大规模非线性最优控制的自适应并行非精确牛顿方法
An Adaptive, Parallel, and Inexact Newton Method for Large-scale Nonlinear Optimal Control
- University of California, San Diego（加利福尼亚大学圣迭戈分校）
- University of California, Berkeley（加利福尼亚大学伯克利分校）
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出自适应重叠时间分解（AOTD），一种并行SQP方法，在线调整重叠大小和求解精度，无需预设重叠，在电力系统与Burgers PDE最优控制问题上，计算量较最佳固定重叠变体减少约2.5倍。
AI中文摘要：
我们设计了自适应重叠时间分解（AOTD），一种用于长时域非线性最优控制问题（OCPs）的并行序列二次规划（SQP）方法。AOTD是首个在线调整重叠大小的重叠时间分解（OTD）算法，无需事先固定合适的重叠大小。具体而言，在每次迭代中，它将长时域划分为重叠的非线性子问题，并行地对每个子问题执行一步SQP，并自适应地选择重叠大小以及求解所得局部KKT系统的精度。两个条件控制接受的步长：一个自适应残差条件，限制拼接的非精确方向与完整问题的精确牛顿方向之间的距离；一个关于自适应惩罚参数的下降条件，证明对精心选择的精确增广拉格朗日函数的下降性。残差条件与求解器无关，既适用于确定性迭代求解器，也适用于随机草图求解器。在标准正则性假设下，我们证明KKT残差从任意初始化收敛到零，并建立收敛到严格局部极小值的序列的局部线性速率。我们在电力系统频率调节OCP和Burgers偏微分方程OCP上验证了AOTD，在单一参数设置下，估计的FLOPs相比最佳固定重叠变体减少了约2.5倍，同时在多个时域长度上收敛。
英文摘要：
We design adaptive overlapping temporal decomposition (AOTD), a parallel sequential quadratic programming (SQP) method for long-horizon nonlinear optimal control problems (OCPs). As designed, AOTD is the first overlapping temporal decomposition (OTD) algorithm that adapts the overlap size online, eliminating the need to fix a suitable overlap a priori. Specifically, at each iteration, it partitions the long horizon into overlapping nonlinear subproblems, performs one SQP step on each in parallel, and adaptively selects both the overlap size and the accuracy to which the resulting local KKT systems are solved. Two conditions govern the accepted step: an adaptive residual condition bounding the distance from the concatenated inexact direction to the exact Newton direction of the full problem, and a descent condition on the adaptive penalty parameters that certifies descent for a carefully chosen exact augmented Lagrangian. The residual condition is solver independent and accommodates both deterministic iterative solvers and randomized sketching solvers. Under standard regularity assumptions, we prove that the KKT residual converges to zero from any initialization and establish a local linear rate for sequences converging to a strict local minimizer. We validate AOTD on a power-system frequency regulation OCP and a Burgers PDE OCP, attaining a reduction in estimated FLOPs of approximately $2.5\times$ to the best fixed-overlap variant while converging across multiple horizon lengths with a single parameter setting.
