发表机构
University of Alberta; University of Ottawa（阿尔伯塔大学; 渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过动力学系统方法刻画凸成本在线分配的最优储备函数设计空间，证明最优算法源于非线性特征值问题，存在无限多个最优设计，并扩展到供应不可知等设置以改进竞争比界限。
AI 中文摘要
我们研究具有凸成本的在线分配（\OACC），这是有限供应模型的一种推广，其中额外资源可以以凸成本动态生产。先前的工作主要集中于构造能够实现强竞争保证的单一算法。相比之下，我们的主要贡献是对\OACC的最优设计空间（即归一化定价规则）的结构性刻画，从而产生一族最优在线算法。通过一种有原则的动力学系统方法，我们证明最优在线算法作为非线性特征值问题的解出现：最优竞争比对应于主特征值，而相关的特征函数决定了最优储备函数。这一视角确立了存在无限多个达到最佳可能竞争比的最优设计，从而统一并推广了一大类先前算法。除了统一之外，我们的刻画揭示了新的结构现象，包括在不损失最优性的情况下灵活混合动态和静态定价的能力，以及在未知或随机成本下构造普遍竞争算法。我们进一步将分析扩展到更具结构性的设置，如供应不可知到达和硬供应约束，获得了优于先前界限的更精确保证。
英文摘要
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.
