Quantile optimization in semidiscrete optimal transport

2026-02-11

AI Summary

This study addresses the semi-discrete optimal transport problem under risk-sensitive settings by extending the classical objective of minimizing expected transport cost to minimizing a quantile of the transport cost distribution. By integrating quantile optimization, semi-discrete optimal transport theory, and geometric analysis, the work provides the first complete characterization of quantile-optimal transport plans that respect prescribed marginal constraints. An efficient simulation-based algorithm is proposed to compute such plans, and a novel tie-breaking rule is introduced to ensure solution uniqueness. Furthermore, the analysis reveals new geometric structures in spatial partitioning induced by the quantile objective, offering both theoretical foundations and computational tools for risk-sensitive transportation and regional segmentation.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• quantile optimization
• optimal transport
• semidiscrete
• cost quantile
• marginal distributions

Innovation

• quantile optimization
• semidiscrete optimal transport
• tie-breaking rule
• geometric structure
• simulation-based computation