A Multi-Objective Interpretation of Optimal Transport

Abstract: This paper connects discrete optimal transport to a certain class of multi-objective optimization problems. In both settings, the decision variables can be organized into a matrix. In the multi-objective problem, the notion of Pareto efficiency is defined in terms of the objectives together with non-negativity constraints and with equality constraints that are specified in terms of column sums. A second set of equality constraints, defined in terms of row sums, is used to single out particular points in the Pareto efficient set which are referred to as "balanced solutions". Examples from several fields are shown in which this solution concept appears naturally. Balanced solutions are shown to be in one-to-one correspondence with solutions of optimal transport problems. As an example of the use of alternative interpretations, the computation of solutions via regularization is discussed.