Estimate then Predict: Convex Formulation for Travel Demand Forecasting

Youngseo Kim; Gioele Zardini; Samitha Samaranayake; Soroosh Shafiee

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Estimate then Predict: Convex Formulation for Travel Demand Forecasting

Youngseo Kim, Gioele Zardini, Samitha Samaranayake, Soroosh Shafiee

arXiv:2609.17919v1Sep 15, 2026cs.GT

TL;DR

The paper targets inconsistencies in sequential four-step travel demand forecasting and the limited behavioral richness and computational practicality of existing combined models. It develops a convex hierarchical destination–mode–route model whose primal solution characterizes equilibrium and whose dual variables recover behavioral parameters. Experiments show efficient solution across benchmark networks, while parameter-recovery validation remains outside the study’s scope.

Problem
Sequential four-step forecasting can produce inconsistent travel patterns, while combined models have lacked behavioral richness, computational efficiency, or unified parameter estimation.
Method
A two-stage convex program integrates hierarchical destination, mode, and route choices, uses moment and conditional-entropy constraints, and calibrates route dispersion separately from high-dimensional parameter recovery.
Results
The model solved all benchmark networks within a couple of minutes, including Chicago Sketch with 93,513 OD pairs and 225,507 routes.
Takeaways & Limitations
The formulation provides a scalable convex approach for combined travel-demand equilibrium modeling with richer mode and route-choice representations.
Takeaways & Limitations
Behavioral parameter recovery was not empirically validated because the benchmark datasets lacked independent travel-pattern and link-count observations.

Abstract

from arXiv · show

Travel demand forecasting is essential for evaluating large-scale infrastructure projects, yet the traditional sequential four-step process can produce inconsistencies across trip distribution, mode choice, and traffic assignment. Although combined models address these inconsistencies, their practical use has been limited by simplified behavioral assumptions, computational burden, and the lack of a unified parameter-estimation framework. We propose a convex programming approach that integrates destination, mode, and route choices within a hierarchical extended logit model. The optimal primal solution characterizes the joint travel-demand equilibrium, while the optimal dual variables recover taste coefficients, alternative-specific constants, and destination- and mode-level scale parameters. The model captures mode correlations through nested logit and route overlap through path-size logit. Its convex structure provides global optimality guarantees and enables efficient solution using off-the-shelf conic solvers. Observed travel patterns are incorporated through moment and conditional-entropy constraints, while the route-level dispersion parameter is calibrated separately using partial link counts. Numerical experiments on eight benchmark networks demonstrate the scalability and computational efficiency of the formulation.

1 Introduction

The paper addresses inconsistencies and limited practicality in sequential four-step travel demand forecasting by jointly modeling destination, mode, and route choices in a convex, estimable framework.

Problem: The traditional four-step process separates trip generation, distribution, modal split, and traffic assignment, allowing inconsistent inputs and outputs to propagate through the forecast.Travel times used in distribution and modal split can differ from congested times produced by final assignment, compromising internal consistency and optimality.
Problem: Combined models jointly equilibrate destination, mode, and route choices, better representing how congestion influences the full spectrum of travel decisions.This contrasts with sequential models that equilibrate only route choice and can produce less coherent outcomes.
Contribution: Joint estimation of behavioral parameters and network flows addresses a long-standing gap in travel demand modeling.The formulation uses duality to recover taste coefficients, alternative-specific constants, and scale parameters without bilinear terms that would destroy convexity.
Method: The proposed framework uses moment and conditional-entropy constraints, with their dual variables recovering behavioral parameters during the estimation stage.It follows a two-stage estimate-then-predict structure for calibrating parameters and forecasting future demand distributions.
Method: Nested logit captures correlations among modes, while path-size logit accounts for overlap among routes within the hierarchical choice model.The behavioral framework represents destination, mode, and route decisions in a unified structure.
Computational approach: Compact convex and exponential-cone formulations enable conic solvers and demonstrate computational scalability across benchmark networks of varying sizes.The literature review identifies a gap between tractable equilibrium modeling and behavioral parameter estimation that this formulation targets.

2 Modeling Assumptions

The paper models destination, mode, and route choices in a hierarchical extended-logit framework, combining MNL, nested logit, and path-size logit components. These choices account for congestion interactions, mode correlations, and route overlap within a unified behavioral structure.

