Constrained Flow Matching via Lagrangian Dual Flows

Flow matching is a powerful tool for generative modeling, but emerging applications in robotics, planning, and physics require inference-time constraints on generated outputs. Such constraints are often complex and highly nonlinear. As a result, methods designed for linear constraints like image inpainting are rarely sufficient, and projection or optimization-based alternatives can be prohibitively expensive. In this paper, we introduce Lagrangian Dual Flows, a new family of constrained generation techniques based on Lagrangian dual dynamics. By simply flowing a dual co-state alongside generated samples, we can guarantee nonlinear constraint satisfaction without expensive optimization subproblems, pseudoinverses, or projection steps during the denoising process. The resulting constrained generation algorithms are simple, effective, and open new theoretical connections between flow matching and primal-dual methods in numerical optimization.