Knowing When to Stop Matters: A Unified Algorithm for Online Conversion under Horizon Uncertainty

TL;DR

This work studies online conversion with a fixed resource budget under dynamically changing prices, addressing horizon uncertainty through four models: OC-Known, OC-Notice, OC-Unknown, and OC-Prediction. It introduces a unified algorithm, PseudoMax, built on horizon-adaptive pseudo-cost functions and analyzed via online primal-dual techniques, capable of handling non-trivial box constraints. The paper derives tight competitive ratios for OC-Known with box constraints and optimal or best-known guarantees for OC-Notice and OC-Unknown, while presenting a learning-augmented extension that leverages horizon predictions to balance robustness and consistency. A practical energy-trading case study illustrates the framework’s potential for real-world decision-making under horizon uncertainty, with future directions including bi-directional trading and broader applicability of the pseudo-cost approach.

Abstract

This paper investigates the online conversion problem, which involves sequentially trading a divisible resource (e.g., energy) under dynamically changing prices to maximize profit. A key challenge in online conversion is managing decisions under horizon uncertainty, where the duration of trading is either known, revealed partway, or entirely unknown. We propose a unified algorithm that achieves optimal competitive guarantees across these horizon models, accounting for practical constraints such as box constraints, which limit the maximum allowable trade per step. Additionally, we extend the algorithm to a learning-augmented version, leveraging horizon predictions to adaptively balance performance: achieving near-optimal results when predictions are accurate while maintaining strong guarantees when predictions are unreliable. These results advance the understanding of online conversion under various degrees of horizon uncertainty and provide more practical strategies to address real world constraints.

Figures (6)

• Figure 1: Comparison of CR across different settings. OC-Known refers to the setting without box constraints, while OC-Known-b represents the same setting but with non-trivial box constraints.
• Figure 2: Illustration of the two-phase trading with box constraints. Step $\tau$ (black) separates the entire trading process into the proactive phase (white) and forced phase (gray).
• Figure 3: Comparison of the unified algorithm across three settings with rate limit.
• Figure 4: Comparison of the learning-augmented algorithm across different settings of $T_{\textsf{pred}}$ and $\lambda$.
• Figure 5: Performance of Algorithm \ref{['alg_RBP']} under various $\lambda$ and $T_{\textsf{pred}}$.

Theorems & Definitions (20)

• Definition 1: Pseudo-Cost Functions
• Theorem 1
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2: Earliest-Possible Switching Step $\tau_{\min}$
• proof
• Proposition 1: Structure of Worst-Case Instances
• Proposition 2: Reduction to Single-Period Force Trading
• Theorem 2