Sampling-Based Global Optimal Control and Estimation via Semidefinite Programming
For researchers in control and robotics, this work makes global optimization via SOS practical for non-polynomial problems, though it is an incremental application of existing theory.
This paper takes Kernel Sum of Squares (KernelSOS) from theory to practice, demonstrating its effectiveness on challenging control and robotics problems. It achieves competitive performance in robot localization and uncovers higher-quality solutions in trajectory optimization without compromising runtimes.
Global optimization has gained attraction over the past decades, thanks to the development of both theoretical foundations and efficient numerical routines. Among recent advances, Kernel Sum of Squares (KernelSOS) provides a powerful theoretical framework, combining the expressivity of kernel methods with the guarantees of SOS optimization. In this paper, we take KernelSOS from theory to practice and demonstrate its use on challenging control and robotics problems. We identify and address the practical considerations required to make the method work in applied settings: restarting strategies, systematic calibration of hyperparameters, methods for recovering minimizers, and the combination with fast local solvers. As a proof of concept, the application of KernelSOS to robot localization highlights its competitiveness with existing SOS approaches that rely on heuristics and handcrafted reformulations to render the problem polynomial. Even in the high-dimensional, non-parametric setting of trajectory optimization with simulators treated as black boxes, we demonstrate how KernelSOS can be combined with fast local solvers to uncover higher-quality solutions without compromising overall runtimes.