[Paper Review] Near-Optimal Primal-Dual Algorithms for Quantity-Based Network Revenue Management

Rui Sun, Xinshang Wang+1 more|arXiv (Cornell University)|Nov 12, 2020

Supply Chain and Inventory Management25 references4 citations

TL;DR

This paper proposes a novel primal-dual algorithm for quantity-based network revenue management that achieves an o(√T) regret bound without solving any deterministic linear programs (DLPs), unlike prior methods. The approach uses a hybrid restart scheme combining probabilistic thresholding and primal-dual updates, enabling efficient real-time implementation while maintaining strong theoretical performance guarantees in large-scale settings.

ABSTRACT

We study the canonical quantity-based network revenue management (NRM) problem where the decision-maker must irrevocably accept or reject each arriving customer request with the goal of maximizing the total revenue given limited resources. The exact solution to the problem by dynamic programming is computationally intractable due to the well-known curse of dimensionality. Existing works in the literature make use of the solution to the deterministic linear program (DLP) to design asymptotically optimal algorithms. Those algorithms rely on repeatedly solving DLPs to achieve near-optimal regret bounds. It is, however, time-consuming to repeatedly compute the DLP solutions in real time, especially in large-scale problems that may involve hundreds of millions of demand units. In this paper, we propose innovative algorithms for the NRM problem that are easy to implement and do not require solving any DLPs. Our algorithm achieves a regret bound of $O(\log k)$, where $k$ is the system size. To the best of our knowledge, this is the first NRM algorithm that (i) has an $o(\sqrt{k})$ asymptotic regret bound, and (ii) does not require solving any DLPs.

Motivation & Objective

• To address the computational intractability of exact dynamic programming in large-scale network revenue management (NRM) problems.
• To eliminate the need for repeated DLP solving in existing asymptotically optimal NRM algorithms, which is computationally expensive in real-time systems.
• To design a practical, implementable algorithm that achieves near-optimal regret performance without relying on DLP solutions.
• To establish a new theoretical benchmark by proving an o(√T) regret bound without DLP computation.

Proposed method

• Introduces a hybrid restart algorithm (ALG6) that divides the time horizon into epochs and applies different subroutines in each.
• Uses LP-based thresholding (Algorithm 5) in the first U epochs to make probabilistic acceptance decisions based on a single initial DLP solution.
• Applies a primal-dual algorithm (Algorithm 3) in the final S−U+1 epochs, which dynamically updates dual variables and makes decisions based on current capacity and demand.
• Employs a restart schedule where each epoch begins with updated remaining capacity and runs a localized version of the algorithm to maintain adaptability.
• Leverages a single initial DLP solution to set thresholding probabilities in phase I, avoiding repeated DLP solves throughout the horizon.
• Combines phase I thresholding with phase II primal-dual updates to balance exploration and exploitation in dynamic capacity allocation.

Experimental results

Research questions

• RQ1 Can a primal-dual NRM algorithm achieve o(√T) regret without solving any DLPs?
• RQ2 Is it possible to design a scalable, real-time NRM algorithm that avoids repeated DLP computation while maintaining strong theoretical performance?
• RQ3 How can a hybrid restart strategy combining thresholding and primal-dual updates improve regret bounds in large-scale NRM problems?
• RQ4 What is the impact of using a single DLP solution at the start versus repeated solving on regret and computational efficiency?

Key findings

• The proposed algorithm achieves an o(√T) regret bound, which is the first such result in NRM that does not require solving any DLPs.
• The algorithm is computationally efficient and suitable for real-time deployment in high-frequency environments such as e-commerce platforms.
• By using a single initial DLP solution and avoiding repeated solving, the method significantly reduces computational overhead compared to prior DLP-reliant approaches.
• The hybrid restart strategy enables strong theoretical performance while maintaining practical implementability across diverse problem scales.