Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Various government laws have recently been enacted to alleviate the environmental deterioration of transportation systems. Environmental constraint is a valid means to explicitly reflect various environmental protection requirements imposed by the government. In this paper, we examine the environmentally constrained traffic equilibrium problem (EC-TEP), which is a fundamental tool for modeling and evaluating environmental protection requirements. Specifically, we provide an equivalent reformulation for the EC-TEP. The proposed reformulation adapts the concept of gap function to simultaneously reformulate the nonlinear complementarity conditions associated with the generalized user equilibrium conditions, environmental constraints, and conservation constraints as an equivalent unconstrained optimization problem. This gap function reformulation has two desirable features: (1) it can handle a general environmental constraint structure (linear or nonlinear; link-based or area-based) and a general link and route cost structure, enhancing the modeling adaptability and flexibility; (2) it is smooth and unconstrained, permitting a number of existing efficient algorithms for its solution. A gradient-based solution algorithm with a self-regulated averaging stepsize scheme is customized to solve the reformulated unconstrained optimization problem. Numerical examples are also provided to demonstrate the modeling flexibility of the proposed EC-TEP reformulation.