Some resolvent methods for general variational inclusions

In this paper, we consider and analyze some classes of resolvent-splitting methods for solving the general variational inclusions using the technique of updating the solution. These resolvent-splitting methods are self-adaptive-type methods, where the corrector step size involves the resolvent equation. We prove that the convergence of these new methods only require the pseudomonotonicity, which is a weaker condition than monotonicity. These new methods differ from the previously known splitting and inertial proximal methods for solving the general variational inclusions and related complementarity problems. The proposed methods include several new and known methods as special cases. Our results may be viewed as refinement and improvement of the previous known methods.