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.

翻译：受解决开环最优控制问题对计算效率和可靠约束满足的需求驱动，我们提出了一种结合扩散模型与数值优化求解器的通用框架。最优控制问题很少能以闭式解求解，因此通常被转录为数值轨迹优化问题，这需要初始猜测。在我们的框架中，这些初始猜测由扩散模型提供。为减轻违反问题约束的样本的影响，我们开发了一种新颖的约束扩散模型，通过在训练中加入额外的约束违反损失来逼近局部最优解的真实分布。为进一步增强鲁棒性，扩散样本作为初始猜测被输入数值求解器进行优化，从而得到最终的最优（即可行）解。在三个任务上的实验评估验证了所提框架在约束满足和计算效率方面的改进，实现了4倍至30倍的加速，且该框架能泛化至不同的轨迹优化问题，并随问题复杂度具有良好的扩展性。