareness. In Advances in Neural Information Processing Systems (NeurIPS). [find](https://scholar.google.com/scholar?q=Tri+Dao%2C+Daniel+Y%2E+Fu%2C+Stefano+Ermon%2C+Atri+Rudra%2C+and+Christopher+R%C3%A9%2E+2022%2E+FlashAttention%3A+Fast+and+memory%2Defficient+exact+attention+with+IO%2Dawareness%2E+In+Advances+in+Neural+Information+Processing+Sy)
7. Lauro Langosco Di Langosco, Jack Koch, Lee D Sharkey, Jacob Pfau, and David Krueger. 2022. Goal misgeneralization in deep reinforcement learning. In International Conference on Machine Learning, pages 12004–12019. ICML. [find](https://scholar.google.com/scholar?q=Lauro+Langosco+Di+Langosco%2C+Jack+Koch%2C+Lee+D+Sharkey%2C+Jacob+Pfau%2C+and+David+Krueger%2E+2022%2E+Goal+misgeneralization+in+deep+reinforcement+learning%2E+In+International+Conference+on+Machine+Learning%2C+pages)
8. Tom Everitt, Victoria Krakovna, Laurent Orseau, Marcus Hutter, and Shane Legg. 2017. Reinforcement learning with a corrupted reward channel. In IJCAI. [find](https://scholar.google.com/scholar?q=Tom+Everitt%2C+Victoria+Krakovna%2C+Laurent+Orseau%2C+Marcus+Hutter%2C+and+Shane+Legg%2E+2017%2E+Reinforcement+learning+with+a+corrupted+reward+channel%2E+In+IJCAI%2E)
9. Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, and 1 others. 2024. [The llama 3 herd of models](https://md.research.elamur.ai/free/en/papers/grattafiori-2024-llama-3-herd-models/). In ArXiv e-prints.
10. Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, and 1 others. 2025. Deepseekr1: Incentivizing reasoning capability in llms via reinforcement learning. In ArXiv e-prints. [find](https://scholar.google.com/scholar?q=Daya+Guo%2C+Dejian+Yang%2C+Haowei+Zhang%2C+Junxiao+Song%2C+Ruoyu+Zhang%2C+Runxin+Xu%2C+Qihao+Zhu%2C+Shirong+Ma%2C+Peiyi+Wang%2C+Xiao+Bi%2C+and+1+others%2E+2025%2E+Deepseekr1%3A+Incentivizing+reasoning+capability+in+llms+via+re)
11. Dhiraj Kalamkar, Dheevatsa Mudigere, Naveen Mellempudi, Dipankar Das, Kunal Banerjee, Sasikanth Avancha, Dharma Teja Vooturi, Nataraj Jammalamadaka, Jianyu Huang, Hector Yuen, and 1 others. 2019. A study of bfloat16 for deep learning training. arXiv preprint arXiv:1905.12322. — [arXiv:1905.12322](https://arxiv.org/abs/1905.12322) [find](https://scholar.google.com/scholar?q=Dhiraj+Kalamkar%2C+Dheevatsa+Mudigere%2C+Naveen+Mellempudi%2C+Dipankar+Das%2C+Kunal+Banerjee%2C+Sasikanth+Avancha%2C+Dharma+Teja+Vooturi%2C+Nataraj+Jammalamadaka%2C+Jianyu+Huang%2C+Hector+Yuen%2C+and+1+others%2E+2019%2E+A+st)
12. Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. 2023. [Let’s verify step by step](https://md.research.elamur.ai/free/en/papers/lightman-2023-let-s-verify-step-step/). In ICLR.
13. Liangchen Luo, Yinxiao Liu, Rosanne Liu, Samrat Phatale, Harsh Lara, Yunxuan Li, Lei Shu, Yun Zhu, Lei Meng, Jiao Sun, and 1 others. 2024. [Improve mathematical reasoning in language models by automated process supervision](https://md.research.elamur.ai/free/en/papers/luo-2024-improve-mathematical-reasoning-language-models/). In ArXiv e-prints, volume 2.
