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 require a lengthy manual process to find. 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 manually find redundant constraints. 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 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日益关注。研究表明，许多情况下需要特定手工构造的冗余约束才能获得紧松弛，进而实现全局最优性。这些约束依赖于问题公式化，通常需要冗长的人工搜索过程。本文提出一种自动方法，用于在存在冗余约束的情况下找到一组充分的冗余约束以实现紧性。我们首先提出一种高效的可行性检验，用于判定给定变量集能否形成紧公式化。其次，我们展示了如何将该方法推广至更大规模的问题。整个过程无需人工寻找冗余约束。我们通过仿真和真实数据集，在基于距离的定位与基于立体视觉的位姿估计任务中验证了该方法的有效性。最后，我们重现近期文献中的半定松弛，并证明我们的自动方法能找到比此前研究更小的约束集来保证紧性。