There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems. Iterative methods are particularly well-suited for parallel solves of such systems. However, fast and stable convergence of iterative methods is reliant on the application of a high-quality preconditioner that reduces the spread and increase the clustering of the eigenvalues of the target matrix. To improve the performance of these approaches, we present a new parallel-friendly symmetric stair preconditioner. We prove that our preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum. Numerical experiments with typical trajectory optimization problems reveal that as compared to the best alternative parallel preconditioner from the literature, our symmetric stair preconditioner provides up to a 34% reduction in condition number and up to a 25% reduction in the number of resulting linear system solver iterations.

翻译：近年来，并行求解轨迹优化问题的策略日益受到关注。在许多轨迹优化算法方法中，关键步骤之一是求解中等规模且稀疏的线性系统。迭代方法特别适合此类系统的并行求解。然而，迭代方法快速稳定的收敛依赖于应用高质量的预条件子，以减小目标矩阵特征值的散布并增强其聚类性。为提升这些方法的性能，我们提出了一种新型并行友好的对称阶梯预条件子。我们证明了该预条件子与轨迹优化的迭代方法结合使用时具有有利的理论性质，例如更聚类的特征值谱。通过典型轨迹优化问题的数值实验对比发现，与文献中最佳替代并行预条件子相比，我们的对称阶梯预条件子可降低高达34%的条件数和多达25%的线性系统求解器迭代次数。