Tricks from the Trade for Large-Scale Markdown Pricing: Heuristic Cut Generation for Lagrangian Decomposition

TL;DR

The paper tackles large-scale markdown pricing under linking constraints by enhancing a Lagrangian decomposition framework with novel heuristics. It introduces a maximum-violation cut-generation strategy that efficiently combines past solved solutions to produce effective cuts, integrated into an extended cutting-plane procedure to deliver high-quality near-optimal prices within tight time windows. Through extensive real-world evaluation at Zalando, the approach yields consistent commercial gains, reducing dual gaps rapidly and achieving multi-million euro improvements in weekly profit and GMV, with causal analyses validating observable effects. The work demonstrates practical impact by balancing computational speed with solution quality in a production setting and clarifies trade-offs between disaggregated and aggregated cut formulations.

Abstract

In automated decision making processes in the online fashion industry, the 'predict-then-optimize' paradigm is frequently applied, particularly for markdown pricing strategies. This typically involves a mixed-integer optimization step, which is crucial for maximizing profit and merchandise volume. In practice, the size and complexity of the optimization problem is prohibitive for using off-the-shelf solvers for mixed integer programs and specifically tailored approaches are a necessity. Our paper introduces specific heuristics designed to work alongside decomposition methods, leading to almost-optimal solutions. These heuristics, which include both primal heuristic methods and a cutting plane generation technique within a Lagrangian decomposition framework, are the core focus of the present paper. We provide empirical evidence for their effectiveness, drawing on real-world applications at Zalando SE, one of Europe's leading online fashion retailers, highlighting the practical value of our work. The contributions of this paper are deeply ingrained into Zalando's production environment to its large-scale catalog ranging in the millions of products and improving weekly profits by millions of Euros.

Figures (6)

• Figure 1: After evaluating two multipliers $\lambda^1$. $\lambda^2$, we obtain solutions $X^1$ and $X^2$. Their implied inequalities reside on the convex shape, which is implicitly given and not known. The two generated cuts provide a lower bound for $\mu^\ast$ given by $\mu^2$ (left). An additional heuristic cut can be created by arbitrary solutions (right). If it cuts off the next multiplier candidate $\mu^2$, we improve the gap without evaluating $\textnormal{LR}(\lambda)$. This cut will most likely not be a facet of the unknown boundary.
• Figure 2: Solid lines and left y axis: relative dual gap of the cutting plane problem over number of outer loop cutting plane iterations. Dashed lines and right axis: L2 norm of Lagrangian multipliers $\lambda^{j+1}$ selected in the cutting plane problem. Each iteration is one new solution of the Lagrangian dual. Baseline uses no heuristic to add cuts to the cutting plane problem. For details on the heuristics see Section \ref{['sec:heuristics']}.
• Figure 3: Example of applying the partial aggregation model or the maximum violation strategy to a fixed set of dual solutions. For maximum violation (red), the heuristic is applied repeatedly, and the observed dual gap $d_j$ and elapsed time are noted. For the disaggregated formulation (blue), different levels of aggregation are solved independently, and the elapsed solving time and dual gap is noted.
• Figure 4: Elapsed time to reach different gaps to best dual bound. For 86 different initial problems (a set of solutions to the relaxed dual problems, iterations from the Lagrangian decomposition) the partial aggregation and heuristic strategies were applied to find primal bounds. For partial aggregation, different group sizes were tested and used to determine the minimum time to gap. For the maximum violation heuristic, the heuristic was applied repeatedly, and the cumulative time to reach the target gap was recorded.
• Figure 5: Full Lagrangian descent using different solving strategies. For 3 different problems (different weeks), the maximum violation strategy was applied (blue, 100 times per solving of the LR), and compared to baseline (red, baseline cutting plane), partially aggregated small (orange, creating 2000 random groups), and partially aggregated large (cyan, using 5000 groups). For each problem the dual gap $d_j$ was noted after each solving of the Lagrangian relaxation.