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

Vehicle Routing Problems (VRPs) are fundamental combinatorial optimization problems with widespread applications in various scenarios. The advanced optimization solvers can effectively solve such problems. However, modeling complex VRP variants for solvers often requires substantial domain expertise, which limits the accessibility of advanced optimization technologies. In this paper, we propose Reinforcement Learning Enhanced LLMAgents(RLEA), a multi-agent framework designed to automate the modeling of complex VRPs. RLEA introduces a lightweight neural Planner trained with Soft Q-learning to efficiently orchestrate the actions of LLM-based agents. In addition, we equip the system with an evolutionary memory module and retrieval-augmented generation, enabling the agent to leverage both accumulated experience and external solver knowledge during program generation and refinement for solving VRPs. We evaluated 48 distinct VRP variants across various solvers. The experimental results demonstrate that RLEA outperforms the previous state-of-the-ar method, achieving a 16.67% higher success rate while significantly reducing runtime errors. These results validate that integrating reinforcement learning with LLM-based reasoning is highly effective for automated optimization modeling. The appendix is available at: https://doi.org/10.5281/zenodo.19134435.

Large language models (LLMs) typically approach combinatorial optimization as an inference-time procedure, solving each instance separately through sampling, search, or repeated prompting. We ask whether reinforcement learning can instead shift part of this reasoning cost into the weights of a code LLM, so that the model synthesizes a reusable solver for an entire problem family. We study this question on Synergistic Dependency Selection (SDS), a controlled variant of constrained Quadratic Knapsack designed to expose a specific failure mode: local signals and strict feasibility constraints make greedy heuristics attractive but unreliable. Under identical scaffolding, Best-of-64 base-model sampling saturates at an approximately 28.7% gap to the global Virtual Best Solver (VBS); code audits show that the base model often retrieves Simulated Annealing templates but misimplements the Metropolis acceptance rule. We fine-tune Qwen2.5-Coder-14B-Instruct with Group Relative Policy Optimization (GRPO) using a feasibility-gated reward and light structural scaffolding. The resulting policy converges to a constraint-aware Simulated Annealing template in 99.8% of feasible SDS outputs, achieves a 5.0% gap to that VBS, and is 91 times cheaper in post-generation execution/search cost than cumulative Best-of-64 evaluation. A compile-once check shows that one best frozen solver per seed remains highly competitive when reused unchanged across the SDS test set, while an additional-domain evaluation on Job Shop Scheduling provides narrower but positive evidence that the scaffold transfers beyond SDS. Negative ablations reveal the limits of this recipe: standard stabilizers degrade performance, a soft feasibility gate fails, and results remain sensitive to reward normalization and domain-specific design choices.

The runtime of Constraint Programming (CP) solvers is highly sensitive to modeling choices, such as symmetry breaking, implied constraints, global constraints, constraint reformulation, and variable representation. Improving these constraint models has traditionally required human expertise, and existing automated reformulation systems are restricted to a predefined library of hand-crafted transformation rules. We introduce an agentic framework that instead reformulates a constraint model from an open-ended space and establishes correctness empirically rather than by construction: a Large Language Model (LLM) agent, given a model and three training instances, proposes alternative formulations, validates each by injecting its solution back into the original model, and diagnoses and repairs failures, returning the best variant it finds in a median of about fifteen minutes. The models are expressed in the CPMpy modeling library, and each proposed model is evaluated on three larger test instances. Across nine combinatorial optimization problems, the generated models outperform the originals on 21 of 27 test instances, and on some problems solve more than two orders of magnitude faster. A comparison against non-agentic baselines that reuse the same validation and selection tools indicates that the gains stem from the agent's iterative diagnosis and repair, not merely from sampling several candidates. These results demonstrate that autonomous agentic methods can support the improvement of constraint models.