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 computational performance of such solvers, we introduce a parallel-friendly, parametrized multi-splitting polynomial preconditioner framework that leverages positive and negative factors. Our approach results in improved convergence of the linear systems solves needed in OCPs. We construct and prove the optimal parametrization of multi-splitting theoretically and demonstrate empirically a 76% reduction in condition number and 46% in iteration counts on a series of numerical benchmarks.

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.