# Robust Optimization

Robust Optimization has its origins in modern Decision Theory, which stems from the 1950s. In the 1970s, it became an area of research in its own right within a wide range of academic fields, e.g. in Operations Research, Control Theory and Economics. Nowadays, Robust Optimization is regarded as a branch of Mathematical Optimization.

Classical Mathematical Optimization mostly solves problem formulations under the assumption that all input data is precisely measureable und no fluctuations are possible. In contrast, Robust Optimization takes uncertainties and inconsistencies into account in the optimization process, searching for externality-robust solutions.

For instance, let’s consider the planning problem involved in the construction of the new airport in Berlin. The project was divided into numerous work packages, which were to be processed consecutively. Each work package was allocated a processing duration, on the basis of which the total duration of the project and the relevant costs were calculated. The fact that the original completion date now lies far in the past and the project is nowhere near finished even today, illustrates that the assumption of precise measurement data is frequently unrealistic.

The mathematical methods developed by the Chair of Management Science at RWTH Aachen University and by other universities enable numerous variations of optimization under uncertainty. We distinguish between two fundamentally different approaches to robustness, both of which assume that externalities can be modeled by a (potentially large) number of scenarios;

• For worst case optimization, we are interested in finding solutions to optimization problems which are still feasible even in the worst case scenario. Among these, we look for one with the best objective function value (e.g. highest revenue). This type of optimization process implies a very pessimistic perspective which often forecasts the costs or the expected duration of a project in a very negative way. However, it does always deliver a feasible solution.
• We can modifiy the above approach to compute more optimistic solutions. Instead of demanding that a solution should be feasible for all scenarios, we now consider solutions which have a high probability, e.g. 99 % of being feasible. In this case, we assume that every possible scenario occurs with some probability. Whereas this type of optimization process might yield a solution which is infeasible for some scenarios, the costs and the duration times can often be estimated much more precisely.

Both of these models of robustness find application in real life and pose different challenges for us. In many cases, classical optimization methods are not sufficient and new methods of Combinational Optimization and Mathematical Programming are needed. The Chair of Management Science in the School of Business and Economics at RWTH Aachen University plays an active role in developing such innovative methods and finding solutions to even the most difficult problems.