2 Modeling Assumptions: Destination, mode, and route choices are represented hierarchically through MNL, nested logit, and path-size logit models, respectively.The hierarchy captures interactions such as route congestion influencing destination choice.
2 Modeling Assumptions: The hierarchical utility adds fixed utilities and random errors for destination, mode, and route choices, with scale parameters linked to the corresponding error variances.Destination and route errors are Gumbel, while the mode error embeds within-nest correlation.
2.1 Path size logit model for route choice: Path-size logit accounts for overlapping routes by penalizing shared links according to link dominance within each route.The model uses route length, link length, and link-route incidence to construct the path-size factor.
2.1 Path size logit model for route choice: The route utility combines negative generalized travel cost with a path-size adjustment, with λ̂ controlling sensitivity to route costs.Generalized route cost sums the costs of all arcs, whose latency functions are typically convex and increasing in flow.
2.2 Nested logit model for mode choice: Nested logit relaxes the independence-of-irrelevant-alternatives assumption by allowing alternatives within the same nest to share correlation.This addresses unrealistic substitution patterns such as the blue-bus and red-bus example.
2.2 Nested logit model for mode choice: Nested logit captures correlations among similar modes by grouping alternatives into nests and using nest-specific dissimilarity factors.When nest correlation is zero, the dissimilarity factor equals 1 and nested logit reduces to multinomial logit.
2.2 Nested logit model for mode choice: The nested-logit probability of a mode is the product of its conditional within-nest probability and the probability of choosing that nest.The nest probability uses an inclusive value that aggregates normalized utilities across alternatives in the nest.

3 Proposed Two-Stage Model

The paper proposes a two-stage convex framework that estimates hierarchical travel-choice parameters from observed travel data and then predicts future demand with those parameters. A primal–dual formulation preserves convexity, supports exponential-cone solution, and recovers a behaviorally consistent equilibrium across destination, mode, and route choices.

3 Proposed Two-Stage Model: Observed mode-specific OD flows, aggregate statistics, reconstructed distributions, and partial link counts provide constraints for calibrating the travel-demand model.The formulation can accommodate partially disaggregated data by matching destination-level OD information and mode-level aggregate statistics separately.
3.2 First stage – estimation: A primal–dual construction preserves convexity by recovering taste coefficients, alternative-specific constants, and destination- and mode-level scale parameters through dual variables.Directly introducing these parameters as primal variables would create nonlinear interactions with endogenous choice probabilities.
3.2 First stage – estimation: Replacing conditional-entropy equalities with lower-bound inequalities enables an exponential-cone reformulation solvable by off-the-shelf conic solvers.The relaxation preserves convexity while incorporating congestion-dependent route costs and stochastic user-equilibrium conditions.
3.2 First stage – estimation: The first-stage problem is a convex program whose optimal primal and dual solutions are equivalent to the hierarchical extended logit model.The formulation characterizes equilibrium flows while providing the behavioral parameters through the corresponding dual solution.
3 Proposed Two-Stage Model: The two-stage framework reconstructs current travel patterns to estimate behavioral parameters, then uses them to predict future demand distributions.Estimated parameters are carried from the first-stage calibration problem into the second-stage prediction problem.
3.4 Summary of our contributions: The route-level dispersion parameter is isolated from the convex estimation problem and calibrated through a one-dimensional search using observed link counts.Jointly estimating this parameter with network flows would create bilinear coupling through the flow-dependent Beckmann term.

4 Experimental Results

Experiments examine route-overlap correction, nested-mode sensitivity, detailed equilibrium outputs, and scalability across benchmark transportation networks. The convex formulation captures behavioral effects while solving large instances efficiently, although parameter recovery is not empirically validated and link-count calibration remains illustrative.

4.1 A toy network: Without route overlap, MNL and PSL produce identical flows; with overlap, PSL changes flows by overlap extent while MNL overestimates congested-road arc flow.The toy network varies the overlapped arc length x across 0, 2, and 3.
4.2 Sioux Falls network: The estimated destination scale parameter θdest is 0.1689, below the fixed θmode and θroute values of 1, consistent with stronger destination-level correlation.The estimate is obtained from the dual variable of Eq. (20a).
4.2 Sioux Falls network: Reducing the public-transit nest parameter τ from 1 to 0.5 lowers bus and subway shares, while increasing automobile competitiveness ρ raises automobile share.Bus share falls from 40.2% to 37.3%, subway share from 24.1% to 20.0%, and automobile share rises from 19.6% to 51.7% or from 25.5% to 59.9%, depending on the comparison.
4.2 Sioux Falls network: Primal solutions provide destination- and mode-level demand plus arc traffic counts and road congestion visualized through volume-to-capacity ratios.The case study extracts detailed travel-demand patterns from the optimization solution, including road, bus, and subway flows.
4.3 Various transportation benchmark networks: All benchmark instances, including Chicago Sketch with 93,513 OD pairs and 225,507 routes, were solved within a couple of minutes.This provides numerical evidence of computational efficiency for large-scale combined models using a conic solver.
4.3 Various transportation benchmark networks: Benchmark experiments use representative λ̂ values because independent observed link counts are unavailable, and improper choices can cause solver instability and inaccurate θdest estimates.The experiments demonstrate behavioral properties and tractability rather than empirical validation of behavioral parameter recovery.

