Toward Globally Optimal State Estimation Using Automatically Tightened Semidefinite Relaxations

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, specific handcrafted redundant constraints are added to the relaxation in order to improve its tightness, which is usually a requirement for obtaining or certifying globally optimal solutions. 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 entire process do we have to manually find redundant constraints. We showcase the effectiveness of the approach by providing new insights on two classical robotics problems: 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.