General auction method for real-valued optimal transport

The auction method developed by Bertsekas in the late 1970s is a relaxation technique for solving integer-valued assignment problems. It resembles a competitive bidding process, where unsatisfied persons (bidders) attempt to claim the objects (lots) offering the best value. By transforming integer-valued transport problems into assignment problems, the auction method can be extended to compute optimal transport solutions. We propose a more general auction method that can be applied directly to real-valued transport problems. We prove termination and provide a priori error bounds for the general auction method. Our numerical results indicate that the complexity of the general auction is roughly comparable to that of the original auction method, when the latter is applicable.