A trial-and-error congestion pricing scheme for networks with elastic demand and link capacity constraints

•Hybrid framework for the trial-and-error congestion scheme.•Two level iteration method for the hybrid framework.•Outer level iteration solved by the approximate subgradient projection method.•Inner level iteration solved by the partial linearization method.•Rigorous proofs for convergence of the two level iteration method.

This paper proposes a combination of trial-and-error congestion pricing schemes that have been studied in the literature. It not only considers the minimization of the total system cost but also addresses the capacity constraints. A two-level iteration method is proposed for solving the hybrid problem, in which the approximate subgradient projection method is used for the outer level iteration phase, and the partial linearization method is used for the inner level iteration phase. We prove the convergence of the two-level iteration method, under the condition that the subproblem for the inner level iteration is only solved approximately, which makes the method efficient and practical. A numerical example is presented to illustrate the application of the two-level iteration method to the trial-and-error congestion pricing scheme.