Diffusion Models Solve Mixed-Integer Optimization Problems Faster

Mixed-integer optimization problems (MIPs) are notoriously challenging computationally, requiring the simultaneous determination of discrete and continuous decisions under complex combinatorial constraints. This research proposes a new learning-based method, Constrained Graph Diffusion (CGD), to tackle these problems more efficiently. CGD leverages a graph-based generative diffusion model specifically designed to learn the discrete components of MIPs. A key innovation of CGD is the integration of a training-free feasibility projection operator directly into the reverse diffusion process. This operator guides intermediate samples towards the feasible set during generation, ensuring that the discrete decisions produced are valid. Once these discrete decisions are generated, the remaining problem simplifies to a continuous optimization, which can then be solved rapidly using standard numerical methods. The framework is problem-agnostic, accommodating a wide range of MIPs through adaptable projection operators. Evaluations on optimal transmission switching for ACOPF and discrete portfolio optimization demonstrate CGD's effectiveness. It shows significant improvements in both feasibility and solution quality compared to other learning-based baselines. Furthermore, CGD achieved speedups of up to 425 times over state-of-the-art numerical solvers for mixed-integer nonlinear programs (MINLPs), highlighting its potential for practical applications.