Infeasibility Aware Large Language Models for Combinatorial Optimization

Large Language Models (LLMs) struggle to solve complex combinatorial problems through direct reasoning, so recent neuro-symbolic systems increasingly use them to synthesize executable solvers. A central design question is how the LLM should represent the solver, and whether it should also attempt to optimize search. We introduce CP-SynC-XL, a benchmark of 100 combinatorial problems (4,577 instances), and evaluate three solver-construction paradigms: native algorithmic search (Python), constraint modeling through a Python solver API (Python + OR-Tools), and declarative constraint modeling (MiniZinc + OR-Tools). We find a consistent representational divergence: Python + OR-Tools attains the highest correctness across LLMs, while MiniZinc + OR-Tools has lower absolute coverage despite using the same OR-Tools back-end. Native Python is the most likely to return a schema-valid solution that fails verification, whereas solver-backed paths preserve higher conditional fidelity. On the heuristic axis, prompting for search optimization yields only small median speed-ups (1.03-1.12x) and a strongly bimodal effect: many instances slow down, and correctness drops sharply on a long tail of problems. A paired code-level audit traces these regressions to a recurring heuristic trap. Under an efficiency-oriented prompt, the LLM may replace complete search with local approximations (Python), inject unverified bounds (Python + OR-Tools), or add redundant declarative machinery that overwhelms or over-constrains the model (MiniZinc + OR-Tools). These findings support a conservative design principle for LLM-generated combinatorial solvers: use the LLM primarily to formalize variables, constraints, and objectives for verified solvers, and separately check any LLM-authored search optimization before use.

Generating optimization instances that are both feasible and computationally challenging is crucial for benchmarking solvers and training learning-based optimization algorithms. Existing non-LLM generators rely on seed instances or parameter tuning, resulting in high test-time computational cost, while existing LLM generators lack explicit hardness measures. Recent reinforcement learning methods with verifier feedback evaluate only binary correctness, which is misaligned with generating challenging problems. We note that an optimization solver reports the cost of solving at several stages of its pipeline, and leverage this to design a reward that scores both the solvability and the hardness of generated problems, measured by branch-and-bound nodes and post-cut relaxation gaps. Our key idea is a challenger-solver asymmetric self-play approach, where an LLM challenger generates progressively harder instances and the solver verifies feasibility and hardness, so no seed or training MILP instances are required. We fine-tune Gemma-4-12B and Qwen3.5-4B with GRPO and a size curriculum into OptiScribe-12B and OptiScribe-4B, which generate feasible yet challenging MILP problems from natural language instructions. On capacitated facility location and max-cut, OptiScribe-12B raises median SCIP search nodes by 1.7-5x and post-cut gaps by 1.1-1.7x over its base model and improves the feasibility rate on facility location by 9-19 points, while OptiScribe-4B raises median nodes by up to 15.6x. The problems cover a wider difficulty range than public benchmarks of the same size, follow instructions on density and difficulty, and can tune solver settings for families that public libraries lack. These results indicate that optimization-specific rewards, used in self-play mode, can teach LLMs to generate high-difficulty optimization benchmarks. We will release our code and models publicly on acceptance.

Large language models are increasingly applied to operational decision-making where the underlying structure is constrained optimization. Existing benchmarks evaluate whether LLMs can formulate optimization problems as solver code, but leave open a complementary question. Can LLMs directly produce correct solutions to fully specified constrained optimization problems without access to a solver? We introduce ConstraintBench, a benchmark for evaluating LLMs on direct constrained optimization across 10 operations research domains, with all ground-truth solutions verified by the Gurobi solver. Each task presents a natural-language scenario with entities, constraints, and an optimization objective; the model must return a structured solution that a deterministic verifier checks against every constraint and the solver-proven optimum. We evaluate six frontier models on 200 tasks and find that feasibility, not optimality, is the primary bottleneck. The best model achieves only 65.0% feasibility, yet feasible solutions average 89 to 96% of the Gurobi-optimal objective. No model exceeds 30.5% on joint feasibility and optimality within 0.1% of the solver reference. Per-domain analysis shows large variation in difficulty, with average feasibility spanning from 85.0% in the facility location domain to 0.8% in the crew assignment domain. Further, systematic failure modes include duration constraint misunderstanding, entity hallucination, and a feasibility-optimality decoupling in facility location and vehicle routing where models achieve high feasibility but 0% optimality. ConstraintBench and all evaluation infrastructure will be publicly released.