# A Multi-Objective Interpretation of Optimal Transport

## TL;DR

This paper links discrete optimal transport to multi-objective optimization, introducing balanced solutions that correspond to optimal transport solutions, and discusses their computation via regularization.

## Contribution

It introduces the concept of balanced solutions in multi-objective optimization and establishes their equivalence with optimal transport solutions.

## Key findings

- Balanced solutions correspond one-to-one with optimal transport solutions.
- Examples from various fields illustrate the natural occurrence of balanced solutions.
- Regularization methods can be used to compute these solutions.

## 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.