Mathematical programs with complementarity constraints (MPCCs) are non-linear optimization problems over a possibly non-convex and non-connected domain. Recent progress on optimality conditions allows to build efficient relaxation methods that converge to MPCC-kind stationary point. However in 2015, Kanzow \& Schwarz pointed out that the implementation of those methods may not guarantee anymore the strong convergence properties.
We introduce here an algorithm that tackle this issue by computing a specific approximate solution of the regularized sub-problems. We provide theoretical results on convergence and existence of the approximate solutions in a general framework that include all the relaxation methods from the literature. We also present numerical results in Julia.