# Elliptic Mathematical Programs with Equilibrium Constraints (MPECs) in Function Space: Optimality Conditions and Numerical Realization

### From DFG-SPP 1253

## Project leader

Institut für Mathematik

HU Berlin

Tel (secr.): +49 30 2093-5844
Fax (secr.): +49 30 2093-5859

Email: hint@math.hu-berlin.de

Homepage: [1]

## DFG funded assistants

Dr. Thomas Surowiec (Humboldt Universität zu Berlin)

surowiec@math.hu-berlin.de

Homepage: [2]

Dr. Antoine Laurain (Humboldt Universität zu Berlin)

laurain@math.hu-berlin.de

Homepage: [3]

## Description

project webpage [4]

Many phenomena in engineering, life sciences, mathematical finance or physics result in a mathematical model of variational or quasi-variational inequality type. Applications comprise contact problems (with friction) in elasticity, torsion problems in plasticity, option pricing in finance, the magnetization of superconductors or ionization problems in electrostatics. Often, one is interested in influencing the system under consideration by some control means in order to optimize a certain output quantity. The resulting optimization problem falls into the realm of Mathematical Programs with Equilibrium Constraints (MPECs), which are challenging due to constraint degeneracy.

The project work concentrates on the development of a suitable optimality theory as well as the design and implementation of efficient solution algorithms for classes of MPECs in function space which are governed by elliptic (quasi)variational inequalities. Among others, the results will be applied to the following processes:

* Stationary magnetization of type-II superconductors; * Control of torsion phenomena with variable plasticity threshold; * Ionization problems in electrostatics.

Concerning stationarity conditions, the project aims at developing new mathematical techniques for deriving and categorizing adequate optimality conditions in function space for such problems. The employed methodology relies on constraint relaxation to satisfy constraint qualifications, so called path-following approaches for constraints with low multiplier regularity and a subsequent asymptotic study for deriving a first order system for the original problem.

Since discretized MPECs result in large scale problems, tailored numerical solution techniques relying on adaptive finite element methods, semismooth Newton and multilevel techniques are developed within this project. The semi- or non-smoothness aspect arises due to the equivalence of the first order optimality systems to non-smooth operator equations.