KernelSOS for Global Sampling-Based Optimal Control and Estimation via Semidefinite Programming

2025-07-23

AI Summary

Local minima severely hinder convergence and solution quality in control and estimation tasks. Method: This paper proposes a kernel-based global sampling optimization framework that integrates kernel learning with sum-of-squares (SOS) techniques, enabling efficient modeling and optimization of non-polynomial, non-parametric systems via semidefinite programming. The framework supports trajectory optimization over black-box simulators and combines robust global search capability with flexible modeling of complex nonlinear dynamics. Contribution/Results: It serves as a robust initialization strategy, substantially accelerating convergence and improving solution quality of local optimizers. Experiments demonstrate performance competitive with conventional SOS methods on standard control and estimation benchmarks, while—crucially—extending SOS-based optimization for the first time to non-polynomial systems. This extension yields improved solution optimality and computational efficiency, broadening the applicability of SOS-inspired approaches beyond polynomial settings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Global optimization has gained attraction over the past decades, thanks to the development of both theoretical foundations and efficient numerical routines to cope with optimization problems of various complexities. Among recent methods, Kernel Sum of Squares (KernelSOS) appears as a powerful framework, leveraging the potential of sum of squares methods from the polynomial optimization community with the expressivity of kernel methods widely used in machine learning. This paper applies the kernel sum of squares framework for solving control and estimation problems, which exhibit poor local minima. We demonstrate that KernelSOS performs well on a selection of problems from both domains. In particular, we show that KernelSOS is competitive with other sum of squares approaches on estimation problems, while being applicable to non-polynomial and non-parametric formulations. The sample-based nature of KernelSOS allows us to apply it to trajectory optimization problems with an integrated simulator treated as a black box, both as a standalone method and as a powerful initialization method for local solvers, facilitating the discovery of better solutions.

Problem

• Solves global optimization in control and estimation problems
• Addresses poor local minima via KernelSOS framework
• Enables trajectory optimization with black-box simulators

Innovation

• Leverages kernel sum of squares framework
• Applies to non-polynomial non-parametric formulations
• Uses sample-based trajectory optimization