Motivated by the statistical analysis of the discrete optimal transport problem, we prove distributional limits for the solutions of linear programs with random constraints. Such limits were first obtained by Klatt, Munk, & Zemel (2022), but their expressions for the limits involve a computationally intractable decomposition of $\mathbb{R}^m$ into a possibly exponential number of convex cones. We give a new expression for the limit in terms of auxiliary linear programs, which can be solved in polynomial time. We also leverage tools from random convex geometry to give distributional limits for the entire set of random optimal solutions, when the optimum is not unique. Finally, we describe a simple, data-driven method to construct asymptotically valid confidence sets in polynomial time.

翻译：受离散最优传输问题统计分析启发，我们证明了具有随机约束的线性规划解的分布极限。这类极限最早由Klatt、Munk和Zemel（2022）给出，但其表达式涉及将$\mathbb{R}^m$分解为指数级数量凸锥的计算不可解过程。我们提出一种基于辅助线性规划的新极限表达式，可在多项式时间内求解。同时利用随机凸几何工具，当最优解不唯一时，给出随机最优解全集的分布极限。最后，我们描述一种简单数据驱动方法，用于在多项式时间内构造渐近有效的置信集。