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Constrained Assortment and Price Optimization under Generalized Nested Logit Models

Hoang Giang Pham, Tien Mai

Abstract

We study assortment and price optimization under the generalized nested logit (GNL) model, one of the most general and flexible modeling frameworks in discrete choice modeling. Despite its modeling advantages, optimization under GNL is highly challenging: even the pure assortment problem is NP-hard, and existing approaches rely on approximation schemes or are limited to simple cardinality constraints. In this paper, we develop the first exact and near-exact algorithms for constrained assortment and joint assortment--pricing optimization (JAP) under GNL. Our approach reformulates the problem into bilinear and exponential-cone convex programs and exploits convexity, concavity, and submodularity properties to generate strong cutting planes within a Branch-and-Cut framework (B\&C). We further extend this framework to the mixed GNL (MGNL) model, capturing heterogeneous customer segments, and to JAP with discrete prices. For the continuous pricing case, we propose a near-exact algorithm based on piecewise-linear approximation (PWLA) that achieves arbitrarily high precision under general linear constraints. Extensive computational experiments demonstrate that our methods substantially outperform state-of-the-art approximation approaches in both solution quality and scalability. In particular, we are able to solve large-scale instances with up to 1000 products and 20 nests, and to obtain near-optimal solutions for continuous pricing problems with negligible optimality gaps. To the best of our knowledge, this work resolves several open problems in assortment and price optimization under GNL.

Constrained Assortment and Price Optimization under Generalized Nested Logit Models

Abstract

We study assortment and price optimization under the generalized nested logit (GNL) model, one of the most general and flexible modeling frameworks in discrete choice modeling. Despite its modeling advantages, optimization under GNL is highly challenging: even the pure assortment problem is NP-hard, and existing approaches rely on approximation schemes or are limited to simple cardinality constraints. In this paper, we develop the first exact and near-exact algorithms for constrained assortment and joint assortment--pricing optimization (JAP) under GNL. Our approach reformulates the problem into bilinear and exponential-cone convex programs and exploits convexity, concavity, and submodularity properties to generate strong cutting planes within a Branch-and-Cut framework (B\&C). We further extend this framework to the mixed GNL (MGNL) model, capturing heterogeneous customer segments, and to JAP with discrete prices. For the continuous pricing case, we propose a near-exact algorithm based on piecewise-linear approximation (PWLA) that achieves arbitrarily high precision under general linear constraints. Extensive computational experiments demonstrate that our methods substantially outperform state-of-the-art approximation approaches in both solution quality and scalability. In particular, we are able to solve large-scale instances with up to 1000 products and 20 nests, and to obtain near-optimal solutions for continuous pricing problems with negligible optimality gaps. To the best of our knowledge, this work resolves several open problems in assortment and price optimization under GNL.
Paper Structure (53 sections, 9 theorems, 98 equations, 6 figures, 9 tables)

This paper contains 53 sections, 9 theorems, 98 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Paper Contributions.
  3. Paper outline:
  4. Literature Review
  5. Assortment Optimization under GNL Model
  6. Solution Methods for Assortment Optimization under GNL Model
  7. Bisection
  8. Convexity and submodularity.
  9. A tractable bisection subproblem and convergence.
  10. Convexification via Log-Transformation Reformulation
  11. Convexity and feasibility.
  12. Submodularity.
  13. Cutting Plane method.
  14. Joint Assortment and Pricing Optimization
  15. Problem Formulation
  16. ...and 38 more sections

Key Result

Proposition 1

Let $\beta>\max_{i\in[m]} \{r_i\}$ and define $r'_i:=\beta-r_i>0$ for all $i\in[m]$, then problem prob:GNL-assort is equivalent (optimization-wise) to the minimization problem

Figures (6)

  • Figure 1: Example of the non-unimodal revenue function $F(y)$ under the GNL model with fixed assortment.
  • Figure 2: PWLA of the exponential function $e^{w_{in}}$ for an error tolerance of $\epsilon = 0.05$.
  • Figure 3: Runtime provided by B&C approach (with OA cuts) when varying cross rate
  • Figure 4: Objective and runtime provided by B&C approach (with OA cuts) when varying assortment capacity on the GNL instances
  • Figure 5: Objective and runtime provided by B&C approach (with OA cuts) when varying assortment capacity on the MGNL instances
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 1: Minimization reformulation
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Example 1: Non-unimodality with One Product and Two Nests
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9