Optimal Transport: A Multi-Objective View

• The paper reinterprets discrete optimal transport as a multi-objective matrix allocation problem characterized by Pareto efficient solutions.
• Balanced solutions enforce both feasibility and fairness by simultaneously satisfying primary (column) and secondary (row) constraints.
• Entropic regularization is applied to smooth the optimization landscape, linking classical linear programming with a multi-objective framework for efficient computation.

The optimal transport objective is a central mathematical framework for modeling allocation and assignment problems, characterized by finding a mapping (or transport plan) between elements of two spaces that minimizes an associated cost under specified constraints. The paper "A Multi-Objective Interpretation of Optimal Transport" (Schumacher, 2017) provides a distinctive reinterpretation of the discrete optimal transport objective in terms of multi-objective optimization, introduces the concept of balanced solutions, establishes precise mathematical links, and discusses computation through regularization. This perspective not only elucidates the structure of classical optimal transport but also unifies it with paradigms in multi-criteria decision making, economics, statistics, and beyond.

1. Multi-Objective Matrix Allocation and the Optimal Transport Objective

The paper interprets the discrete optimal transport problem as a special case of a multi-objective matrix allocation problem (MOMA). In this setting, the decision variables are arranged into an $n \times m$ matrix $X = (x_{ij})$, with $n$ agents (e.g., producers, criteria, classes) and $m$ tasks, consumers, or alternative outcomes.

The MOMA’s formal structure is:

• Each row $i$ (agent) has its own objective function,

$\Psi_{(i)}(X_{i\cdot}) = \sum_j g_{ij}(x_{ij}).$

• The primary (feasibility) constraints correspond to column sums,

$\sum_i x_{ij} = c_j, \qquad x_{ij} \geq 0,$

ensuring each task or outcome is fully allocated.

• Pareto efficiency is defined relative to these column constraints: a solution $X$ is Pareto efficient if no feasible $Y$ improves one $\Psi_{(i)}$ without worsening others.

Rather than reducing to a single aggregate objective by scalarization, the paper demonstrates that Pareto efficient solutions for the multi-objective problem correspond, via a scalarization and logarithmic transformation, to standard optimal transport solutions. Notably, for linear settings, using the change of variables $a_{ij} = \log b_{ij}$ connects the classical linear optimal transport cost to the multi-objective version, with $f_{ij}'(x) = \log g_{ij}'(x)$.

2. Balanced Solutions and Constraint Structure

Balanced solutions arise from the imposition of an additional (secondary) set of equality constraints, capturing fairness or exogenous rights. The precise problem definition is:

• Primary (column) constraints: for all $j$, $\sum_i x_{ij} = c_j$, $x_{ij} \geq 0$.
• Secondary (row) constraints: for all $i$, $\sum_j x_{ij} = r_i$.
• Global feasibility: $\sum_i r_i = \sum_j c_j$, ensuring the total assignment is possible.

A "balanced solution" is a Pareto efficient matrix $X$ (with respect to the primary constraints) that also satisfies the secondary constraints. In many applications, $c_j$ encodes feasibility requirements (e.g., resource, object, or workload totals imposed by external factors), and $r_i$ encodes individual entitlements, workloads, or fairness objectives.

Balanced solutions are unique in that they single out a point (or points) on the Pareto front that simultaneously fulfill all hard resource constraints and fairness expectations. In the context of optimal transport, every balanced solution (for appropriate linear $g_{ij}$) is in one-to-one correspondence with a solution to the discrete optimal transport problem.

3. Applications and Paradigmatic Examples

This theoretical framework unifies multiple paradigms:

• Task/Chore Allocation: Columns encode tasks to be fully assigned; rows encode agents’ predetermined workloads. The objectives $g_{ij}$ model (dis)utility, producing assignments that are both optimal and fair.
• Statistical Classification: Assignment of objects (columns) to categories (rows), with constraints on class proportions and full classification, realizing optimal assignments under probabilistic or cost-based objectives.
• Risk Sharing: Jointly allocating uncertain payoffs (columns) among participants (rows), with constraints representing available amounts per outcome and fixed entitlements, as in cooperative insurance.

Thus, the theory of balanced solutions encompasses problems in economics (resource and income allocation), operations research (workload balancing), and statistics (classification with proportional constraints).

