Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021.

最优控制理论提供了对复杂资源分配决策的洞察。前向-后向扫描法 (FBSM) 是一种迭代技术，通常用于解决由庞特里亚金最大值原理 (PMP) 在最优控制中的应用引起的两点边值问题。FBSM 在系统生物学中很受欢迎，因为它可以很好地适应系统规模并且易于实施。在这篇综述中，我们讨论了最优控制的 PMP 方法和 FBSM 的实现。通过将 FBSM 概念化为定点迭代过程，我们利用和调整现有的加速技术来提高其收敛速度。我们表明，无需代价高昂的加速技术调整即可实现收敛改进。此外，我们证明了这些方法可以在底层 FBSM 无法收敛的情况下引起收敛。这项工作中用于实现 FBSM 和加速技术的所有代码都可以在 GitHub 上获得，网址为 https://github.com/Jesse-Sharp/Sharp2021。