Constrained Flow Matching via Lagrangian Dual Flows

2026-07-05

AI Summary

Existing generative models struggle to efficiently satisfy highly nonlinear constraints during inference, often requiring costly optimization or projection steps. This work introduces Lagrangian dual dynamics into the flow matching framework for the first time, enabling simultaneous evolution of dual costate variables alongside the denoising process. The proposed approach strictly enforces complex nonlinear constraints without resorting to pseudoinverses, projections, or nested sub-optimizations. By establishing a theoretical connection between flow matching and primal-dual methods from numerical optimization, the method yields a simple yet efficient constrained generation algorithm. Empirical results demonstrate significantly reduced computational overhead across diverse nonlinear constraint tasks while maintaining high sample compliance.

Abstract

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.

Problem

• constrained generation
• nonlinear constraints
• flow matching
• inference-time constraints
• generative modeling

Innovation

• Constrained Flow Matching
• Lagrangian Dual Flows
• Nonlinear Constraints
• Generative Modeling
• Primal-Dual Methods