Andreas Horländer

• Bilevel problems are optimization problems for which parts of the variables are constrained to be an optimal solution to another nested optimization problem. This structure renders bilevel problems particularly well-suited for modeling hierarchical decision-making processes. They are widely applicable in areas such as energy markets, transportation systems, security planning, and pricing. However, the hierarchical nature of these problems also makes them inherently challenging to solve, both in theory and in practice. In this thesis, we study different nonlinear problem settings for the nested optimization problem. First, we focus on nonlinear but convex bilevel problems with purely integer variables. We propose a solution algorithm that uses a branch-and-cut framework with tailored cutting planes. We prove correctness and finite termination of the method under suitable assumptions and put it into context of existing literature. Moreover, we provide an extensive numerical study to showcase the applicability of our method and we compare it to the state-of-the-art approach for a less general setting on suitable instances from the literature. Furthermore, we discuss challenges that arise when we try to generalize our approach to the mixed-integer setting. Next, we study mixed-integer bilevel problems for which the nested problem has a nonconvex and quadratic objective function, linear constraints, and continuous variables. We state and prove a complexity-theoretical hardness result for this problem class and develop a lower and upper bounding scheme to solve these problems. We prove correctness and finite termination of the proposed method under suitable assumptions and test its applicability in a numerical study. Finally, we consider bilevel problems with continuous variables, where the nested problem has a convex-quadratic objective function and linear constraints. We reformulate them as single-level optimization problems using necessary and sufficient optimality conditions for the nested problem. Then, we explore the family of so-called P-split reformulations for this single-level problem and test their applicability in a preliminary numerical study.