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，机器人领域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。研究表明，在许多情况下，需要特定手工设计的冗余约束才能获得紧松弛进而实现全局最优性。这些约束依赖于具体问题表述，通常需要通过冗长的手动过程来识别。相反，本文提出了一种自动方法，用于寻找一组充分的冗余约束以实现紧性（若存在的话）。我们首先提出一种高效的可行性检验方法，用于判断给定变量集是否能够形成紧表述。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，我们无需手动寻找冗余约束。通过基于距离的定位与立体视觉位姿估计的仿真及真实数据集实验，我们验证了该方法的有效性。最后，我们复现了近期文献中提出的半定松弛，并表明我们的自动方法总能找到比先前研究更小的一组足以实现紧性的约束集。