Sampling-Based Global Optimal Control and Estimation via Semidefinite Programming

TL;DR

This paper advances Kernel Sum of Squares (KernelSOS) for practical control and robotics applications, demonstrating its effectiveness in high-dimensional, black-box trajectory optimization and localization tasks.

Contribution

It bridges the gap between KernelSOS theory and practice, introducing strategies for hyperparameter calibration, minimizer recovery, and integration with local solvers in applied settings.

Findings

KernelSOS competes with heuristic SOS methods in robot localization.

The method uncovers higher-quality solutions in black-box trajectory optimization.

Practical strategies enable KernelSOS to be effective in high-dimensional control problems.

Abstract

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.