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We propose a novel exact method to solve the probabilistic catalog matching problem faster than previously possible. Our new approach uses mixed integer programming and introduces quadratic constraints to shrink the problem by multiple orders of magnitude. We also provide a method to use a feasible solution to dramatically speed up our algorithm. This gain in performance is dependent on how close to optimal the feasible solution is. Also, we are able to provide good solutions by stopping our mixed integer programming solver early. Using simulated catalogs, we empirically show that our new mixed integer program with quadratic constraints is able to be set up and solved much faster than previous large linear formulations. We also demonstrate our new approach on real-world data from the Hubble Source Catalog. This paper is accompanied by publicly available software to demonstrate the proposed method.

Building on previous Bayesian approaches, we introduce a novel formulation of probabilistic cross-identification, where detections are directly associated to (hypothesized) astronomical objects in a globally optimal way. We show that this new method scales better for processing multiple catalogs than enumerating all possible candidates, especially in the limit of crowded fields, which is the most challenging observational regime for new-generation astronomy experiments such as the Rubin Observatory Legacy Survey of Space and Time. Here we study simulated catalogs where the ground truth is known and report on the statistical and computational performance of the method. The paper is accompanied by a public software tool to perform globally optimal catalog matching based on directional data.