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Discreteness to Convexity: Promotion Planning via Simplotope Triangulation

Taotao He, Mohit Tawarmalani

TL;DR

The paper tackles price promotion optimization with discrete price ladders across many products and periods by introducing a simplotope-based convex-hull reformulation that yields ideal MILP representations. By lifting univariate functions to the vertices of a unary simplotope and applying staircase inequalities, the authors obtain tight convex relaxations for supermodular compositions, enabling exact linear representations for complex cross-item and cross-period interactions as well as multiplicative historic effects. They develop both unary and logarithmic binarization schemes, provide fast separation procedures, and demonstrate substantial computational gains over nonlinear baselines, achieving speeds such as reducing average solve times from hundreds of seconds to seconds for multi-product instances. The results extend the tractability of price promotion optimization to large-scale, real-world settings and establish broad applicability to nonlinear discrete optimization, including connections to $\mathop{\mathrm{L^\natural}}$-convexity and related discrete-convex analysis. Overall, the work offers a rigorous, scalable framework for convexifying a wide class of nonlinear discrete objectives, with concrete implications for revenue management in retail operations.

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.

Discreteness to Convexity: Promotion Planning via Simplotope Triangulation

TL;DR

The paper tackles price promotion optimization with discrete price ladders across many products and periods by introducing a simplotope-based convex-hull reformulation that yields ideal MILP representations. By lifting univariate functions to the vertices of a unary simplotope and applying staircase inequalities, the authors obtain tight convex relaxations for supermodular compositions, enabling exact linear representations for complex cross-item and cross-period interactions as well as multiplicative historic effects. They develop both unary and logarithmic binarization schemes, provide fast separation procedures, and demonstrate substantial computational gains over nonlinear baselines, achieving speeds such as reducing average solve times from hundreds of seconds to seconds for multi-product instances. The results extend the tractability of price promotion optimization to large-scale, real-world settings and establish broad applicability to nonlinear discrete optimization, including connections to -convexity and related discrete-convex analysis. Overall, the work offers a rigorous, scalable framework for convexifying a wide class of nonlinear discrete objectives, with concrete implications for revenue management in retail operations.

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.
Paper Structure (33 sections, 21 theorems, 143 equations, 5 figures, 3 tables, 4 algorithms)

This paper contains 33 sections, 21 theorems, 143 equations, 5 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Description and Contributions
  3. Opportunity: tighter formulations via simplotope
  4. Discrete domains and the unary binarization simplotope
  5. Modeling cross-item effects: advantage of simplotope over hypercube
  6. Modeling Composite Effects
  7. From the unary simplotope to an ideal formulation
  8. Supermodular composition
  9. Modeling via a logarithmic binarization
  10. Strong Formulations for Price Promotion Optimization
  11. Compact formulations
  12. Linear demand model
  13. Log-log demand model
  14. Stronger formulations via fast separations and their theoretical performance
  15. Computational Results
  16. ...and 18 more sections

Key Result

Proposition 1

If $d_2 = 1$ then an MIP formulation of eq:example is given by $\bigl\{(\boldsymbol{x},\mu,\boldsymbol{z}) \in \mathop{\mathrm{\mathbb{R}}}\nolimits^2 \times \mathop{\mathrm{\mathbb{R}}}\nolimits \times \{0,1\}^{d_1+d_2} \bigm|eq:unary-example\text{ and }eq:generalziedMC \bigr\}$.

Figures (5)

  • Figure 1: The grid representation of a discrete lattice and vertices of a simplotope
  • Figure 2: Grid: $\{0,3,5\}\times\{0,2\}$. Function (resp. convex envelope) at $(x,y) = (2,1)$ evaluate to $22$ (resp. $12$).
  • Figure 3: The geometry behind the inequalities \ref{['eq:generalziedMC1']} and \ref{['eq:generalziedMC2']}.
  • Figure 4: Subdivision of prism into three pyramids, where the blue one does not appear in Figure \ref{['fig:staircase-ineq']}.
  • Figure 5: Simplicial subdivisions of a box domain associated with the locally Lovász extension and staircase extension

Theorems & Definitions (28)

  • Example 1
  • Example 2
  • Proposition 1
  • Example 3
  • Proposition 2
  • Lemma 1
  • Definition 1: topkis2011supermodularity
  • Example 4
  • Theorem 1
  • Proposition 3
  • ...and 18 more