Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications, such as online auction, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former one typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithm only requires first-order computations but induces a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm, that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only O(log log T) times over the time horizon T. Moreover, when we are allowed to solve LPs only M times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly O(T^((1/2)^(M-1)) regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.

翻译：在线线性规划（OLP）因其广泛的应用（如在线拍卖、网络收益管理、订单履行和广告投放）而受到研究者和从业者的广泛关注。现有的OLP算法可分为两类：基于线性规划（LP）的算法和无线性规划（LP-free）算法。前者通常能保证更好的性能，但需要求解大量线性规划问题，计算成本较高。相比之下，无线性规划算法仅需一阶计算，但性能较差。本研究通过提出一种性能优异的算法，弥合了这两种极端方法之间的差距：该算法仅在少数选定的时间点求解线性规划，而在其他时间点进行一阶计算。具体而言，当输入数据来自未知的有限支撑分布时，所提算法在时间范围T内仅需O(log log T)次求解线性规划，即可实现常数级遗憾（即使对于困难的“退化”情况）。此外，当仅允许求解线性规划M次时，我们设计了相应的调度方案，使算法能保证近乎O(T^((1/2)^(M-1)))的遗憾。本研究强调了在销售周期开始和结束时进行求解的价值，并提供了一个新颖的框架，以证明所提策略在不同不频繁求解调度下的性能保证。数值实验验证了所提算法的有效性。