A Smooth Approach to the solution of Nonlinear Complementarity Problems involving $\mathcal{P}_{0}$-function

We present a family of smoothing methods to solve nonlinear complementarity problems (NCPs) involving
$\mathcal{P}_{0}$
-function. Several regularization or approximation techniques like Fisher-Burmeister's method, interior-point methods (IPMs) approaches, or smoothing methods already exist. All the corresponding methods solve a sequence of nonlinear systems of equations and depend on parameters that are difficult to drive to zero. The main novelty of our approach is to consider the smoothing parameters as variables that converge by themselves to zero. We do not need any complicated updating strategy, and then obtain nonparametric algorithms. We prove some global and local convergence results and present several numerical experiments, comparisons, that show the efficiency of our approach.