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Numerical Mathematics and Optimization

Projects

Overview of various projects of the last years in our research area. Please click on the respective link to get more information.

Project management: Prof. Dr. Christian Kanzow

Assistant: Dr. Yekini Shehu

Project period: 2016-2019

Funding institution: Alexander von Humboldt Foundation

Funding code: 1163904 - NGA - HFST - P

Project description: The post-doctoral research proposal focuses on the following two parts:
(a) Fixed-point algorithms for optimization, quasi-variational inequalities and (generalized) Nash Equilibrium Problems
(b) Numerical analysis and implementation of fixed-point algorithms considered in (a)

Project management: Prof. Dr. Christian Kanzow

Project partner: Prof. Dr. Daniel Wachsmuth, Julius-Maximilians-Universität Würzburg

Assistant: Daniel Steck

Project period: 2016-2019

Funding institution: DFG

Authorization sum: 189.700,00 €

Funding code: KA 1296/24-1

Project description: This project aims at the development and analysis of numerical algorithms for certain classes of quasi-variational inequalities. Such problems arise, for instance, from generalized Nash equilibria in multi-player optimal control problems, or from value functions in stochastic control problems.
The project pursues two principal approaches:
(a) Transfer of solution methods from finite to infinite dimensions;
(b) Development of specialized approaches which exploit the specific structure of certain quasi-variational problems.
All methods are analyzed from a theoretical perspective and tested extensively on suitable application examples.

Project management: Prof. Dr. Christian Kanzow

Project partner: Prof. Dr. Oliver Stein, Karlsruher Institut für Technologie (KIT)

Assistant: Nadja Harms

Project period: 2011-2013

Funding institution: DFG

Authorization sum: 171.700,00 €

Funding code: KA 1296/18-1

Project description: Generalized Nash equilibrium problems (GNEPs) are an extension of classical Nash equilibrium problems (NEPs) whereby the strategy sets of each player are allowed to depend on the rivals' decision variables. Such problems occur in several application contexts such as traffic control or telecommunications. The practical solution of GNEPs is a highly challenging task, and several methods have been developed in recent years, each using different approaches and exhibiting certain advantages and disadvantages. This project proposes a novel approach which seeks to overcome some of the existing drawbacks by formulating the GNEP as a mathematical program with equilibrium constraints (MPEC). This is a class of optimization problems which has enjoyed substantial progress in recent years, both theoretically and numerically. Applying the MPEC framework to the special case of GNEPs gives rise to several important issues and questions, and the treatment of these is one of the main subjects of the project.

Project management: Prof. Dr. Christian Kanzow

Project partner: Prof. Dr. Oliver Stein, Karlsruher Institut für Technologie (KIT)

Assistant: Axel Dreves

Project period: 2009 - 2012

Funding institution: DFG

Authorization sum: 171.700,00 €

Funding code: KA 1296/17-1

Project description: This project was concerned with non-cooperative games where each player aims to maximize their respective profit unilaterally, that is, by choosing their strategy individually and not cooperating with rivals. In addition, each player needs to satisfy certain constraints which may depend on the rivals' actions. Such problems are usually called *generalized Nash equilibrium problems*, and their solutions are characterized as situations where no player can improve his payoff by unilaterally changing their strategy. Various techniques were constructed which reformulate these problems as standard optimization problems. It was shown that the resulting problems can be tackled by suitably modified standard methods which then ultimately compute a solution of the underlying Nash problem. This resulted in multiple algorithms with different advantages, some finding either as many solutions as possible or specific solutions with certain properties. In addition, some of the methods achieve locally rapid convergence, whereas others achieve significant robustness.

Project management: Prof. Dr. Christian Kanzow

Project partner: Prof. Dr. Peter Knabner, Friedrich-Alexander-Universität Erlangen-Nürnberg

Assistant: Hannes Buchholzer

Project period: 2010-2011

Funding institution: DFG

Funding code: KA 1296/16-1 / KA 1296/16-2

Project description: The project focuses on the accurate and efficient numerical treatment of time-dependent reactive transport problems with many species (in porous media) in 2 or 3 space dimensions with local complementarity conditions as essential ingredient.

Project management: Prof. Dr. Christian Kanzow

Project partner: Prof. Dr. Wolfgang Achtziger, TU Dortmund

Assistant: Tim Hoheisel

Project period: 2007 - 2010

Funding institution: DFG

Funding code: KA 1296/15-1