Recent advances in technology for abductive reasoning, or inference to the best explanation, encourage the application of abduction to real-life commonsense reasoning problems. This paper describes Etcetera Abduction, a new implementation of logical abduction that is both grounded in probability theory and optimized using contemporary linear programming solvers. We present a Weighted Max-SAT formulation of Etcetera Abduction, which allows us to exploit highly advanced technologies developed in the field of SAT and Operations Research. Our experiments demonstrate the scalability of our proposal on a large-scale synthetic benchmark that contains up to ten thousand axioms, using one of the stateof-the-art mathematical optimizers developed in these fields. This is the first work to evaluate a SAT-based approach to abductive reasoning at this scale. The inference engine we developed has been made publicly available.