14. Paulius Micikevicius, Sharan Narang, Jonah Alben, Gregory Diamos, Erich Elsen, David Garcia, Boris Ginsburg, Michael Houston, Oleksii Kuchaiev, Ganesh Venkatesh, and 1 others. 2017. Mixed precision training. arXiv preprint arXiv:1710.03740. [find](https://scholar.google.com/scholar?q=Paulius+Micikevicius%2C+Sharan+Narang%2C+Jonah+Alben%2C+Gregory+Diamos%2C+Erich+Elsen%2C+David+Garcia%2C+Boris+Ginsburg%2C+Michael+Houston%2C+Oleksii+Kuchaiev%2C+Ganesh+Venkatesh%2C+and+1+others%2E+2017%2E+Mixed+precision+tr)
15. Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, and 1 others. 2022. [Training language models to follow instructions with human feedback](https://md.research.elamur.ai/free/en/papers/ouyang-2022-training-language-models-follow-instructions/). Advances in neural information processing systems, 35:27730–27744.
16. Samyam Rajbhandari, Jeff Rasley, Olatunji Ruwase, and Yuxiong He. 2020. Zero: Memory optimizations toward training trillion parameter models. In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1– 16. IEEE. [find](https://scholar.google.com/scholar?q=Samyam+Rajbhandari%2C+Jeff+Rasley%2C+Olatunji+Ruwase%2C+and+Yuxiong+He%2E+2020%2E+Zero%3A+Memory+optimizations+toward+training+trillion+parameter+models%2E+In+SC20%3A+International+Conference+for+High+Performance+Com)
17. John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. 2017. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347. [find](https://scholar.google.com/scholar?q=John+Schulman%2C+Filip+Wolski%2C+Prafulla+Dhariwal%2C+Alec+Radford%2C+and+Oleg+Klimov%2E+2017%2E+Proximal+policy+optimization+algorithms%2E+arXiv+preprint+arXiv%3A1707%2E06347%2E)
18. Amrith Setlur, Chirag Nagpal, Adam Fisch, Xinyang Geng, Jacob Eisenstein, Rishabh Agarwal, Alekh Agarwal, Jonathan Berant, and Aviral Kumar. 2025. [Rewarding progress: Scaling automated process verifiers for llm reasoning](https://md.research.elamur.ai/free/en/papers/setlur-2024-rewarding-progress-scaling-automated-process/). In ICLR.
19. Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, and 1 others. 2024. [Deepseekmath: Pushing the limits of mathematical reasoning in open language models](https://md.research.elamur.ai/free/en/papers/shao-2024-deepseekmath-pushing-limits-mathematical-reasoning/). In arXiv preprint arXiv:2402.03300.
20. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, and 1 others. 2022. [Chain-of-thought prompting elicits reasoning in large language models](https://md.research.elamur.ai/free/en/papers/wei-2022-chain-thought-prompting-elicits-reasoning/). Advances in neural information processing systems, 35:24824– 24837.
21. Lilian Weng. 2024. Reward hacking in reinforcement learning. [find](https://scholar.google.com/scholar?q=Lilian+Weng%2E+2024%2E+Reward+hacking+in+reinforcement+learning%2E)
22. An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, and 1 others. 2024a. Qwen2. 5 technical report. In ArXiv e-prints. [find](https://scholar.google.com/scholar?q=An+Yang%2C+Baosong+Yang%2C+Beichen+Zhang%2C+Binyuan+Hui%2C+Bo+Zheng%2C+Bowen+Yu%2C+Chengyuan+Li%2C+Dayiheng+Liu%2C+Fei+Huang%2C+Haoran+Wei%2C+and+1+others%2E+2024a%2E+Qwen2%2E+5+technical+report%2E+In+ArXiv+e%2Dprints%2E)
23. An Yang, Beichen Zhang, Binyuan Hui, Bofei Gao, Bowen Yu, Chengpeng Li, Dayiheng Liu, Jianhong Tu, Jingren Zhou, Junyang Lin, and 1 others. 2024b. Qwen2. 5-math technical report: Toward mathematical expert model via self-improvement. arXiv preprint arXiv:2409.12122. — [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://md.research.elamur.ai/free/en/papers/yang-2024-qwen2-5-math-technical-report/)
24. Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L. Griffiths, Yuan Cao, and Karthik Narasimhan. 2023. Deliberate problem solving with large language models. In NeurIPS. [find](https://scholar.google.com/scholar?q=Shunyu+Yao%2C+Dian+Yu%2C+Jeffrey+Zhao%2C+Izhak+Shafran%2C+Thomas+L%2E+Griffiths%2C+Yuan+Cao%2C+and+Karthik+Narasimhan%2E+2023%2E+Deliberate+problem+solving+with+large+language+models%2E+In+NeurIPS%2E)
25. Dan Zhang, Sining Zhoubian, Ziniu Hu, Yisong Yue, Yuxiao Dong, and Jie Tang. 2024. [Rest-mcts*: Llm self-training via process reward guided tree search](https://md.research.elamur.ai/free/en/papers/zhang-2024-rest-mcts-llm-self-training/). Advances in Neural Information Processing Systems, 37:64735–64772.
