Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement

Recent advances in diffusion models show promising potential to accelerate nonconvex problem solving by leveraging their multimodality. However, most existing diffusion-based optimization approaches rely on supervised learning and lack a mechanism to enforce constraint satisfaction, which is required in real-world applications. In that case, we investigate and theoretically analyze the inherent problem of supervised diffusion solvers and identify the distributional misalignment problem, i.e., the generated solution distribution often exhibits low probability mass on the feasible region. To resolve this issue, we propose DiOpt, a new diffusion-based learning framework for constrained nonconvex optimization, which effectively learns the mapping from noise to the constraint region. Specifically, this framework operates in two distinct phases: an initial warm-start phase, implemented via supervised learning, followed by a bootstrapping training phase. This dual-phase architecture is designed to iteratively refine solutions, thereby improving the objective function with high constraint satisfaction. Finally, we also employ a solution selection technique in inference for better optimality. Notably, DiOpt is the first successful integration of the diffusion solver in constrained nonconvex optimization. Evaluations on diverse nonconvex tasks demonstrate the superiority of DiOpt in both optimality and constraint satisfaction.