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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.

In this paper, we propose an inexact version of proximal gradient algorithm with extrapolation for solving a class of nonconvex nonsmooth optimization problems. Specifically, the subproblem in proximal gradient algorithm with extrapolation is allowed to be solved inexactly by certain relative error criterion, in the sense that the criterion can be updated adaptively in each iteration. Under the assumption that an auxiliary function satisfies the Kurdyka-Łojasiewicz (KL) inequality, we prove that the iterative sequence generated by the inexact proximal gradient algorithm with extrapolation converges to a stationary point of the considered problem. Furthermore, the convergence rate of the proposed algorithm can be established when the KL exponent is known. Moreover, we illustrate the advantage by applying the algorithm to solve a nonconvex optimization problem.