  The paper starts with a concise description of the recently developed
semismooth* Newton method for the solution of general inclusions. This method
is then applied to a class of variational inequalities of the second kind. As a
result, one obtains an implementable algorithm exhibiting a local superlinear
convergence. Thereafter we suggest several globally convergent hybrid
algorithms in which one combines the semismooth* Newton method with selected
splitting algorithms for the solution of monotone variational inequalities.
Their efficiency is documented by extensive numerical experiments.

Optimization and Control