The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

Proximal Point Methods (PPM) can be traced to the pioneer works of Moreau [16], Martinet [14,15] and Rockafellar [19, 20] who used as regularization function the square of the Euclideannorm. In this work, we study PPM in the context of optimization and we derive a class of suchmethods which contains Rockafellar's result. We also present a less stringent criterion to theacceptance of an approximate solution to the subproblems that arise in the inner loops of PPM.Moreover, we introduce a new family of augmented Lagrangian methods for convex constrainedoptimization, that generalizes the PE+ class presented in [2].