ADMM-based Continuous Trajectory Optimization in Graphs of Convex Sets

2026-03-11

AI Summary

This work addresses the challenge of continuous trajectory optimization in non-convex environments by proposing a joint discrete–continuous optimization framework based on the Alternating Direction Method of Multipliers (ADMM). The approach parameterizes trajectories as polynomials and introduces a spatiotemporal allocation graph to model coupled spatiotemporal constraints. By integrating mixed-integer programming with shortest-path search, the method enables efficient solution computation. In contrast to conventional decoupled strategies, the proposed framework substantially expands the feasible search space and achieves stable convergence from arbitrary initial conditions without requiring complex warm-start procedures. Experimental results demonstrate significant improvements in both solution quality and robustness.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

This paper presents a numerical solver for computing continuous trajectories in non-convex environments. Our approach relies on a customized implementation of the Alternating Direction Method of Multipliers (ADMM) built upon two key components: First, we parameterize trajectories as polynomials, allowing the primal update to be computed in closed form as a minimum-control-effort problem. Second, we introduce the concept of a spatio-temporal allocation graph based on a mixed-integer formulation and pose the slack update as a shortest-path search. The combination of these ingredients results in a solver with several distinct advantages over the state of the art. By jointly optimizing over both discrete spatial and continuous temporal domains, our method accesses a larger search space than existing decoupled approaches, enabling the discovery of superior trajectories. Additionally, the solver's structural robustness ensures reliable convergence from naive initializations, removing the bottleneck of complex warm starting in non-convex environments.

Problem

• trajectory optimization
• non-convex environments
• continuous trajectories
• spatio-temporal optimization

Innovation

• ADMM
• trajectory optimization
• spatio-temporal allocation graph
• polynomial parameterization
• shortest-path search