A Bundle Method to Solve Multivalued Variational Inequalities Genevi Eve Salmon, Jean-jacques Strodiot, and Van

In this paper we present a bundle method for solving a generalized variational inequality problem. This problem consists in nding a zero of the sum of two multivalued operators de ned on a real Hilbert space. The rst one is monotone and the second one is the subdi erential of a lower semicontinuous proper convex function. The method is based on the auxiliary problem principle due to Cohen and the strategy is to approximate, in the subproblems, the nonsmooth convex function by a sequence of convex piecewise linear functions as in the bundle method in nonsmooth optimization. This makes these subproblems more tractable. Moreover to ensure the existence of subgradients at each iteration, we also introduce a barrier function in the subproblems. This function prevents the iterates to go outside the interior of the feasible domain. First we explain how to build, step by step, a suitable piecewise linear approximation and we give conditions to ensure the boundedness of the sequence generated by the algorithm. Then we study the properties that a gap function must satisfy to obtain that each weak limit point of this sequence is a solution of the problem. In particular, we give existence theorems of such a gap function when the rst multivalued operator is paramonotone, weakly closed and Lipschitz continuous on bounded subsets of its domain and when it is the subdi erential of a convex function. When it is strongly monotone, we obtain that the sequence generated by the algorithm strongly converges to the unique solution of the problem.