26. Jonathan Uesato, Nate Kushman, Ramana Kumar, Francis Song, Noah Siegel, Lisa Wang, Antonia Creswell, Geoffrey Irving, and Irina Higgins. 2022. [Solving math word problems with process-and outcome-based feedback](https://md.research.elamur.ai/free/en/papers/uesato-2022-solving-math-word-problems-process/). In ArXiv e-prints.
27. Peiyi Wang, Lei Li, Zhihong Shao, RX Xu, Damai Dai, Yifei Li, Deli Chen, Yu Wu, and Zhifang Sui. 2023. [Math-shepherd: Verify and reinforce llms step-by-step without human annotations](https://md.research.elamur.ai/free/en/papers/wang-2024-math-shepherd-verify-reinforce-llms/). In ACL.
28. Teng Wang, Zhenqi He, Wing-Yin Yu, Xiaojin Fu, and Xiongwei Han. 2024a. Large language models are good multi-lingual learners: When llms meet cross-lingual prompts. In COLING. [find](https://scholar.google.com/scholar?q=Teng+Wang%2C+Zhenqi+He%2C+Wing%2DYin+Yu%2C+Xiaojin+Fu%2C+and+Xiongwei+Han%2E+2024a%2E+Large+language+models+are+good+multi%2Dlingual+learners%3A+When+llms+meet+cross%2Dlingual+prompts%2E+In+COLING%2E)
29. Teng Wang, Wing-Yin Yu, Zhenqi He, Zehua Liu, Hailei Gong, Han Wu, Xiongwei Han, Wei Shi, Ruifeng She, Fangzhou Zhu, and 1 others. 2024b. Bpp-search: Enhancing tree of thought reasoning for mathematical modeling problem solving. arXiv preprint arXiv:2411.17404. [find](https://scholar.google.com/scholar?q=Teng+Wang%2C+Wing%2DYin+Yu%2C+Zhenqi+He%2C+Zehua+Liu%2C+Hailei+Gong%2C+Han+Wu%2C+Xiongwei+Han%2C+Wei+Shi%2C+Ruifeng+She%2C+Fangzhou+Zhu%2C+and+1+others%2E+2024b%2E+Bpp%2Dsearch%3A+Enhancing+tree+of+thought+reasoning+for+mathematic)
30. Teng Wang, Wing-Yin Yu, Ruifeng She, Wenhan Yang, Taijie Chen, and Jianping Zhang. 2024c. Leveraging large language models for solving rare mip challenges. arXiv preprint arXiv:2409.04464. — [arXiv:2409.04464](https://arxiv.org/abs/2409.04464) [find](https://scholar.google.com/scholar?q=Teng+Wang%2C+Wing%2DYin+Yu%2C+Ruifeng+She%2C+Wenhan+Yang%2C+Taijie+Chen%2C+and+Jianping+Zhang%2E+2024c%2E+Leveraging+large+language+models+for+solving+rare+mip+challenges%2E+arXiv+preprint+arXiv%3A2409%2E04464%2E+%E2%80%94+%5BarXiv%3A24)

*Updated 2 November 2026*