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 paper 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. Secondly, 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. Finally, we 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.

翻译：近年来，机器人学中常见优化问题的半定松弛因其能够提供全局最优解而受到越来越多的关注。在许多情况下，研究表明需要特定的手工构建冗余约束以获得紧松弛，从而实现全局最优性。这些约束依赖于具体的问题表述，通常需要通过冗长的手动过程来识别。本文则提出了一种自动方法来寻找一组足以获得紧松弛的冗余约束（如果它们存在的话）。我们首先提出了一种高效的可行性检查方法，用于判断给定变量集是否能够产生紧的松弛形式。其次，我们展示了如何将该方法扩展到更大规模的问题。在整个过程中，我们无需手动寻找任何冗余约束。我们通过仿真和真实数据集，在基于距离的定位和基于立体的姿态估计问题上展示了该方法的有效性。最后，我们复现了近期文献中提出的几种半定松弛，并表明我们的自动方法总能找到比先前考虑的更小的、足以保证紧性的约束集。