A generalized duality method for solving variational inequalities Applications to some nonlinear Dirichlet problems

In [2] Bermúdez and Moreno introduced a duality algorithm for the numerical solution of variational inequalities; this algorithm is based on some properties of the Yosida regularization of maximal monotone operators. The performances of this algorithm strongly depend on the choice of two constant parameters. A generalization of the algorithm with automatic choice of parameters was discussed in [13], where the constant parameters were replaced by scalar functions, thus improving the convergence of the algorithm. In this article we present a generalization of the Bermúdez-Moreno algorithm that allows the use of very general operators as parameters, extending some of the results in [2], [13] and [14]. As a particular case, we analyze the use of scalar and matrix-valued parameters in a Lp(Ω)M context. We apply the results developed to some boundary value problems involving the p-Laplacian operator, where it is shown that the use of matrix-valued parameters improves the convergence of the algorithm.