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.
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