Exploiting Chordal Sparsity for Fast Global Optimality with Application to Localization

• Main Authors: Dümbgen, Frederike, Holmes, Connor, Barfoot, Timothy D
• Format: Journal Article
• Language: English
• Published: 04-06-2024
• Subjects: Computer Science - Robotics
• DOI: 10.48550/arxiv.2406.02365

In recent years, many estimation problems in robotics have been shown to be
solvable to global optimality using their semidefinite relaxations. However,
the runtime complexity of off-the-shelf semidefinite programming (SDP) solvers
is up to cubic in problem size, which inhibits real-time solutions of problems
involving large state dimensions. We show that for a large class of problems,
namely those with chordal sparsity, we can reduce the complexity of these
solvers to linear in problem size. In particular, we show how to replace the
large positive-semidefinite variable with a number of smaller interconnected
ones using the well-known chordal decomposition. This formulation also allows
for the straightforward application of the alternating direction method of
multipliers (ADMM), which can exploit parallelism for increased scalability. We
show for two example problems in simulation that the chordal solvers provide a
significant speed-up over standard SDP solvers, and that global optimality is
crucial in the absence of good initializations.