A Cardinality-Constrained Approach to Combinatorial Bilevel Congestion Pricing

TL;DR

This paper introduces a scalable, integer-free algorithm for combinatorial bilevel congestion pricing that efficiently minimizes total travel time by limiting toll locations, solving large-scale problems with convergence guarantees.

Contribution

It proposes a novel cardinality-constrained approach that transforms CBCP into a single-level problem, enabling efficient solutions without integer variables and providing convergence guarantees.

Findings

Solved a CBCP instance with 3,000 links in about 20 minutes.

First algorithm capable of handling large-scale CBCP with convergence assurance.

Eliminated integer variables using a cardinality constraint, improving scalability.

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.