The topic of Discrete Optimization is to optimize functions over discrete sets of feasible solutions. Often the feasible solutions are certain structures in combinatorial objects such as graphs or subsets of the integer-lattice in Euclidean space specified by certain constraints. The focus of our research is on the interplay between those two types of problems, called Polyhedral Combinatorics, with strong ties to Discrete Geometry and in particular to Polyhedral Theory. Parts of our research are embedded into the DFG Research Training Group 2297 Mathematical Complexity Reduction. For more details, please follow the links on top.