Title: Lagrangian Meets Diffusion: Feasibility-aware Generative Modeling for Mixed Integer Linear Programming
Primary area: optimization
Keywords: MILP;Optimal;

Abstract: End-to-end Predict-and-Search (PaS) methods show promise for Mixed Integer Linear Programming (MILP), but they typically assume variables independence and provide only deterministic single-point predictions, limiting solution diversity and demanding extensive search for high-quality solutions. We propose \textbf{VRG}, a feasibility-aware generative framework that operates in visual space. It transforms MILP solution vectors into image representations, which are in turn processed by a U-Net-based score network with Lagrangian relaxation guidance. The visual encoding enables convolutional kernels to capture interdependencies among variables while Lagrangian relaxation guides sampling toward feasible, near-optimal regions. The guided generator produces diverse, high-quality candidates rather than a single point estimate. The resulting candidates define compact and effective trust-region subproblems for standard MILP solvers. Across various public benchmarks, VRG consistently outperforms PaS baselines in solution quality and, while maintaining competitive optimality with state-of-the-art solvers such as SCIP and Gurobi, achieves markedly lower computational effort (reduced search time and explored nodes). Our source code is available at https://anonymous.4open.science/r/VRG-E09E/.