Sampling-Based Global Optimal Control and Estimation via Semidefinite Programming

TL;DR

This paper identifies and addresses the practical considerations required to make the KernelSOS method work in applied settings: restarting strategies, systematic calibration of hyperparameters, methods for recovering minimizers, and the combination with fast local solvers.

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.

Figures (9)

• Figure 1: Overview of the approach. This paper demonstrates the effectiveness of KernelSOS for optimization problems in estimation and control. Based on function evaluations only, KernelSOS can address non-polynomial and non-parametric problems beyond the reach of classic Sum of Squares methods. Each iteration solves a single semidefinite program, and multiple restarts yield an approximately globally optimal solution, which can be refined by a local solver.
• Figure 2: Illustration of KernelSOS with different kernels. Given samples (black crosses) of an arbitrary (possibly unknown) function $f$ (orange), KernelSOS minimizes a kernel-defined surrogate function (blue tones), by solving a simple SDP. When the kernel is chosen such that $f$ is in its RKHS, a good estimate of the global minimum is found, even for a low number of samples. Here, $f$ is a polynomial of 4th degree, and therefore the polynomial kernel of degrees 2 and 3, and the Gaussian kernel with $\sigma>1$, lead to good results. The width of the plot is 2.
• Figure 3: Calibration results for range-only localization. The localization error on a calibration dataset for different choices of kernel (Laplacian or Gaussian), kernel scale $\sigma$, and regularization parameter $\lambda$ is plotted. The baseline corresponds to picking the lowest-cost sample. While both kernel choices lead to similar best-case errors, the performance of the Gaussian kernel is less sensitive to the correct choice of $\lambda$.
• Figure 4: Noise and time study for range-only localization. Top: Distance to ground truth as a function of the noise level in range-only localization, using the non-squared (left) and squared (right) formulation. The local solver often converges to local minima, while the (parametric) global EquationSOS and SampleSOS solvers perform consistently across noise levels and formulations. Notably, the (non-parametric) KernelSOS method achieves similar performance for a wide range of noise levels, and significantly outperforms the best-sample baseline. Bottom: Run-time comparison. Dotted line use the MOSEK solver, solid lines use custom solvers, in particular the open-sourced Newton solver.
• Figure 5: Illustration of the surrogate cost and the benefit of restarts. The cost of the range-only problem is computed either using non-squared distances (left half) or squared distances (right half). In each block, we have on the left: real cost; center: surrogate function found by the first KernelSOS step; right: surrogate function found by the first restart (second step). Black crosses represent the known landmarks, black x-marks represent the used samples. Orange and green x-marks denote the minimizers found by KernelSOS initially and after restarting, respectively.