OR-Diffusion: Structural Data Synthesis and Hierarchical Verification for Optimization Modeling

Large language models can translate problem descriptions into optimization models, but two challenges limit their utility. First, existing synthesis pipelines mainly paraphrase or perturb parameters, altering descriptions rather than constraint topology. Second, evaluation and training collapse distinct failure modes into a scalar reward, ignoring the natural priority: executability, structural fidelity, feasibility, and optimality, in that order. We introduce OR-Diffusion, which couples structural synthesis with hierarchical training. It synthesizes structurally novel instances via discrete diffusion over graph specifications, each feasible by construction and carrying a solver-certified gold optimum. A Hierarchical Verification Engine (HVE) certifies candidates along four ordered dimensions and reports the weakest failing layer. It guides supervised fine-tuning on executability, structural fidelity, and feasibility, and a preference stage on optimality. A pair enters that stage only when its rejected response passes the first three layers. On six public benchmarks, OR-Diffusion raises an 8B backbone from 57.8% to 78.2% macro optimal-solution rate, surpassing a 671B generalist (DeepSeek-V3, 73.3%) and specialized fine-tuned models (e.g., ORLM-8B, 65.5%), while also improving executability and feasibility. Ablations show that the two components are complementary, yielding a joint gain of 6.5 percentage points, and that diffusion outperforms textual synthesis by 4.2 percentage points. Weakest-layer diagnosis confirms this division of labor: fine-tuning removes most executability failures and leaves structural mismatches as the main residual, and the gated stage raises optimality without reducing feasibility.