Solving a Class of Variational Inequality Problems with a New Inexact Strategy

We consider a class of variational inequality problems with linear constraints, where the mapping is unknown and the system is an oracle. The capacitated traffic congestion pricing problem of transportation is such an application, and many classical methods cannot deal with this class of problems. Note that the cost of the observation (observe the exact solution of the subproblem) is very expensive. It is important to get an inexact solution instead of an exact solution, especially when the iteration is far from the solution set. In this paper, we propose a modified inexact prediction–correction method. Under the mild condition that the underlying mapping is strongly monotone, we prove the global convergence. Some numerical examples are presented to illustrate the efficiency of the inexact strategy.