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A Cardinality-Constrained Approach to Combinatorial Bilevel Congestion Pricing

Lei Guo, Jiayang Li, Yu Marco Nie, Jun Xie

TL;DR

This work tackles combinatorial bilevel congestion pricing (CBCP), where a tolling authority must choose both toll locations and levels to minimize total travel time. It introduces a cardinality-constrained formulation that replaces binary toll decisions with a constraint $|\mathrm{supp}(u)|\le\kappa$, and reformulates the problem into a block-separable, single-level program via a penalty and decomposition approach. The authors prove convergence to an approximate KKT point and demonstrate scalability by solving CBCP instances with up to about 3,000 links in roughly 20 minutes, outperforming heuristic methods on large networks. This yields a practically effective framework for joint toll allocation and levying, with clear guidance on toll-location selection and substantial congestion-reduction potential in large-scale networks.

Abstract

Combinatorial bilevel congestion pricing (CBCP), a variant of the mixed (continuous/discrete) network design problems, seeks to minimize the total travel time experienced by all travelers in a road network, by strategically selecting toll locations and determining toll charges. Conventional wisdom suggests that these problems are intractable since they have to be formulated and solved with a significant number of integer variables. Here, we devise a scalable local algorithm for the CBCP problem that guarantees convergence to an approximate Karush-Kuhn-Tucker point. Our approach is novel in that it eliminates the use of integer variables altogether, instead introducing a cardinality constraint that limits the number of toll locations to a user-specified upper bound. The resulting bilevel program with the cardinality constraint is then transformed into a block-separable, single-level optimization problem that can be solved efficiently after penalization and decomposition. We are able to apply the algorithm to solve, in about 20 minutes, a CBCP instance with up to 3,000 links. To the best of our knowledge, no existing algorithm can solve CBCP problems at such a scale while providing any assurance of convergence.

A Cardinality-Constrained Approach to Combinatorial Bilevel Congestion Pricing

TL;DR

This work tackles combinatorial bilevel congestion pricing (CBCP), where a tolling authority must choose both toll locations and levels to minimize total travel time. It introduces a cardinality-constrained formulation that replaces binary toll decisions with a constraint , and reformulates the problem into a block-separable, single-level program via a penalty and decomposition approach. The authors prove convergence to an approximate KKT point and demonstrate scalability by solving CBCP instances with up to about 3,000 links in roughly 20 minutes, outperforming heuristic methods on large networks. This yields a practically effective framework for joint toll allocation and levying, with clear guidance on toll-location selection and substantial congestion-reduction potential in large-scale networks.

Abstract

Combinatorial bilevel congestion pricing (CBCP), a variant of the mixed (continuous/discrete) network design problems, seeks to minimize the total travel time experienced by all travelers in a road network, by strategically selecting toll locations and determining toll charges. Conventional wisdom suggests that these problems are intractable since they have to be formulated and solved with a significant number of integer variables. Here, we devise a scalable local algorithm for the CBCP problem that guarantees convergence to an approximate Karush-Kuhn-Tucker point. Our approach is novel in that it eliminates the use of integer variables altogether, instead introducing a cardinality constraint that limits the number of toll locations to a user-specified upper bound. The resulting bilevel program with the cardinality constraint is then transformed into a block-separable, single-level optimization problem that can be solved efficiently after penalization and decomposition. We are able to apply the algorithm to solve, in about 20 minutes, a CBCP instance with up to 3,000 links. To the best of our knowledge, no existing algorithm can solve CBCP problems at such a scale while providing any assurance of convergence.

Paper Structure

This paper contains 23 sections, 5 theorems, 60 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related studies
  3. Bilevel programming
  4. Bilevel congestion pricing
  5. Problem setting
  6. Cardinality-constrained formulation
  7. Algorithm
  8. Reformulation
  9. Description
  10. Convergence
  11. Numerical study
  12. Hearn's network
  13. Sioux-Falls network
  14. Chicago-Sketch network
  15. Concluding remarks
  16. ...and 8 more sections

Key Result

Proposition 1

Given Assumption ass, we have the following properties: (i) The lower-level solution set ${\cal S}(u)$ is a singleton for any $u$, and the value function ${\cal V}(u)$ is continuously differentiable with (ii) The value function ${\cal V}(u)$ is concave and the gap function $f(u,v) - {\cal V}(u)$ is block-wise convex, i.e., it is convex with respect to the upper-level variables $u$ given $v$ and c

Figures (2)

  • Figure 1: Topology of the Hearn's network.
  • Figure 2: Toll links selected by PBCD (Algorithm \ref{['mainalg1']}) and H1--H4.

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3