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Constrained Pricing under Finite Mixtures of Logit

Hoang Giang Pham, Tien Mai

TL;DR

The paper addresses constrained pricing under logit-based demand, introducing globally reliable optimization methods for MNL and finite mixtures of logit (FMNL) models. It develops a PTAS for constrained pricing under MNL via an exponential-cone reformulation and a bisection scheme, and extends to FMNL with a bilinear-convex reformulation that enables a specialized branch-and-bound algorithm whose complexity is exponential only in the number of segments $T$. Theoretical guarantees are complemented by extensive experiments showing that the proposed methods outperform baselines and that incorporating constraints and heterogeneity yields substantial revenue gains, while unconstrained pricing is a poor proxy under constraints. Overall, the work bridges practical pricing requirements with expressive demand models and provides provably near-optimal solutions for constrained pricing in realistic settings.

Abstract

The mixed logit model is a flexible and widely used demand model in pricing and revenue management. However, existing work on mixed-logit pricing largely focuses on unconstrained settings, limiting its applicability in practice where prices are subject to business or regulatory constraints. We study the constrained pricing problem under multinomial and mixed logit demand models. For the multinomial logit model, corresponding to a single customer segment, we show that the constrained pricing problem admits a polynomial-time approximation scheme (PTAS) via a reformulation based on exponential cone programming, yielding an $\varepsilon$-optimal solution in polynomial time. For finite mixed logit models with $T$ customer segments, we reformulate the problem as a bilinear exponential cone program with $O(T)$ bilinear terms. This structure enables a Branch-and-Bound algorithm whose complexity is exponential only in $T$. Consequently, constrained pricing under finite mixtures of logit admits a PTAS when the number of customer segments is bounded. Numerical experiments demonstrate strong performance relative to state-of-the-art baselines.

Constrained Pricing under Finite Mixtures of Logit

TL;DR

The paper addresses constrained pricing under logit-based demand, introducing globally reliable optimization methods for MNL and finite mixtures of logit (FMNL) models. It develops a PTAS for constrained pricing under MNL via an exponential-cone reformulation and a bisection scheme, and extends to FMNL with a bilinear-convex reformulation that enables a specialized branch-and-bound algorithm whose complexity is exponential only in the number of segments . Theoretical guarantees are complemented by extensive experiments showing that the proposed methods outperform baselines and that incorporating constraints and heterogeneity yields substantial revenue gains, while unconstrained pricing is a poor proxy under constraints. Overall, the work bridges practical pricing requirements with expressive demand models and provides provably near-optimal solutions for constrained pricing in realistic settings.

Abstract

The mixed logit model is a flexible and widely used demand model in pricing and revenue management. However, existing work on mixed-logit pricing largely focuses on unconstrained settings, limiting its applicability in practice where prices are subject to business or regulatory constraints. We study the constrained pricing problem under multinomial and mixed logit demand models. For the multinomial logit model, corresponding to a single customer segment, we show that the constrained pricing problem admits a polynomial-time approximation scheme (PTAS) via a reformulation based on exponential cone programming, yielding an -optimal solution in polynomial time. For finite mixed logit models with customer segments, we reformulate the problem as a bilinear exponential cone program with bilinear terms. This structure enables a Branch-and-Bound algorithm whose complexity is exponential only in . Consequently, constrained pricing under finite mixtures of logit admits a PTAS when the number of customer segments is bounded. Numerical experiments demonstrate strong performance relative to state-of-the-art baselines.
Paper Structure (52 sections, 4 theorems, 41 equations, 10 figures, 1 algorithm)

This paper contains 52 sections, 4 theorems, 41 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Technical Contributions.
  4. Literature Review
  5. Managerial Insights.
  6. Problem Formulations
  7. Model Setup
  8. Constrained Price Optimization Problem
  9. PTAS for Constrained Pricing under MNL
  10. PTAS via Bisection and Exponential Cone Programming
  11. Complexity Analysis
  12. Constrained Pricing under FMNL Models
  13. Bilinear-convex Reformulation
  14. A B&B Scheme for the Bilinear--Convex Program
  15. Node Relaxation (ECP & McCormick Envelope)
  16. ...and 37 more sections

Key Result

Proposition 1

Problem prob:PO-MNL-Sub is equivalent to the following ECP: where the exponential cone is defined as

Figures (10)

  • Figure 1: Revenue function under the FMNL model when $m=1$ and $T \in \{2,3,4\}$.
  • Figure 2: Our proposed methods vs. the gradient-based algorithm on the C&P-FMNL dataset.
  • Figure 3: Objective loss incurred by approximating multi-segment pricing problems with a single-segment (MNL) model.
  • Figure 4: Objective gap between projected unconstrained solutions and our B&B method.
  • Figure 5: B&B execution flow. Left: The tree structure. Right: Queue operations.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Proposition 1
  • Remark 1: Quasiconcavity under variable transformation
  • Proposition 2
  • Proposition 3
  • Remark 2: Pairwise constraints on prices
  • Theorem 1: Complexity of B&B for the constrained pricing under the FMNL model
  • Remark 3: Practical complexity
  • Remark 4: Comparison to the bisection--ECP scheme for MNL-based pricing
  • Remark 5: Comparison to the B&B complexity in ruben2022mnsc