# Beyond Objective Equivalence: Constraint Injection for LLM-Based Optimization Modeling on Vehicle Routing Problems

arXiv: https://arxiv.org/abs/2606.04816 · HTML: https://mlsanity.com/paper/2606.04816/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [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)
- [LLMOPT: Learning to Define and Solve General Optimization Problems from Scratch](https://mlsanity.com/paper/2410.13213/) (2024)
- [OptiBench Meets ReSocratic: Measure and Improve LLMs for Optimization Modeling](https://mlsanity.com/paper/2407.09887/) (2024)
- [LLMs for Mathematical Modeling: Towards Bridging the Gap between Natural and Mathematical Languages](https://mlsanity.com/paper/2405.13144/) (2024)
- [OptMATH: A Scalable Bidirectional Data Synthesis Framework for Optimization Modeling](https://mlsanity.com/paper/2502.11102/) (2025)
- [Solver-Informed RL: Grounding Large Language Models for Authentic Optimization Modeling](https://mlsanity.com/paper/2505.11792/) (2025)
- [Large Language Models as End-to-end Combinatorial Optimization Solvers](https://mlsanity.com/paper/2509.16865/) (2025)
- [StepORLM: A Self-Evolving Framework With Generative Process Supervision For Operations Research Language Models](https://mlsanity.com/paper/2509.22558/) (2025)
- [OR-R1: Automating Modeling and Solving of Operations Research Optimization Problem via Test-Time Reinforcement Learning](https://mlsanity.com/paper/2511.09092/) (2025)
- [OptiMind: Teaching LLMs to Think Like Optimization Experts](https://mlsanity.com/paper/2509.22979/) (2025)
- [Reasoning in a Combinatorial and Constrained World: Benchmarking LLMs on Natural-Language Combinatorial Optimization](https://mlsanity.com/paper/2602.02188/) (2026)

## Built on by (most cited first, 2 of 2)

- [Falsification-Based Verification of LLM-Generated Optimization Models: Sound Test Batteries and Their Detection Limits](https://mlsanity.com/paper/2607.16646/) (2026)
- [OR for AI That Does OR: Routing LLMs up the Escalator inside the OSCAR Framework](https://mlsanity.com/paper/2610.00912/) (2026)

## Abstract (arXiv)

Large language models (LLMs) can generate executable solver code from natural-language descriptions of optimization problems. However, existing verification signals focus on whether the generated formulation produces the correct objective value. Such signals can overlook constraint-level errors: incorrect formulations may still produce the same optimum when extra or missing constraints do not affect the tested instance. We propose constraint injection, a verification method that directly tests whether the generated code implements the intended constraints. We use feasible solutions to detect incorrectly added constraints and one-constraint-violating solutions to detect omissions. Together with the optimal value check, these tests verify both the objective and the constraint set. We evaluate this approach on vehicle routing problems (VRPs), which provide a challenging testbed with diverse, tightly coupled operational constraints. We develop VRPCoder, an 8B model for generating Gurobi code from natural-language VRP descriptions, and VRPBench, an expert-verified benchmark spanning 21 VRP variants across four subsets. The verifier serves as a rejection-sampling criterion for synthetic data construction and as a rollout-level reward for reinforcement learning (RL). On VRPBench, VRPCoder-RL reaches 93% overall Pass@1, outperforms Gemini-3.1-Pro Preview on three subsets, exceeds Claude-Sonnet-4.5 by 28 percentage points, and exceeds the strongest prior OR-LLM by 78 percentage points.