Discreteness to Convexity: Promotion Planning via Simplotope Triangulation

TL;DR

This paper presents a novel convex hull-based reformulation for promotion planning that significantly accelerates computation, enabling large-scale supermarket pricing optimization with improved accuracy and efficiency.

Contribution

It introduces a new convexification approach using simplotope triangulation for supermodular functions, extending previous methods to multiple price levels and nonlinear effects.

Findings

Average solve time reduced from 434 to 0.06 seconds for 10 products and 5 price levels

Extended tight LP relaxation conditions from two to multiple price levels

Applicable to nonlinear discrete optimization and revenue management problems

Abstract

Price promotion optimization is a computationally challenging problem central to supermarket operations, requiring simultaneous pricing decisions across multiple products and periods. This paper introduces a novel formulation for supermodular functions and univariate compositions using explicit convex hull descriptions derived from simplotope triangulations, departing from prior reliance on rectangular domains. Leveraging this reformulation with Gurobi, we achieve substantial performance gains, with average solve times for problems with 10 products and 5 price levels reducing from 434 to 0.06 seconds, enabling significant instance scaling. We demonstrate conditions for a tight linear programming relaxation extending previous results from two to multiple price levels and from additive to multiplicative historical effects. Our approach is broadly applicable to nonlinear discrete optimization, and we contribute techniques for convexifying compositions of arbitrary univariate functions and a framework for convexifying a superclass of L natural functions, providing powerful tools for revenue management. This work advances the tractability of price promotion optimization, offering a practical and theoretically grounded solution for large-scale supermarket operations.