Competitive Online Optimization under Inventory Constraints

TL;DR

This paper introduces extsf{CR-Pursuit}, a new algorithmic framework for online optimization with inventory constraints, achieving near-optimal competitive ratios and unifying previous results while extending to more general revenue functions.

Contribution

The paper presents extsf{CR-Pursuit}, a novel algorithmic approach that attains minimal competitive ratios for inventory-constrained online optimization, simplifying and extending prior work.

Findings

Achieves near-optimal competitive ratio within 1/3 of the lower bound for one-way trading with price elasticity.

Unifies and simplifies existing results for standard one-way trading problems.

Establishes new bounds for generalized revenue functions, including concave cases.

Abstract

This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization that is a generalization of the classic one-way trading problem and has a wide range of applications. We present a new algorithmic framework, \textsf{CR-Pursuit}, and prove that it achieves the minimal competitive ratio among all deterministic algorithms (up to a problem-dependent constant factor) for inventory-constrained online optimization. Our algorithm and its analysis not only simplify and unify the state-of-the-art results for the standard one-way trading problem, but they also establish novel bounds for generalizations including concave revenue functions. For example, for one-way trading with price elasticity, the \textsf{CR-Pursuit} algorithm achieves a competitive ratio that is within a small additive constant (i.e., 1/3) to the lower bound of $\ln \theta+1$, where $\theta$ is the ratio between the maximum and minimum base prices.