Semantic Exploration for Underspecification in LLM-Based Optimization Formulation

Large language models (LLMs) can translate natural-language optimization problems into mathematical formulations. However, natural-language descriptions are often underspecified or ambiguous, and successful solver execution can leave these issues undetected: a solvable formulation need not capture the intended problem. We introduce DIVERSEFORM, a formulation assistant that explores alternative mathematical interpretations through history-conditioned diverse sampling. Each sampling step includes previous formulations and requests a different defensible interpretation. We design a declarative intermediate representation that simplifies modeling by directly expressing mathematical variables, constraints, and objectives. This representation makes differences between candidate formulations explicit, facilitating comparison during the search. Experiments show that DIVERSEFORM outperforms existing prompting-based approaches using frontier models and surpasses a state-of-the-art open model fine-tuned specifically for optimization modeling, without requiring additional training. Further analysis shows that semantic exploration helps identify distinct interpretations and uncover underspecification that calls for clarification.