# Semismooth* Newton Methods for Quasi-Variational inequalities and Contact Problems with friction

## DI Michael Mandlmayr

**Oct. 5, 2022, 3:30 p.m. BA 9909**

Motivated by many applications with real world Background, we consider semismooth$^*$ Newton methods for generalized equations. A generalized equations is typically described via a multifunction $F$, and we are interested in finding a point $x$ where $0\in F(x)$. We present the theoretical and algorithmic framework for semismooth$^*$ Newton methods, which enable us to solve many challenging generalized equations efficiently. This abstract procedure is Illustrated on two simple examples.

We introduce the new concept of semismoothness$^*$ of a certain order, which yields more precise convergence results. Moreover we present an inexact version of the semismooth$^*$ Newton method with results on convergence (speed).

Moreover we present the application of the semismooth$^*$ Newton method to Quasi Variational Inequalities in two different formulations(where one also covers generalized nonlinear programming) and Contact problems with Coulomb friction.

For each of these interesting problems we construct a specified version of the semismooth$^*$ Newton method which converges locally (quadratic) superlinear.

Lastly Show numerical results that support the claims from the theory.