Exploiting Chordal Sparsity for Fast Global Optimality with Application to Localization

TL;DR

This paper introduces a method leveraging chordal sparsity to significantly speed up semidefinite relaxations in estimation problems, enabling real-time solutions and parallel computation, demonstrated through localization applications.

Contribution

It presents a novel approach to reduce SDP complexity using chordal decomposition and ADMM, improving scalability and speed for large-scale estimation problems.

Findings

Chordal decomposition reduces SDP complexity from cubic to linear.

Parallel ADMM enhances scalability and speed.

Global optimality is essential without good initial guesses.

Abstract

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.