In recent years, semidefinite relaxations of common optimization problems in robotics have attracted growing attention due to their ability to provide globally optimal solutions. In many cases, it was shown that specific handcrafted redundant constraints are required to obtain tight relaxations, and thus global optimality. These constraints are formulation-dependent and typically identified through a lengthy manual process. Instead, the present article suggests an automatic method to find a set of sufficient redundant constraints to obtain tightness, if they exist. We first propose an efficient feasibility check to determine if a given set of variables can lead to a tight formulation. Second, we show how to scale the method to problems of bigger size. At no point of the process do we have to find redundant constraints manually. We showcase the effectiveness of the approach, in simulation and on real datasets, for range-based localization and stereo-based pose estimation. We also reproduce semidefinite relaxations presented in recent literature and show that our automatic method always finds a smaller set of constraints sufficient for tightness than previously considered.

中文翻译：

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使用自动收紧的半定松弛进行全局最优状态估计


近年来，机器人技术中常见优化问题的半定放宽因其能够提供全局最优解而引起了越来越多的关注。在许多情况下，结果表明需要特定的手工冗余约束才能获得紧密松弛，从而获得全局最优性。这些限制取决于配方，通常通过漫长的手动过程来确定。相反，本文提出了一种自动方法来查找一组足够的冗余约束来获得紧密性（如果存在）。我们首先提出了一个有效的可行性检查，以确定一组给定的变量是否会导致一个紧密的公式。其次，我们展示了如何将该方法扩展到更大尺寸的问题。在这个过程的任何时候，我们都不必手动查找冗余的约束。我们展示了该方法在仿真和真实数据集中对基于距离的定位和基于立体的姿态估计的有效性。我们还重现了最近文献中提出的半定松弛，并表明我们的自动方法总是找到比以前认为的足够紧密的约束集更小的约束。