Competitive Online Optimization under Inventory Constraints
Qiulin Lin, Hanling Yi, John Pang, Minghua Chen, Adam Wierman, Michael Honig, Yuanzhang Xiao
TL;DR
The paper addresses online optimization with inventory constraints (OOIC), where a decision maker must allocate a fixed inventory Delta over an unknown horizon T in the face of sequential concave revenue functions g_t. It introduces the CR-Pursuit framework, which at each round selects actions to maintain an offline-to-online revenue ratio close to a chosen constant pi, and proves that the unique pi solving Φ_Δ(pi)=Δ yields the optimal competitive ratio among deterministic algorithms up to problem-dependent factors. The analysis shows a fundamental lower bound of ln θ + 1 (with θ = M/m) and provides upper bounds that scale by a problem-dependent constant c, making CR-Pursuit optimal up to multiplicative factors for general OOIC, including one-way trading and price-elasticity generalizations. The results unify and extend prior work, yield near-optimal online algorithms for classic and generalized one-way trading, and offer a path toward beyond-worst-case and randomized refinements with practical applications in power markets, spectrum trading, and online advertising.
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 $\ln θ+1$, where $θ$ is the ratio between the maximum and minimum base prices.