An adaptation of the proximal algorithm for the traffic assignment problem under a user equilibrium formulation for a general asymmetric traffic network is presented in this paper. It follows the recently published results of Pennanen regarding convergence under non monotonicity. As it is well known the problem can be formulated as a variational inequality and the algorithmic solutions developed up to date guarantee convergence only under too restrictive conditions which are difficult to appear in practice. In this paper it is also discussed the possibility of including the algorithm on a demand adjustment problem formulated as a bilevel program with lower level traffic equilibrium constraints expressed as a variational inequality.

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