Project Title

Elliptic Mathematical Programs with Equilibrium Constraints (MPECs) in function space: optimality conditions and numerical realization.

This project is part of the DFG-Priority Program [SPP 1253] ''Optimization with Partial Differential Equations''.
People
Project Description [detailed summary]
HU Berlin
Schematic of Inexact
Path-Following
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:

HU Berlin
Bingham-Fluid
in a Pipe
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
HU Berlin
Results of Descent Algorithm
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.

[project website] at DFG
Guests
Project Related Publications
Author Title Journal / Publisher Bibtex
M. Hintermüller,
T. Surowiec
First Order Optimality Conditions for Elliptic Mathematical Programs with Equilibrium Constraints via Variational Analysis To Appear in SIAM J. Opt. 2011
M. Hintermüller,
V. Kovtunenko,
K. Kunisch
Obstacle Problems with Cohesion: A Hemi-Variational Inequality Approach and its Efficient Numerical Solution. SIAM Journal on Optimization 21 (2), 2011, pp. 491-516 [bib]
M. Hintermüller,
A. Laurain
Optimal shape design subject to elliptic variational inequalities SIAM Journal on Control and Optimization, Vol.49 (2011), No.3, pp. 1015-1047 [bib]
M. Hintermüller,
I. Kopacka
A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Computational Optimization and Applications.
DOI: 10.1007/s10589-009-9307-9 (2009)
[bib]
M. Hintermüller,
I. Kopacka
Mathematical Programs with Complementarity Constraints in Function Space: C- and Strong Stationarity and a Path-Following Algorithm SIAM J. Optim. Volume 20,
Issue 2,
pp. 868-902 (2009)
[bib]
M. Hintermüller,
K. Kunisch
PDE-Constrained Optimization Subject to Pointwise Constraints on the Control, the State and its Derivative SIAM J. Optim., Vol. 20, Issue 3, pp. 1133-1156 (2009) [bib]
Preprints
Author Title
M. Hintermüller,
B.S. Mordukhovich,
T. Surowiec
Several Approaches for the Derivation of Stationarity Conditions for Elliptic MPECs with Upper-Level Control Constraints
M. Hintermüller,
C.N. Rautenberg
A Sequential Minimization Technique for Elliptic Quasi-Variational Inequalities with Gradient Constraints
M. Freiberger,
M. Hintermueller,
A. Laurain,
H. Scharfetter
Topological sensitivity analysis in fluorescence optical tomography
C. Conca,
A. Laurain,
R. Mahadevan
Minimization of the ground state for two phase conductors in low contrast regime
M. Hintermüller,
C.Y. Kao,
A. Laurain
Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions
M. Hintermüller,
J.C. de los Reyes
A Duality-Based Semismooth Newton Framework for Solving Variational Inequalities of the Second Kind.