# Fast and Certifiable Trajectory Optimization

arXiv: https://arxiv.org/abs/2406.05846 · HTML: https://mlsanity.com/paper/2406.05846/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2024.

## Built on by (most cited first, 5 of 5)

- [Building Rome with Convex Optimization](https://mlsanity.com/paper/2502.04640/) (2025)
- [SIMPACT: Simulation-Enabled Action Planning using Vision-Language Models](https://mlsanity.com/paper/2512.05955/) (2025)
- [Differentiable Model Predictive Control on the GPU](https://mlsanity.com/paper/2510.06179/) (2025)
- [Learning Multiple Initial Solutions to Optimization Problems](https://mlsanity.com/paper/2411.02158/) (2024)
- [Tempered Sequential Monte Carlo for Trajectory and Policy Optimization with Differentiable Dynamics](https://mlsanity.com/paper/2604.21456/) (2026)

## Abstract (arXiv)

We propose semidefinite trajectory optimization (STROM), a framework that computes fast and certifiably optimal solutions for nonconvex trajectory optimization problems defined by polynomial objectives and constraints. STROM employs sparse second-order Lasserre's hierarchy to generate semidefinite program (SDP) relaxations of trajectory optimization. Different from existing tools (e.g., YALMIP and SOSTOOLS in Matlab), STROM generates chain-like multiple-block SDPs with only positive semidefinite (PSD) variables. Moreover, STROM does so two orders of magnitude faster. Underpinning STROM is cuADMM, the first ADMM-based SDP solver implemented in CUDA and runs in GPUs (with C/C++ extension). cuADMM builds upon the symmetric Gauss-Seidel ADMM algorithm and leverages GPU parallelization to speedup solving sparse linear systems and projecting onto PSD cones. In five trajectory optimization problems (inverted pendulum, cart-pole, vehicle landing, flying robot, and car back-in), cuADMM computes optimal trajectories (with certified suboptimality below 1%) in minutes (when other solvers take hours or run out of memory) and seconds (when others take minutes). Further, when warmstarted by data-driven initialization in the inverted pendulum problem, cuADMM delivers real-time performance: providing certifiably optimal trajectories in 0.66 seconds despite the SDP has 49,500 variables and 47,351 constraints.