Huge-Scale Assortment Optimization with Customer Choice: A Parallel Primal-Dual Approach

TL;DR

This work proposes a first-order primal-dual method, SPFOM, which requires only a small computational cost per iteration, achieves a provably near-optimal convergence rate, and can be readily extended to parallel computing environments.

Abstract

We study huge-scale assortment optimization problems to maximize expected revenue under customer choice, addressing a fundamental challenge in industries such as transportation, retail, and healthcare. The choice-based linear programming (CBLP) formulation provides a powerful framework for optimizing sales allocations across customer segments, yet traditional approaches often fail to solve CBLPs of huge scale (involving millions of customer choices) due to the lack of algorithmic designs that exploit problem structure. To overcome this computational bottleneck, we propose a first-order primal-dual method, SPFOM, which requires only a small computational cost per iteration, achieves a provably near-optimal convergence rate, and can be readily extended to parallel computing environments. Computational experiments demonstrate the computational and practical superiority of SPFOM over state-of-the-art solvers for large-scale linear programs. The framework is extended to a multi-period assortment optimization setting with inventory constraints, where SPFOM estimates global shadow prices that enhance classical bid-price control policies compared with benchmark methods such as market segment decomposition. Numerical experiments and a case study using real-world data from the ZOZOTOWN platform validate the practical effectiveness of SPFOM, highlighting its advantages in improving revenue performance while maintaining balanced inventory levels.

Figures (7)

• Figure 1: Framework of our modeling and algorithmic approach.
• Figure 2: Illustration of convergence and optimality behavior of SPFOM: The left figure (a) shows the convergence of SPFOM with $m=100$ and $n=10,000$. The right figure (b) compares the optimum values of SPFOM and PDLP for $m=100$ and $n \in [1000, 10000.0]$.
• Figure 3: Computing speed of SPFOM: The figure (a) compares the computing speed of SPFOM and PDLP in a small-size market where $m=100$ and $n \in [1000, 10000.0]$. The figure (b) shows the logarithmic computing time ($\log_{10}$) of both SPFOM and PDLP, averaged over several runs. The figure (c) displays the computing time in hours for SPFOM in a large-sized market where $n \in [10000.0, 1000000.0]$ and $m= \max \{100, n/1000\}$.
• Figure 4: We simulate a market with 100 batches of 1000 customers each, where 2 products are recommended per batch. The platform offers 40 products whose prices are drawn uniformly from $[1, 20]$ and initial inventories from $[1, 2000]$. Each customer's preference toward each product is independently drawn from $[0, 1]$. In every batch, the MSD policy partitions customers into 10 segments, whereas the global policy operates on the batch as a whole. All results reported in the figure are averaged over 30 independent simulation runs.
• Figure 5: Comparison of GO and MSD across 96 multi-period batches.

Theorems & Definitions (11)

• Lemma 1
• Theorem 1: Optimality Preservation
• Theorem 2: Linear Convergence
• Remark 1
• Corollary 1
• Lemma 2
• Corollary 2
• Theorem EC.1
• Lemma EC.1
• Lemma EC.2