Beyond a Single Optimal Design: A Dynamical Systems Characterization of Online Allocation with Convex Costs

2026-09-18

AI Summary

本文研究在线分配的凸成本问题，通过动力系统方法表征最优设计空间，揭示了多种最优定价策略，包括动态和静态定价混合，并在不同场景下提供更优保证。

Abstract

We study online allocation with convex costs (\OACC), a generalization of limited-supply models in which additional resources can be produced dynamically at convex cost. Prior work largely focuses on constructing a single algorithm that achieves strong competitive guarantees. In contrast, our main contribution is a structural characterization of the optimal design space of reserve functions (i.e., normalized pricing rules) for \OACC, yielding a family of optimal online algorithms. Using a principled dynamical-systems approach, we show that optimal online algorithms arise as solutions of a nonlinear eigenvalue problem: the optimal competitive ratio corresponds to the dominant eigenvalue, and the associated eigenfunction determines the optimal reserve function. This perspective establishes the existence of infinitely many optimal designs attaining the best possible competitive ratio, thereby unifying and generalizing a broad class of prior algorithms. Beyond unification, our characterization uncovers new structural phenomena, including the ability to flexibly mix dynamic and static pricing without loss of optimality and the construction of universally competitive algorithms under unknown or stochastic costs. We further extend the analysis to more structured settings such as supply-oblivious arrivals and hard supply constraints, obtaining sharper guarantees that improve upon previous bounds.

Problem

• online allocation
• convex costs
• optimal design space
• competitive ratio
• eigenvalue problem

Innovation

• dynamical-systems approach
• nonlinear eigenvalue problem
• optimal reserve function
• competitive ratio