Modern Operational Research and Its Mathematical Methods (2009)

Course coordinator — Prof. Dr. Gerhard Wilhelm Weber, Institute of Applied Mathematics, Middle East Technical University.

Language of the course — English.

Plan of the course

Modern Operational Research and Its Mathematical Methods

1. Introductions into Operational Research and Its Mathematical Problems.
2. Methods from Mathematical Data Mining (Supported by Optimization).
   1. Clustering Theory.
   2. Classification Theory.
   3. Regression Theory.
3. Further Advanced Methods from Mathematical Optimization.
   1. On Foundations of Continuous Optimization.
   2. Nonsmooth Optimization.
   3. Elements of Semi-Infinite Optimization.
4. Applications of Mathematical Operational Research.
   1. Quality Control and Improvement in Manufacturing.
   2. Prediction of Credit Default and of Financial Processes.
   3. Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks.
5. Conclusion.

This lecture series begins with a more informal and motivation part which displays some classical and modern application areas of Operational Research (OR). In fact, in its early years, OR was introduced to resolve "operational" problems of the protection of countries and their freedom and, then, to give answers to decision problems from the worlds of business, economics and, more and more, technology related. In fact, OR has always been following an interdisciplinary approach, and such that the use of quantitative mathematical methods could become helpful for the handling of real-world problems. During the last decade, urgent questions and challenges from environmental protection, health care, the entire sectors of computational, biology and medicine, of social complexity, of development, in particular, of the less privileged ones on earth, and of sustainability, widened and enriched the scope of OR. In this first part of the lecture series, we shall present some "historical" documents, provided by witnesses of the history of OR and of its international scientific community. Since OR is a huge and rapidly growing area, in our lectures, we may spend some extra care on the employment of mathematical methods, in particular, of optimization theory, which are considered to be a modern and exciting scientific "key technology".

Indeed, Operational Research offers techniques of decision aid to scientists and practitioners. Those representatives from the academical and the nonacademical world need to establish their decision-making upon given data; for this reason they have to first of all learn from the data which are usually affected with noise. This statistical and mathematical kind of learning is also called Data Mining. We distinguish between unsupervised learning, where the responses (observations) do not carry any label, such as yes or no, 1 or 0, or any real number, and supervised learning, where such an additional labeling is given. The most famous class of unsupervised problems and methods is Clustering Theory, while Classification Theory is a very important domain in supervised learning. Sometimes and with a special focus, one also speaks about Machine Learning. If the observations are real-valued, we study the well-know Regression Theory. All these three classes within data mining are intimately related and often used very concertedly in order to solve real-world problems. We will pay a special attention to the tradeoff which exists between the goals of accuracy and of stability. So we shall also involve a regularization part into objective functions which are controlling the model complexity. In this part of our lecture, we introduce and discuss how methods from statistics and probability theory, from the theories of mathematical modeling and inverse problems and, in particular, optimization, can serve well for treating the various OR problems. We introduce recently developed techniques and present numerical computations by using Nonsmooth, Infinite and Semi-Infinite Programming, Conic Quadratic Programming as well as of Nonlinear Regression, before in the next part of the lecture series we shall present closer details.

We all have become educated in some optimization theory during our days of school and undergraduate studies. Step by step, we turned to higher dimensional cases of problems and, especially, to problems under constraints. These equality but also inequality constraints jointly define the feasible set. We shall represent how, in analytical and topological terms, these sets look like and behave, and how their interplay with the objective function is. By this very attractive and "higher" viewpoint, we shall repeat and extend our knowledge in optimization theory, where we mainly focus on the case of continuous optimization. An important tool and research area is Nonsmooth Optimization which has various practical motivations but does not allow a direct use of our school and undergraduate knowledge. In fact, whenever some objective or constraint function is not differentiable, we have to use our terms from calculus in a generalized way, i.e., we have to employ the so-called Nonsmooth Calculus. Whenever the extension from basic optimization consists in the permission of an infinite number of inequality contraints, then we speak about Semi-Infinite Optimization. There are many applications for this class of problems, e.g., in robotics and uniform (in time) kind of optimization (optimal control). We are interested again in understanding how such problems behave, especially, under perturbations, for better concepts of solution concepts for them. Moreover, we go one more step ahead by discussing the so-called Generalized Semi-Infinite Optimization, where the set of constraints may even depend on the decision variables. Here, we will point out some phenomena and approaches. However, for a sound understanding of all these problems and methods we need sound and interesting practical motivations; we will provide them in the next part of our lecture.

Those motivations chosen by us from the world of modern OR are, firstly, Quality analysis, control and improvement in manufacturing, where we use both our topological methods and our nonsmooth optimization techniques in order to deeply understand and then overcome problems from the world of Production, but also of the Service and the Financial sectors. After this, we focus on finance issues by using our various methods from data mining to predict credit default and to model financial processes in the form of stochastic differential equations. Here, so-called Tikhonov regularization, conic quadratic programming and nonlinear regression come into play. Finally, we will use mixed integer and semi-infinite optimization to model the new Gene-Environment and Eco-Finance Networks and investigate their dynamical and stability behaviours.

All our various lectures do not aim at an exhaustive completeness: the lecturers do not want to overwhelm the listeners with information. Instead, we look forward to a common learning which is well motivated and should become benefitial and, if possible, practically useful for all participants. For this reason, we try to balance between theory, methods and applications, and shall enjoy a dialogue form of common exchange. We try to bring us all into a position and mood to pose questions, to apply new insights by ourselves and to make proposals. In fact, we wish to invite us all to join a dynamics of collaboration, of an enjoyable scientific culture and to an experience in carrying over responsibility. Here, we all are givers and takers and more and more prepared to help building up our economies, societies and countries, by the offers of modern OR and of its scientific community, the friendship in EURO and IFORS.

Very many literature sources will become involved into our presentations and discussions. We shall provide some soft and hard copies of papers and of our presentations. For further information we recommend to have a look into Curriculum Vitae with Scientific Development of Gerhard-Wilhelm Weber and to send us e-mails, with any kinds of questions and proposals to This e-mail address is being protected from spambots. You need JavaScript enabled to view it and This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Thank you very much for your interest!

Other course tutors:

Mrs. Cand. PhD. Başak Akteke-Öztürk — Institute of Applied Mathematics, METU.