A Fast Semidefinite Convex Relaxation for Optimal Control Problems With Spatio-Temporal Constraints

2026-01-06

arXiv.org

AI Summary

This work addresses the nonconvexity arising from the coupling between system dynamics and event times in optimal control of autonomous systems subject to spatiotemporal constraints. To tackle this challenge, the authors propose a novel approach that integrates a time-scaling direct multiple-shooting method with a sparsity-aware semidefinite programming (SDP) convex relaxation. By segmenting the prediction horizon according to characteristic time constraints and constructing a lifted-form convex relaxation that exploits sparsity, the method avoids both the need for prespecified waypoint timings and the pitfalls of nonconvex optimization that often lead to suboptimal solutions. Numerical simulations and real-world quadrotor experiments demonstrate that the proposed framework achieves near-optimal performance while significantly improving computational efficiency, making it well-suited for real-time control under complex spatiotemporal constraints.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Solving optimal control problems (OCPs) of autonomous agents operating under spatial and temporal constraints fast and accurately is essential in applications ranging from eco-driving of autonomous vehicles to quadrotor navigation. However, the nonlinear programs approximating the OCPs are inherently nonconvex due to the coupling between the dynamics and the event timing, and therefore, they are challenging to solve. Most approaches address this challenge by predefining waypoint times or just using nonconvex trajectory optimization, which simplifies the problem but often yields suboptimal solutions. To significantly improve the numerical properties, we propose a formulation with a time-scaling direct multiple shooting scheme that partitions the prediction horizon into segments aligned with characteristic time constraints. Moreover, we develop a fast semidefinite-programming-based convex relaxation that exploits the sparsity pattern of the lifted formulation. Comprehensive simulation studies demonstrate the solution optimality and computational efficiency. Furthermore, real-world experiments on a quadrotor waypoint flight task with constrained open time windows validate the practical applicability of the approach in complex environments.

Problem

• optimal control problems
• spatio-temporal constraints
• nonconvex optimization
• autonomous agents
• trajectory optimization

Innovation

• semidefinite programming
• convex relaxation
• time-scaling
• direct multiple shooting
• spatio-temporal constraints