A Dynamic Near-Optimal Algorithm for Online Linear Programming

TL;DR

This paper introduces a dynamic, near-optimal online algorithm for linear programming problems where constraints are revealed sequentially, improving decision-making in resource allocation under uncertainty.

Contribution

It proposes a learning-based, dynamic threshold algorithm that adapts over time, achieving near-optimal performance in online LPs with random arrival order.

Findings

Algorithm's competitiveness improves over previous methods.

Performance is near-optimal in worst-case scenarios.

Dynamic learning enhances decision accuracy.

Abstract

A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) in which the constraint matrix is revealed column by column along with the corresponding objective coefficient. In such a model, a decision variable has to be set each time a column is revealed without observing the future inputs and the goal is to maximize the overall objective function. In this paper, we provide a near-optimal algorithm for this general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Specifically, our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from the revealed columns in the previous period are used to determine the sequential decisions in the current period. Due to the feature of dynamic learning, the competitiveness of our algorithm improves over the past study of the same problem. We also present a worst-case example showing that the performance of our algorithm is near-optimal.