Research Areas

Nonlinear and Nonsmooth Optimization

Our working group is studying nonlinear, continuous optimization problems. We are concerned with the characterization of extremal points via necessary and sufficient optimality conditions as well as the construction of solution algorithms. Special emphasis is laid on optimization problems which are modeled with the aid of nonsmooth functions, as these are of particular interest in the field of data science. Furthermore, we are interested in disjunctive optimization problems and irregular optimization problems.

Bilevel Optimization

Optimization problems which are constrained by the (parameter-dependent) solution set of yet another optimization problem are referred to as bilevel optimization problems. These are inherently nonconvex, nonsmooth, and irregular optimization problems which play, nevertheless, an important role in the fields of data, engineering,  and natural sciences as well as in economic applications as they can be used to model cooperative and competitive behavior of the decision makes. There is a strong connection between bilevel optimization and mathematical game theory.

Optimal Control and Optimization in Abstract Spaces

Optimization problems that are constrained by ordinary or partial differential equations are referred to as optimal control problems. Our working group is interested in optimal control problems with disjunctive constraints (such as switching constraints on control functions) or sparsity-promoting terms in the objective function or the constraints. Obviously, optimal control problems are optimization problems with decision variables in a function space. More generally, we are concerned with optimization problems in abstract spaces over functions (e.g. image processing) or matrices (low rank matrix recovery).

Nonsmooth Variational Analysis

In order to differentiate nonsmooth mappings, generalized notions of derivatives are essential. Our working group is concerned with the investigation of existing and the formal construction of new variational objects which can be used for that purpose. Particularly, we focus on the derivation of calculus rules and the (algorithmic) applicability of these tools. Special emphasis is laid on generalized derivatives of higher order.