Associate Professor Dirk Oliver Theis, Institute of Computer Science UT
Professor Nicolas Gillis, University of Mons, Belgium
PhD Matthias Walter, Aachen University, Germany
Mathematical Optimization is finding the optimal solution of an optimization problem, i.e., the maximum or minimum value of a real function (objective function) subjected to a set of constraints. Mathematical optimization is divided into classes depending on the nature of the objective function as well as the constraints and these classes vary in their computational complexity. In practice, robust and fast processes are needed to tackle the problems, and so optimization problems which can be modeled into classes of polynomial complexity are applied directly. Several algorithms that solve different optimization problems exist, in particular, efficient ones for the classes with polynomial complexity. But, in some cases, even the efficient algorithms face difficulties in solving optimization problems such as taking immensely long time to find the optimal solution. In this case, studying the optimization, in depth, leads to the reasons behind these pitfalls and thus modify the algorithms to avoid them. This thesis studies the solution of a Convex Program which was used to obtain partial information decomposition, a tool used recently to analyze particular complex systems. Many difficulties arises in solving the Convex Program and so the study done in this thesis resulted in a fast and robust algorithm to tackle the problem. Then it studies the situation, which appeared recently in neuroscience, when the obtained partial information decomposition is optimized subject to some constraints. Finally, it studies a fundamental optimization problem in the control of automated systems that is APX-hard. The problem is modeled IP- and CNF- models which can be used, in future, to design good heuristics which tackles the problem in reasonable time.