论文标题
路径网络最佳控制问题的共轭梯度的结构化预处理
Structured preconditioning of conjugate gradients for path-graph network optimal control problems
论文作者
论文摘要
为一类约束网络最佳控制问题的二阶方法的牛顿步骤开发了一个结构化的预处理共轭梯度（PCG）求解器。特定感兴趣的是由$ n $异质子系统的路径互连引起的离散时间动力学问题。每个PGC步骤的计算复杂性被证明为$ O（nt）$，其中$ t $是时间范围的长度。提出的预处理涉及每个PCG步骤固定数量的jacobi迭代。根据这个数字给出了对有效条件的分析约束。这些计算可以在最佳控制问题的空间和时间维度上分解，以分为大小的子问题，该大小独立于$ n $和$ t $。为质量 - 弹簧抑制链提供了数值结果。
A structured preconditioned conjugate gradient (PCG) solver is developed for the Newton steps in second-order methods for a class of constrained network optimal control problems. Of specific interest are problems with discrete-time dynamics arising from the path-graph interconnection of $N$ heterogeneous sub-systems. The computational complexity of each PGC step is shown to be $O(NT)$, where $T$ is the length of the time horizon. The proposed preconditioning involves a fixed number of block Jacobi iterations per PCG step. A decreasing analytic bound on the effective conditioning is given in terms of this number. The computations are decomposable across the spatial and temporal dimensions of the optimal control problem, into sub-problems of size independent of $N$ and $T$. Numerical results are provided for a mass-spring-damper chain.
