Penalty Decomposition Methods for Second-Best Congestion Pricing Problems on Large-Scale Networks

The second-best congestion pricing (SBCP) problem is one of the most challenging problems in transportation because of its two-level hierarchical structure. In spite of various intriguing attempts at solving SBCP, existing solution methods are either heuristic without a convergence guarantee or suitable for solving SBCP on small networks only. In this paper, we first reveal some convexity-based structural properties of the marginal value function reformation of SBCP, and then, by effectively exploiting these structural properties, we propose two dedicated decomposition methods for solving SBCP on large-scale networks, which are different from existing methods in that they avoid linearizing nonconvex functions. We establish the convergence of the two decomposition methods under commonly used conditions and provide the maximum number of iterations for deriving an approximate stationary solution. The computational experiments based on a collection of real road networks show that in comparison with three existing popular methods, the two proposed methods are capable of solving SBCP on larger-scale networks, and for instances that can be solved by existing methods, the two proposed methods are substantially faster.