Preconditioning

The efficient solution of moderately large-scale linear systems arising from the KKT conditions in optimal control problems (OCPs) is a critical challenge in robotics. With the stagnation of Moore’s law, there is growing interest in leveraging GPU-accelerated iterative methods, and corresponding parallel preconditioners, to overcome these computational challenges. To improve the performance of such solvers, we introduce a parallel-friendly, parametrized multi-splitting polynomial preconditioner framework. We first construct and prove the optimal parametrization theoretically in terms of the least amount of distinct eigenvalues and the narrowest spectrum range. We then compare the theoretical time complexity of solving the linear system directly or iteratively. We finally show through numerical experiments how much the preconditioning improves the convergence of OCP linear systems solves.

In this work 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.