A Bundle Method for Solving Variational Inequalities

In this paper, we present a bundle method for solving a generalized variational inequality problem. This problem consists of finding a zero of the sum of two multivalued operators defined on a real Hilbert space. The first one, F, is monotone and the second is the subdifferential of a lower semicontinuous proper convex function. Our method is based on the auxiliary problem principle due to Cohen, and our strategy is to approximate, in the subproblems, the nonsmooth convex function by a sequence of convex piecewise linear functions, as in the bundle method for nonsmooth optimization. This makes the subproblems more tractable. First, we explain how to build, step by step, suitable piecewise linear approximations by means of a bundle strategy, and we present a new stopping criterion to determine whether the current approximation is good enough. This criterion is the same as that commonly used in the special case of nonsmooth optimization. Second, we study the convergence of the algorithm for the case when the stepsizes are chosen going to zero and for the case bounded away from zero. In the first case, the convergence can be proved under rather mild assumptions: the operator F is paramonotone and possibly multivalued. In the second case, the convergence needs a stronger assumption: F is single-valued and satisfies a Dunn property. Finally, we illustrate the behavior of the proposed algorithm by some numerical tests.