Discrete Optimal Transport

Abstract

We introduce the Assignment Problem, which involves minimizing the cost of transporting goods from a finite number of sources to a finite number of targets. Due to the discrete nature of the assignment problem, a solution might be computed using a brute-force numerical approach; however, these are not efficient. In this thesis, we relax the assignment problem, and connect it to the infamous Kantorovich Problem and its Dual. Theoretically, the problem consists of maximizing concave functional under linear inequality constraints. We develop the needed theoretical background from Functional and Convex analysis to solve the assignment problem which allowed us to present more efficient numerical methods such as the Bertsekas’ auction algorithm, and the network simplex algorithm.