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.

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，现成的半定规划求解器的时间复杂度可达问题规模的三次方，这阻碍了涉及高维状态量的实时求解。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至与问题规模呈线性关系。具体而言，我们展示了如何通过经典的弦图分解方法，将大型半正定变量替换为多个相互关联的较小变量。该公式体系还能直接应用交替方向乘子法，该方法可利用并行计算提升可扩展性。通过两个仿真算例，我们证明弦图求解器相比标准半定规划求解器能实现显著加速，且在缺乏良好初始化的情况下，全局最优性具有关键意义。