Automated Inverse Optimization with LLMs via Verification-Guided Hypothesis Search

Inverse optimization (IO) infers latent decision-making models from observed decisions, but classically requires experts to rigidly specify the objective, constraints, and behavioral assumptions prior to estimate model parameters. While recent advances in large language models (LLMs) have enabled automated construction of optimization models from natural language, they struggle in the inverse setting, where the latent model structure must be iteratively inferred from behavioral demonstrations. We formulate automated IO as a structure-aware learning problem, jointly identifying the objective components, latent constraints, behavioral assumptions, and numerical parameters from observed decisions. To address this, we propose VeriHy, an LLM-based framework that performs verification-guided search over competing model hypotheses. VeriHy organizes hypotheses in a directed acyclic graph and uses evidence from successful candidates and counterexamples to guide model refinement and recombination. Experiments across convex, combinatorial, and real-world IO problems show that VeriHy achieves performance competitive with expert-crafted classical methods and significantly outperforms LLM baselines under matched search budgets.