On the application of the SCD semismooth* Newton method to variational inequalities of the second kind

Abstract

The paper starts with a description of the SCD (subspace containing derivative) mappings and the SCD 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 locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth * Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequalities of the second kind, where one admits really large numbers of players (firms) and produced commodities.

On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction

“…To improve the global convergence properties, we use a non-monotone line search heuristic as introduced in [14]. Recall that the quantity ŷ(k) given by (5.18) acts as a residual for GE (5.16) at the point (x (k) , d(k) ).…”

“…Moreover, the "regularity" requirement, needed to ensure the (locally) superlinear convergence, could have been substantially relaxed. In [14] this so-called SCD semismooth * Newton method has been implemented in a class of variational inequalities (VIs) of the second kind that includes, among various problems of practical importance, also a class of discretized contact problems with Tresca friction ( [15]). The very good performance of the new method, when applied to those problems, has led us to consider more complicated problems, in which Q does not amount to the subdifferential of a convex function.…”

“…The very good performance of the new method, when applied to those problems, has led us to consider more complicated problems, in which Q does not amount to the subdifferential of a convex function. This clearly requires a generalization of the theory from [14]. As a concrete representative problem of this type, we have chosen a discretized static contact problem, where the Tresca friction is replaced by the (physically more realistic) Coulomb friction.…”