A New Algorithm of the Variational Inequality Problems with Application on the Asymmetric Traffic Equilibrium Problem

In this paper, we introduce a double projection algorithm based on extragradient method for solving the variational inequality problems and we prove the convergence theorem of proposed algorithm. One of the parameters that determine the efficiency and accuracy of the projection method is a properly selected step size. This selection is based on the contractive properties of operator which is projected on the feasible region. For example, if the Lipschitz constant is not known, we have trouble choosing step size of the algorithm. The proposed algorithm eliminates the need to know the Lipschitz constant and provides a method that facilitates step size selection.We formulate the asymmetric traffic equilibrium problem as a variational inequality on the path flows space. According to the decomposable structure of the feasible set of this model, we obtain the traffic network equilibrium state, by using the proposed algorithm. Finally, we present the numerical results of using this algorithm on the Sioux-Falls test traffic network.