标题： 路径图网络最优控制问题的共轭梯度结构化预处理 显示英文标题

标题： Structured preconditioning of conjugate gradients for path-graph network optimal control problems

摘要： 为一类约束网络最优控制问题的二阶方法中的牛顿步骤开发了一个结构化预处理共轭梯度（PCG）求解器。 特别感兴趣的是由$N$个异构子系统路径图互联产生的离散时间动力学问题。 每个 PGC 步骤的计算复杂度被证明是$O(NT)$，其中$T$是时间范围的长度。 所提出的预处理方法在每个 PCG 步骤中涉及固定数量的块雅可比迭代。 给出了关于此数量的有效条件性的递减解析界。 计算可以分解为最优控制问题的空间和时间维度，得到与$N$和$T$无关的子问题。 提供了质量-弹簧-阻尼器链的数值结果。 显示英文摘要

摘要： 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.