5 Conclusion

The paper proposes a convex-programming formulation that unifies travel-demand equilibrium and behavioral-parameter estimation, while identifying scope limits and future extensions.

The convex formulation integrates travel-demand choices while ensuring solution existence and tractability.It is presented as equivalent to a hierarchical extended logit model.
Observed travel patterns enter through moment and conditional-entropy constraints, whose dual variables recover behavioral parameters.The framework provides an alternative parameter-estimation architecture based on the travel distribution.
Complementary relationship between our framework and activity-based models: The framework offers compact aggregate parameter estimates but sacrifices the detailed activity information represented by activity-based models.Its outputs can also serve as inputs to activity-based models for richer trip-activity simulation.
Future research: Validation with multi-source real-world data and comparisons against sequential, activity-based, and machine-learning approaches remain future work.The paper also proposes applying the model across transportation networks in different cities.
Future research: Future extensions include multimodal trip chains, dynamic demand, and methods for non-separable cross-mode link costs.These extensions are intended to broaden the model’s applicability and behavioral coverage.
The extended model could evaluate policy interventions such as tolling and congestion pricing, as well as investments in autonomous and electric vehicles.

A.1 Proof of Theorem 1

The proof establishes convexity and derives hierarchical logit relationships from the formulation’s KKT conditions and dual variables.

The formulation maximizes a concave objective subject to convex inequality constraints and linear equality constraints, making it a convex optimization problem.The concavity comes from conditional entropy and the negative Beckmann term, while entropy constraints define convex feasible sets.
The KKT conditions characterize optimal solutions under standard constraint qualifications and recover the hierarchical extended logit model.
The Lagrangian introduces dual variables for the formulation’s constraints, enabling first-order conditions to be taken with respect to conditional probabilities.The derivation applies the chain rule and omits terms constant in the joint probabilities.
At optimality, joint probabilities satisfy the formulation’s equality constraints and induce the corresponding conditional probabilities.
The first-order conditions are obtained by differentiating with respect to route, mode, destination, and higher-level conditional probabilities.

A.2 Proof of Theorem 2

The proof converts entropy and congestion terms into conic representations, yielding an exponential-cone formulation of the first-stage problem.

Conditional-entropy terms are exponential-cone representable, while the integrated latency term is represented through power-cone constructions.These representations introduce auxiliary variables for the entropy and latency components.
The entropy hypograph is represented by exponential-cone constraints using inequalities of the form y_i ≤ -p_i ln p_i.
The epigraph of a convex increasing rational power function is exponential-cone representable through geometric-cone representations.
Auxiliary variables t_ij, u_ijN, v_ijm, and w_ijmr encode the conditional-entropy terms in the conic reformulation.
The resulting construction reformulates the full first-stage problem as an exponential cone program.
For automobile travel, the latency function uses the standard BPR form, whereas subway and bus travel are assumed unaffected by congestion.For unaffected modes, the latency becomes constant under the stated parameter setting.

A.3 Relationship between the dual formulation and maximum likelihood

The appendix connects the destination-choice dual formulation to maximum-likelihood estimation by rewriting its logit probability terms and deriving the dual problem.

The derivation forms a Lagrangian for destination-choice entropy terms and replaces ln p_j|i using the model’s defined functional form.
The appendix then derives the corresponding dual problem from the destination-choice formulation.
The destination-choice component of the dual formulation is connected to the log-likelihood objective.

B Background: Sequential Modeling Approach and Equivalent Convex Models

Entropy maximization underlies convex formulations of trip distribution, modal split, and traffic assignment. These formulations connect the traditional stages to unified optimization problems.

Entropy maximization is central to combined convex modeling of travel demand.It originated in thermodynamics and was adapted by Shannon for information theory; trip distribution and modal split generalize entropy maximization.
The traditional stages correspond to entropy-based or convex formulations for gravity modeling, MNL modal split, and Wardrop equilibrium.The section examines these three correspondences in sequence.

B.1 Convex formulation for trip distribution

The trip-distribution formulation derives the conventional gravity model from entropy maximization. Observed productions, attractions, total trips, and budget constraints shape the resulting nonuniform distribution.

