In this paper, we introduce a new Lagrangian-like algorithm which is based on the combination of a nonsmooth Newton-type algorithm based on NCP-functions and an algorithm of Uzawa-type.
The lack of differentiability of NCP-functions only arises from critical indices which belong to the contact nodes.
The main idea is to apply an Uzawa-type algorithm for all equations fulfilling the contact condition, whereas
the nonsmooth Newton-type is only used on the remaining equations.
As we can see in numerical test calculations, this procedure
leads to a fast and robust algorithm.