Research Interests, Application Areas, Education

Main Research Areas

Application Areas

Current Research and Application Challenges

Education

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Main Research Areas:

My main research areas are optimization, discrete mathematics, and operations research. I am particularly interested in integer programming and the geometric approach to combinatorial optimization problems (polyhedral combinatorics, convex geometry, cutting plane algorithms, branch&cut methods). On the combinatorial side, graph and matroid theory come into play here. Specific combinatorial optimization problems I have worked on are the travelling salesman problem, the max-cut problem, the stable set problem and perfect graphs, the linear ordering problem, clustering, routing and scheduling, online optimization, and various connectivity and network flow problems.

Application Areas:

I have a deep interest in (real) applications. I have worked with scientists from other disciplines and, in particular, with engineers and economists from industry to mathematically model challenging problems of their domain of expertise. The theoretical analysis of such models and the development of efficient algorithms for their numerical solution are driving forces of my research. I have contributed to the following application areas:

• chip design and very large scale integration
• printed circuit board design and production, control of CNC machines
• simulation and optimization of computer manufacturing
• flexible manufacturing, job scheduling
• production planning in the chemical industry
• logistics, planning of public transportation systems
• vehicle routing, duty scheduling
• online dispatching of vehicles, online elevator control
• control of (semi-)automatically guided vehicles
• energy
• location planning in the telecommunication industry
• design and dimensioning of telecommunication networks
• design of optical networks
• frequency assignment in cellular phone networks
• spin glasses
• input-output analysis

Current Research and Application Challenges:

Many of the technical and operational problems of the type mentioned above can nowadays be solved to the satisfaction of practitioners. My research team at ZIB contributes to the general "tool development" in linear, integer, and mixed-integer programming by constantly improving our ZIB Optimization Suite that contains ZIMPL, a modeling language, SoPLex, an LP code, and SCIP a solver for mixed-integer programs. In general, however, the "real challenges from practice" cannot be solved with "automatic tools" yet; mathematical effort and expertise is still needed.

My research team keeps contributing to these issues in a wide range of application areas and by developing new solution methods, but we think that the time seems ripe to address the following "big issues" with higher intensity:

1.

Online and real time optimization problems, particularly those arising in practice, need to be investigated more thoroughly. Algorithmic paradigms, mathematical tools and concepts of performance analysis have to be developed that satisfy both theory and practice. Oversimplified, the big questions are:   - When is an online algorithm "good"? - What are typical ingredients of "good" online algorithms?

2.

Mathematics usually adresses quantitatively specifiable questions; it has hardly contributed to more general (often more important) issues such as the design of "good" systems. Top management is, in general, not too interested in the optimization of some machine or the like. It wants to know whether a company is positioned well to meet the challenges of the future. Here are typical questions of this kind: - Is the public transportation system of a city good? - Can the railroad system of a country be improved by introducing competition? - How does one regulate or deregulate energy supply systems? - Is the supply chain of a company of good quality? - What is a good telecommunication network? It is clear that these are problems of high economic impact. A hard part here is their "mathematical modelling". For a mathematician, the questions above are very impricisely stated. Can we mathematicians (in close cooperation with specialists) come up with methods to formalize and quantify these questions more precisely and develop methods to answer them (somehow)?

Education:

I like teaching and try to bring current research into the university class room. For details click here.

I also participate in education programs for PhD students. I was a member of the European Graduate Program "Combinatorics, Geometry and Computation" that was a joint endeavor by the Freie Universität, Humboldt-Universität and Technische Umiversität as well as the Konrad-Zuse-Zentrum für Informationstechnik Berlin together with the Eidgenössische Technische Hochschule Zürich. This European Graduate Program "CGC" was sponsored by the Deutsche Forschungsgemeinschaft/German Research Foundation (DFG) until the end of 2005.

Now, I am involved in a new Graduate Program "Methods for Discrete Structures" supported by the German Research Foundation (DFG). The newly established Research Training Group offers scholarships for Ph.D. students and one post-doc. The program started on October 1, 2006.

I participated in a DFG funded Graduate Program "MAGSI"with the title "Stochastic Modelling and Quantitative Analysis of Complex Systems in Engineering". This was a Graduate Program directed to PhD students, in particular in Engineering and Computer Science to which some mathematicians from TU and HU Berlin contributed their modelling experience. This program ended in 2006.

I helped to organize a Ph.D. program in Applied Mathematics in Ecuador as a joint project between TU Berlin and Escuela Politécnica Nacional (EPN) in Quito (financed by DAAD from 2002-2006).

Furthermore, I am involved in the activities of the Berlin Mathematical School (joint graduate program of three Berlin universities HU, FU, TU and several mathematical institutes).

But I also think that high school education (at least in Germany) needs some overhaul. I therefore started a project "Diskrete Mathematik für die Schule"  (sponsored by VolkswagenStiftung) to bring discrete mathematics and optimization into high school. I do believe that quite a number of topics in this area can be taught to high school students. These new topics broaden their knowledge range (the current focus in Germany is on geometry and analysis) and show that interesting issues of the (modern) daily life of an ordinary person (such as car navigation, mobile phone operation and planning, cryptography, garbage truck or postman routing, etc.) can be tackled with mathematics. This may result in a more friendly perception of mathematics in school and in the public domain.

I organized several Summer Schools (in Germany, in China) to teach modern methods of discrete optimization, linear and integer programming.

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