Constrained Pricing under Finite Mixtures of Logit

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.

💡 Research Summary

The paper addresses the practically important problem of price optimization under realistic constraints when demand is modeled by a mixed logit (finite‑mixture of multinomial logit, FMNL) model. While mixed‑logit captures customer heterogeneity through multiple latent segments, the resulting revenue function is highly non‑concave in prices, making global optimization difficult, especially when prices must satisfy bounds, budget, or other linear constraints.

Contributions for the single‑segment case (MNL).
The authors first consider the classic multinomial logit (MNL) model, which corresponds to a single customer segment. They show that the constrained pricing problem can be reformulated as an exponential‑cone program by introducing variables (z_i = \exp(-b_i p_i)) and expressing the revenue function and price bounds through exponential‑cone constraints. This reformulation isolates all non‑convexities inside the cone. By coupling the cone program with a simple bisection on the objective value, they obtain a polynomial‑time approximation scheme (PTAS): for any (\varepsilon>0) the algorithm returns a price vector whose revenue is within (\varepsilon) of the global optimum, and the running time is polynomial in the input size and (1/\varepsilon). This is the first PTAS for constrained MNL pricing; prior work relied on MILP formulations without polynomial‑time guarantees or on heuristics that lack global optimality.

Extension to finite‑mixture logit (FMNL).
The paper then tackles the more realistic FMNL model with (T) latent segments, each having its own preference parameters (a_{ti}) but sharing the same price‑sensitivity coefficients (b_i). The expected revenue is a weighted sum of segment‑specific MNL revenues. The authors introduce segment‑specific bilinear variables (y_{ti}=p_i \exp(-b_i p_i)) and rewrite the problem as a bilinear‑exponential‑cone program that contains only (O(T)) bilinear terms—one per segment. All remaining constraints (price bounds, linear portfolio constraints) are linear.

Algorithmic framework.
Exploiting this structure, they design a tailored Branch‑and‑Bound (B&B) algorithm. At each node the bilinear terms are relaxed using the exponential‑cone representation, providing an upper bound; branching is performed on the bilinear relationships of each segment. The worst‑case complexity is exponential only in the number of segments (T) and polynomial in the number of products (m), the number of linear constraints (K), and (1/\varepsilon). Consequently, when (T) is bounded (a realistic assumption, as managers typically identify a modest number of meaningful segments), the algorithm yields a PTAS for the constrained FMNL pricing problem.

Theoretical significance.
Compared with van de Geer and den Boer (2022), who gave a PTAS for unconstrained FMNL pricing, this work is the first to incorporate general linear price constraints while preserving polynomial‑time approximation guarantees. It also improves upon recent MILP‑based approaches (e.g., Shao et al., 2026) that cannot guarantee scalability. The key technical insight is that, under the assumption of segment‑independent price sensitivities, all non‑convexities can be captured by a small number of bilinear terms, enabling efficient convex relaxations via exponential cones.

Computational experiments.
The authors conduct extensive numerical tests on synthetic instances with up to 100 products, 2–5 segments, and various linear constraints (price caps, weighted‑average price limits, budget constraints). They compare against state‑of‑the‑art baselines: MILP formulations, gradient‑based heuristics, and the unconstrained FMNL PTAS. Results show that the proposed B&B algorithm consistently attains revenues 5–20 % higher than heuristics and 30–50 % closer to the true optimum than MILP solutions, while requiring substantially less computation time (often an order of magnitude faster). The algorithm remains robust as constraints become tighter, where other methods frequently fail to converge.

Managerial implications.
The paper highlights three practical takeaways: (1) Ignoring constraints or applying them post‑hoc can cause significant revenue loss; integrating constraints directly into the optimization yields superior outcomes. (2) Because algorithmic complexity scales with the number of segments, firms should focus on identifying a parsimonious set of meaningful segments rather than over‑segmenting. (3) The methodology demonstrates that rich demand models need not be computationally prohibitive; with modest segment counts, near‑optimal pricing can be obtained in polynomial time.

Conclusion.
Overall, the study delivers the first polynomial‑time approximation scheme for constrained price optimization under both single‑segment MNL and multi‑segment FMNL demand models. By leveraging exponential‑cone reformulations and a bilinear‑structure‑aware Branch‑and‑Bound, it bridges the gap between expressive demand modeling and tractable, globally reliable pricing optimization, offering both theoretical advances and actionable tools for practitioners.