Exploiting Chordal Sparsity for Fast Global Optimality with Application to Localization

TL;DR

The paper tackles the computational burden of obtaining globally optimal solutions for estimation problems in robotics via semidefinite relaxations, typically scaling cubically with problem size $n$. It exploits chordal sparsity through chordal decomposition to replace a single large PSD variable with several smaller interconnected PSD variables, reducing solver complexity to linear in $n$, i.e., from $O(n^3)$ to $O(n)$. This formulation also enables applying the alternating direction method of multipliers (ADMM) to exploit parallelism and improve scalability. Two simulated problems show substantial speed-ups over standard SDP solvers, and the authors demonstrate that preserving global optimality is crucial when initializations are poor, enabling scalable, globally optimal localization and related estimation tasks in robotics.

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.