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Network Design for the Traffic Assignment Problem with Mixed-Integer Frank-Wolfe

Kartikey Sharma, Deborah Hendrych, Mathieu Besançon, Sebastian Pokutta

TL;DR

The paper addresses the network design problem for centralized traffic assignment by formulating it as a mixed-integer convex optimization problem and applying the Boscia framework to solve it via branch-and-bound with Frank-Wolfe relaxations. It develops and compares multiple formulations—mixed-integer conic, mixed-integer with original constraints, and penalty-based—and solution methods including Integer Frank-Wolfe, penalty-based Frank-Wolfe, Benders-based LMO, and direct conic/MINLP approaches. Empirical results on Transportation Networks Library instances show that a MIP-based Frank-Wolfe approach excels for deterministic problems and limited scenarios, while a penalty-based approach dominates in large-scenario stochastic settings; Benders-based LMOs offer competitive performance and potential improvements. Overall, the study provides scalable, methodologically diverse strategies for jointly designing network edges and routing flows under uncertainty, with practical implications for centralized traffic management and infrastructure planning.

Abstract

We tackle the network design problem for centralized traffic assignment, which can be cast as a mixed-integer convex optimization (MICO) problem. For this task, we propose different formulations and solution methods in both a deterministic and a stochastic setting in which the demand is unknown in the design phase. We leverage the recently proposed Boscia framework, which can solve MICO problems when the main nonlinearity stems from a differentiable objective function. Boscia tackles these problems by branch-and-bound with continuous relaxations solved approximately with Frank-Wolfe algorithms. We compare different linear relaxations and the corresponding subproblems solved by Frank-Wolfe, and alternative problem formulations to identify the situations in which each performs best. Our experiments evaluate the different approaches on instances from the Transportation Networks library and highlight the suitability of the mixed-integer Frank-Wolfe algorithm for this problem. In particular, we find that the Boscia framework is particularly applicable to this problem and that a mixed-integer linear Frank-Wolfe subproblem performs well for the deterministic case, while a penalty-based approach, with decoupled feasible regions for the design and flow variables, dominates other approaches for stochastic instances with many scenarios.

Network Design for the Traffic Assignment Problem with Mixed-Integer Frank-Wolfe

TL;DR

The paper addresses the network design problem for centralized traffic assignment by formulating it as a mixed-integer convex optimization problem and applying the Boscia framework to solve it via branch-and-bound with Frank-Wolfe relaxations. It develops and compares multiple formulations—mixed-integer conic, mixed-integer with original constraints, and penalty-based—and solution methods including Integer Frank-Wolfe, penalty-based Frank-Wolfe, Benders-based LMO, and direct conic/MINLP approaches. Empirical results on Transportation Networks Library instances show that a MIP-based Frank-Wolfe approach excels for deterministic problems and limited scenarios, while a penalty-based approach dominates in large-scenario stochastic settings; Benders-based LMOs offer competitive performance and potential improvements. Overall, the study provides scalable, methodologically diverse strategies for jointly designing network edges and routing flows under uncertainty, with practical implications for centralized traffic management and infrastructure planning.

Abstract

We tackle the network design problem for centralized traffic assignment, which can be cast as a mixed-integer convex optimization (MICO) problem. For this task, we propose different formulations and solution methods in both a deterministic and a stochastic setting in which the demand is unknown in the design phase. We leverage the recently proposed Boscia framework, which can solve MICO problems when the main nonlinearity stems from a differentiable objective function. Boscia tackles these problems by branch-and-bound with continuous relaxations solved approximately with Frank-Wolfe algorithms. We compare different linear relaxations and the corresponding subproblems solved by Frank-Wolfe, and alternative problem formulations to identify the situations in which each performs best. Our experiments evaluate the different approaches on instances from the Transportation Networks library and highlight the suitability of the mixed-integer Frank-Wolfe algorithm for this problem. In particular, we find that the Boscia framework is particularly applicable to this problem and that a mixed-integer linear Frank-Wolfe subproblem performs well for the deterministic case, while a penalty-based approach, with decoupled feasible regions for the design and flow variables, dominates other approaches for stochastic instances with many scenarios.
Paper Structure (18 sections, 14 equations, 2 figures, 1 table)

This paper contains 18 sections, 14 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Network Design for Traffic Assignment.
  3. Stochastic Network Design.
  4. Previous Work.
  5. Contributions.
  6. Solution Methods
  7. Integer Frank-Wolfe
  8. Penalty-based Integer Frank-Wolfe
  9. Benders decomposition of the linear network design
  10. Direct Solution Methods
  11. Computational Experiments
  12. Instances.
  13. Stochastic Network Design.
  14. Results.
  15. Mixed-Integer Conic Formulation.
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Example of a network with an optional purchasable (dashed) edge. Each edge is labeled with the corresponding flow, there is a unit flow from each source $S_{1,2}$ to the destination $D$. Because of the superlinear congestion costs, edges with two units of flow incur a cost greater than twice that of a single unit.
  • Figure 2: Comparison of the Integer LMO, the Shortest path LMO, and the Benders Decomposition based LMO with solutions returned within 13000s.