Constrained Assortment and Price Optimization under Generalized Nested Logit Models

TL;DR

This paper introduces the first exact algorithms for assortment and joint assortment-price optimization under the complex generalized nested logit model, significantly improving solution quality and scalability.

Contribution

Develops novel exact and near-exact algorithms for constrained assortment and joint pricing optimization under GNL, overcoming previous computational limitations.

Findings

Outperforms existing approximation methods in solution quality.

Successfully solves large-scale instances with up to 1000 products.

Achieves near-optimal solutions for continuous pricing with negligible gaps.

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.