# Learning to Optimize through Solver-Grounded Self-Play

arXiv: https://arxiv.org/abs/2609.34205 · HTML: https://mlsanity.com/paper/2609.34205/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Self-play RL for reasoning, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 45 of its 62 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- Synthetic data and self-improvement for LLMs / [Self-play RL for reasoning](https://mlsanity.com/synth/self-play-rl-for-reasoning/)

## Builds on (most cited first, 30 of 45)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [GLM-5: from Vibe Coding to Agentic Engineering](https://mlsanity.com/paper/2602.15763/) (2026)
- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)
- [Absolute Zero: Reinforced Self-play Reasoning with Zero Data](https://mlsanity.com/paper/2505.03335/) (2025)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [Self-Play Preference Optimization for Language Model Alignment](https://mlsanity.com/paper/2405.00675/) (2024)
- [R-Zero: Self-Evolving Reasoning LLM from Zero Data](https://mlsanity.com/paper/2508.05004/) (2025)
- [When Scaling Meets LLM Finetuning: The Effect of Data, Model and Finetuning Method](https://mlsanity.com/paper/2402.17193/) (2024)
- [NL4Opt Competition: Formulating Optimization Problems Based on Their Natural Language Descriptions](https://mlsanity.com/paper/2303.08233/) (2023)
- [ORLM: A Customizable Framework in Training Large Models for Automated Optimization Modeling](https://mlsanity.com/paper/2405.17743/) (2024)
- [OptiMUS: Scalable Optimization Modeling with (MI)LP Solvers and Large Language Models](https://mlsanity.com/paper/2402.10172/) (2024)
- [LLMOPT: Learning to Define and Solve General Optimization Problems from Scratch](https://mlsanity.com/paper/2410.13213/) (2024)
- [Language Models Can Teach Themselves to Program Better](https://mlsanity.com/paper/2207.14502/) (2022)
- [Data Diversity Matters for Robust Instruction Tuning](https://mlsanity.com/paper/2311.14736/) (2023)
- [OptiBench Meets ReSocratic: Measure and Improve LLMs for Optimization Modeling](https://mlsanity.com/paper/2407.09887/) (2024)
- [OptMATH: A Scalable Bidirectional Data Synthesis Framework for Optimization Modeling](https://mlsanity.com/paper/2502.11102/) (2025)
- [Autoformulation of Mathematical Optimization Models Using LLMs](https://mlsanity.com/paper/2411.01679/) (2024)
- [LLaMoCo: Instruction Tuning of Large Language Models for Optimization Code Generation](https://mlsanity.com/paper/2403.01131/) (2024)
- [Solver-Informed RL: Grounding Large Language Models for Authentic Optimization Modeling](https://mlsanity.com/paper/2505.11792/) (2025)
- [Learning to Solve and Verify: A Self-Play Framework for Code and Test Generation](https://mlsanity.com/paper/2502.14948/) (2025)
- [Towards Foundation Models for Mixed Integer Linear Programming](https://mlsanity.com/paper/2410.08288/) (2024)
- [A Survey of Optimization Modeling Meets LLMs: Progress and Future Directions](https://mlsanity.com/paper/2508.10047/) (2025)
- [Large Language Models as End-to-end Combinatorial Optimization Solvers](https://mlsanity.com/paper/2509.16865/) (2025)
- [Diagnosing Infeasible Optimization Problems Using Large Language Models](https://mlsanity.com/paper/2308.12923/) (2023)

## Abstract (arXiv)

Optimization modeling is central to many decision-making scenarios, but traditionally requires extensive domain expertise. While Large Language Models (LLMs) have shown promise in automating this process, current training paradigms mainly rely on human-annotated or teacher-generated datasets. This dependence introduces a Generalization Ceiling, where models overfit to narrow data distributions, and Capability Anchoring, where models' reasoning is bounded by annotator proficiency and teacher model capability. In response, we propose OPT-Zero, the first fully self-play training framework for optimization modeling that requires zero external training data. OPT-Zero employs a single LLM in a dual-role closed loop: a Proposer that synthesizes increasingly challenging optimization problems alongside their mathematical formulations and solving code, and a Solver that attempts to resolve the problems given only natural-language problem descriptions. Grounded in execution feedback from external optimization solvers, we alternately train both roles using reinforcement learning. This process fosters an auto-curriculum in which the Proposer and Solver co-evolve: generating harder valid problems by the Proposer seamlessly enhances the structural reasoning ability of the Solver. Extensive results indicate that with zero curated data, OPT-Zero matches state-of-the-art data-dependent methods while exhibiting substantially stronger generalizability, establishing self-play training as a highly scalable paradigm for advancing LLM reasoning in modeling and solving optimization problems.