4. Computation and Regularization

Classical optimal transport is a linear program, but its direct solution is often computationally expensive. Regularization—typically with entropic smoothing—yields a strictly convex (or strictly concave) objective that is computationally efficient:

• Entropic regularization: For the single-objective (SOMA) optimal transport problem, regularize $f_{ij}(x; \eta) = a_{ij} x + \eta f_1(x)$, choosing $f_1(x) = -x \log x$. This strictly concave perturbation enables iterative methods like IPFP (Iterative Proportional Fitting Procedure) and smooths the optimization landscape.
• Multi-objective regularization: Under the transformation $f_{ij}'(x) = \log g_{ij}'(x)$, entropic regularization for the single-objective version matches isoelastic (constant relative risk aversion, CRRA) regularization for the multi-objective model, yielding $g_{ij}(x; \eta) = b_{ij} \frac{x^{1-\eta}}{1-\eta}$. This correspondence provides a theoretical link between the computational treatment of regularized OT and multi-objective problems.

The paper details the iteration:

$$\beta_j = \max_i(\alpha_i b_{ij}), \qquad \alpha_i^{\mathrm{new}} = \min_j(\beta_j / b_{ij}),$$

and demonstrates, via nonlinear Perron-Frobenius theory, the convergence of regularized algorithms.

5. Mathematical Characterization and Duality

Key mathematical results include:

• Balanced solution conditions (linear case): For given $b_{ij}$, there exist positive dual variables $\alpha_i, \beta_j$ such that

$$\begin{cases} x_{ij} = 0 \implies \alpha_i b_{ij} \leq \beta_j,\ x_{ij} > 0 \implies \alpha_i b_{ij} = \beta_j, \end{cases}$$

together with the column and row sum constraints.

• Translation to OT problem: The identification $a_{ij} = \log b_{ij}$ maps the multi-objective balanced solution to the OT linear program:

$$\max \sum_{i,j} a_{ij} x_{ij} \quad \text{subject to} \quad \sum_i x_{ij} = c_j,\ \sum_j x_{ij} = r_i,\ x_{ij} \geq 0.$$

• Regularization duality: Entropic regularization in the (SOMA) OT problem induces isoelastic regularization in the multi-objective context. The regularized optimality conditions and update schemes are mathematically linked through pointwise transformations.

This dual view exposes a deeper structure: optimal transport is not just an allocation problem, but a special, "balanced" selection within a multi-criterion, Pareto-efficient set.

6. Conceptual and Practical Implications

The multi-objective formulation of OT enables direct reasoning about trade-offs between agents or criteria, especially under complex constraints. The characterization of balanced solutions links the physical assignment of resources to notions of fairness, efficiency, and feasibility that are relevant in diverse domains such as economics, risk management, and statistical inference.

Regularization, particularly entropic regularization, is not only a computational device but is also interpretable in terms of risk, fairness, and elasticity of objectives in the multi-objective framework. The iterative methods and convergence theory developed in the paper provide guidance for practical solution of large-scale allocation and transport problems, connecting advances in numerical optimization to the structure of balanced multi-objective problems.

7. Summary Table: Key Correspondences

Concept	Multi-Objective Formulation	Optimal Transport Interpretation
Decision variables	Matrix $X = (x_{ij})$	Transport plan $X = (x_{ij})$
Primary constraints	Column sums: $\sum_i x_{ij} = c_j$	Marginal constraints for consumers/tasks
Secondary constraints	Row sums: $\sum_j x_{ij} = r_i$	Marginal constraints for agents/suppliers
Objective functions	$\Psi_{(i)}(X_{i\cdot})$	Linear cost or reward function
Balanced solution	Pareto efficient + row constraints	OT solution with both marginals fixed
Regularization	Isoelastic (CRRA) utility	Entropic (–$x$ log $x$) penalty
Dual variables	$(\alpha_i, \beta_j)$ (utilities)	$(\alpha_i, \beta_j)$ (potentials in dual LP)

The paper thus frames optimal transport not only as a single-objective linear program, but as a canonical instance of a balanced solution on the Pareto frontier of multi-objective matrix allocation, with explicit structural, computational, and interpretive consequences. This multi-objective view unifies disparate domains and motivates efficient solution methods leveraging regularization and duality.