B.1.1 Gravity model for trip distribution: The gravity model estimates aggregated trips Tij between origin zone i and destination zone j.Origins and destinations are spatial zones, and individual travelers are not distinguished.
B.1.1 Gravity model for trip distribution: The gravity formulation uses origins, destinations, generalized costs, an impedance function, and balancing factors.Oi and Dj represent trip production and attraction, cij represents generalized cost, and Ai and Bj balance production and attraction.
B.1.2 Entropy maximization principle in trip distribution: The entropy model seeks the most plausible trip distribution by treating probability as proportional to the number of system states producing that distribution.The state count is motivated by combinatorial theory.
B.1.2 Entropy maximization principle in trip distribution: The optimization fixes observed origin productions, destination attractions, and total travel expenditure.The first two constraints match observed trips from each origin and to each destination; the third enforces the regional budget Ĉ.
B.1.2 Entropy maximization principle in trip distribution: Stirling’s approximation converts the combinatorial objective into an entropy expression involving −Σ Tij log Tij.With pij = Tij/T, this becomes the information-theoretic entropy H = −Σ pij log pij.
B.1.2 Entropy maximization principle in trip distribution: Solving the entropy model’s first-order conditions recovers the gravity model with f(cij) = exp(−βcij).The entropy-maximizing distribution with the relevant constraints closely aligns with the conventional gravity-model distribution.
B.1.2 Entropy maximization principle in trip distribution: Without observation constraints, the trip distribution would be uniform; the constraints introduce nonuniformity consistent with observations.The multipliers λi, μj, and β enter through the Lagrangian formulation.
B.1.2 Entropy maximization principle in trip distribution: Wilson’s multimodal extension implicitly incorporates modal split through the generalized gravity formulation.This motivates the relationship between entropy-maximizing models and logit choice models.

B.2 Convex formulation for modal split

The modal-split formulation links entropy maximization with logit choice and parameter estimation. In particular, MNL can be recovered from an entropy problem whose constraints reproduce observed aggregate choices and values.

B.2 Convex formulation for modal split: MNL can be derived from an entropy-maximization model, establishing an equivalence between the two formulations.The family of logit models is presented as the standard framework for traveler mode choice.
B.2.1 Multinomial logit model and maximum likelihood estimation for parameter estimation: The MNL setup represents utility as systematic attributes plus random variation across individuals and alternatives.Attributes include trip-related measures such as travel cost or travel time, with βk as their coefficients and ASCm as alternative-specific constants.
B.2.1 Multinomial logit model and maximum likelihood estimation for parameter estimation: The logit closed form follows from assuming identically distributed independent type-I extreme-value errors.This distribution is also called the Gumbel distribution.
B.2.1 Multinomial logit model and maximum likelihood estimation for parameter estimation: Observed individual choices enter maximum-likelihood estimation through indicators ym, producing estimated coefficients β̂.The formulation focuses on a specific origin-destination pair and omits that pair’s subscript.
B.2.2 Entropy maximization principle in parameter estimations: The entropy formulation constrains choice probabilities to sum to one and predicted choices and deterministic values to match observations.These constraints reproduce aggregate observations across the entire system.
B.2.2 Entropy maximization principle in parameter estimations: A Lagrangian first-order condition produces the MNL form from the entropy-maximization problem.The associated multipliers correspond to the behavioral parameters identified by the equivalent estimation result.
B.2.2 Entropy maximization principle in parameter estimations: Entropy and maximum-likelihood estimation identify the same MNL coefficients and alternative-specific constants.Anas’s result gives β = α and ASC = γ, and describes the maximum-entropy estimators as reproducing rational user behavior.
B.2.2 Entropy maximization principle in parameter estimations: When parameters are predetermined and patterns are aggregated, satisfaction maximization is a special case of the entropy formulation.The deterministic value is V̂m = ASĈm + Σ β̂k Xmk, with the individual index omitted.

B.3 Convex formulation for traffic assignment

The Beckmann formulation represents static traffic assignment as a convex optimization problem. Under increasing latency, its solution characterizes Wardrop user equilibrium, while SOCP reformulation supports scalable computation for suitable latency functions.

The Beckmann optimization framework describes static traffic flow at equilibrium through integrals of link travel time.The objective integrates total travel time across network links.
Increasing link latency makes the Beckmann objective convex and ensures its solution characterizes user equilibrium.At equilibrium, no user can reduce travel time by unilaterally changing routes.
Simulation-based methods provide flexible modeling of realistic traffic interactions and are suitable for large-scale networks.Existing software makes this approach relatively straightforward to use.
The Beckmann problem can be reformulated as SOCP when road latency is a convex power function.This supports scalability because SOCP is nearly as tractable as linear programming with modern interior-point methods.

B.4 Related concepts in other disciplines

The section situates the paper's concepts within related work from information theory and operations research. It introduces terminology from these fields to support interdisciplinary exchange and clarify connections with transportation research.

Related concepts have been explored beyond transportation, including in information theory and operations research.
The section presents terminology from other domains alongside references to contextualize these efforts and facilitate interdisciplinary research exchange.
It notes that the relationship between random utility models and entropy functions has been investigated within transportation research.