Fast and Certifiable Trajectory Optimization

TL;DR

This paper introduces STROM, a fast, GPU-accelerated semidefinite programming framework for certifiably optimal trajectory optimization in complex nonlinear problems, significantly outperforming existing tools.

Contribution

STROM employs a novel sparse second-order Lasserre's hierarchy and GPU-based ADMM solver to achieve rapid, certifiably optimal solutions for nonconvex trajectory problems.

Findings

STROM generates chain-like SDPs with positive semidefinite variables.

cuADMM solves large SDPs in minutes or seconds, outperforming traditional solvers.

Real-time trajectory optimization achieved with warmstarting in inverted pendulum case.

Abstract

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.