It is well known that a general network economic equilibrium problem can be formulated as a variational inequality (VI) and solving the VI will result in a description of network equilibrium state. This paper, however, introduces a new class of normative control problem that requires the network equilibrium link flow to satisfy physical and/or environmental constraints, but without explicit definition of system optimal objectives. By applying the duality principle, this paper will formulate the problem as an inverse variational inequality (IVI) because the variables and the mappings in the IVI are in the opposite positions of a classical VI. The economics meaning of the dual variables (or Larangian multipliers) explains that the optimal solutions for such problems represent the shadow prices of the binding constraints. Since the solution algorithm of IVI only relies on the link-flow information, perfect knowledge of network demand and supply is not required for such problem. The problem is practical appealing since management authority is often required to maintain link-based level of services for paid users and explicit knowledge of network demand and supply is not available. We demonstrate the formulation and solution algorithms of such problems using an example of link-based road pricing.
Abstract