An Approximate Bundle Method for Solving Variational Inequalities

We present an approximate algorithm for solving generalized variational inequality problem (P) by combining the bundle technique with the auxiliary problem method. For subproblems we construct a new lower approximation to the involved function φ, as in the bundle method for nonsmooth optimization, which only requires the approximate function values and approximate subgradients instead of the exact ones. This makes the subproblem easier to solve and more tractable. Besides that, a new stopping criterion is given to determine whether the current approximation is good enough, it not only takes into account the contribution of the operator F in problem (P), but also utilizes the inexact information of the involved function φ in subproblem. Finally, we study the convergence of the proposed algorithm for the case when the operator F is paramonotone, possibly multivalued and the stepsizes are chosen going to zero.