The primal douglas-rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem

We apply the Douglas-Rachford splitting algorithm to a class of multi-valued equations consisting of the sum of two monotone mappings. Compared with the dual application of the same algorithm, which is known as the alternating direction method of multipliers, the primal application yields algorithms that seem somewhat involved. However, the resulting algorithms may be applied effectively to problems with certain special structure. In particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem. This research was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.