  Motivated by the need to solve open-loop optimal control problems with
computational efficiency and reliable constraint satisfaction, we introduce a
general framework that combines diffusion models and numerical optimization
solvers. Optimal control problems are rarely solvable in closed form, hence
they are often transcribed into numerical trajectory optimization problems,
which then require initial guesses. These initial guesses are supplied in our
framework by diffusion models. To mitigate the effect of samples that violate
the problem constraints, we develop a novel constrained diffusion model to
approximate the true distribution of locally optimal solutions with an
additional constraint violation loss in training. To further enhance the
robustness, the diffusion samples as initial guesses are fed to the numerical
solver to refine and derive final optimal (and hence feasible) solutions.
Experimental evaluations on three tasks verify the improved constraint
satisfaction and computational efficiency with 4$\times$ to 30$\times$
acceleration using our proposed framework, which generalizes across trajectory
optimization problems and scales well with problem complexity.
