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 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 by 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 in simulation that the algorithms provide a significant speed up for two example problems: matrix-weighted and range-only localization.

翻译：近年来，机器人学中的许多估计问题已被证明可通过半定松弛方法求得全局最优解。然而，现成半定规划求解器的运行时复杂度可达问题规模的三次方，这阻碍了涉及大状态维度问题的实时求解。本文证明，对于具有弦图稀疏性的一大类问题，我们可以将这些求解器的复杂度降低至问题规模的线性级别。具体而言，我们展示了如何通过著名的弦图分解方法，将大型半正定变量替换为多个相互关联的较小变量。该公式还便于直接应用交替方向乘子法（ADMM），该方法可利用并行性提升可扩展性。仿真实验表明，所提算法在矩阵加权定位和纯距离定位两个示例问题上均实现了显著的加速效果。