Quantile optimization in semidiscrete optimal transport

Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study a variant in which the goal is to minimize a quantile of the cost, rather than the mean. For the semidiscrete setting, where one distribution is continuous and the other is discrete, we derive a complete characterization of the optimal transport plan and develop simulation-based methods to efficiently compute it. One particularly novel aspect of our approach is the efficient computation of a tie-breaking rule that preserves marginal distributions. In the context of geographical partitioning problems, the optimal plan is shown to produce a novel geometric structure.

💡 Research Summary

The paper introduces a novel objective for optimal transport (OT) problems: instead of minimizing the expected transportation cost, it seeks to minimize a chosen quantile (α‑quantile) of the cost distribution. The authors focus on the semidiscrete setting, where the source distribution X is continuous and the target distribution Y is discrete, a scenario common in applications such as geographical partitioning, logistics, and economics.

The first technical contribution is a feasibility analysis for a fixed threshold t and quantile level α. By extending Farkas’ lemma to infinite dimensions, the authors derive a dual system involving Lagrange multipliers (ζ, ϕ, ψ). This dual system can be reduced to a finite‑dimensional, unconstrained, convex optimization problem (equation 11) that depends only on ζ and ψ. The system is feasible if and only if the optimal value of this problem is zero; otherwise it is infeasible. This result provides a clean certificate of feasibility for the original infinite‑dimensional constraints on the joint distribution π.

Building on this, the quantile‑optimization problem is tackled. The goal is to find the smallest t such that the α‑quantile of the cost does not exceed t. The authors show that this is equivalent to solving a root‑finding problem where t is expressed as a function of the optimal ζ and ψ obtained from (11). Unlike classical mean‑minimizing OT, which requires solving a single linear program, the quantile version requires solving a finite‑dimensional convex program at each iteration of the root‑finding routine.

A crucial structural difference from classical OT is the loss of the Monge property. In the optimal plan, the conditional distribution π(x,·) may need to split mass among several target points because the minimizer of the expression ζ · 1{c(x,k)≤t}+ψ_k can be non‑unique. The paper proves that a “tie‑breaking rule”—a probabilistic mechanism that assigns the split while preserving the prescribed marginal p of Y—always exists. This rule is essential to construct a feasible joint distribution that attains the desired quantile.

To compute both the optimal quantile and the tie‑breaking rule, the authors propose simulation‑based algorithms. For the quantile itself they use Sample Average Approximation (SAA), which reduces the problem to solving a linear program based on sampled X values; this yields an estimator that converges at the canonical O(N⁻¹ᐟ²) rate. For the tie‑breaking rule they employ Stochastic Approximation (SA), which efficiently handles the large number of constraints that arise when enforcing the marginal of Y. Both methods require only the ability to sample from X, not its explicit density.

The paper validates the methodology on a geographical partitioning example. In the classical mean‑minimizing semidiscrete OT, the optimal regions form an additively weighted Voronoi diagram. By contrast, the quantile‑optimizing solution generates a partially randomized partition: some region boundaries become stochastic, leading to a novel geometric structure not previously observed in computational geometry. Empirically, this structure reduces the probability of extremely long travel times, illustrating the risk‑averse advantage of quantile optimization.

In summary, the contributions are: (1) the first formalization of quantile‑based objectives in OT; (2) a complete analytical characterization of feasibility and optimality in the semidiscrete case via an infinite‑dimensional duality argument; (3) identification of a root‑finding subproblem and a probabilistic tie‑breaking rule as the two computational bottlenecks; (4) simulation‑based algorithms with provable O(N⁻¹ᐟ²) convergence for both components; and (5) discovery of a new, partially random geometric partitioning structure with practical implications for risk‑averse planning. The work bridges optimal transport theory, stochastic optimization, and computational geometry, offering tools that are likely to be useful for operations researchers, data scientists, and applied mathematicians.