Topics for projects - Elisabeth Köbis

My research lies in the field of set optimization, i.e., the optimization of set-valued mappings. If the mapping is single-valued, then this field recovers the concept of vector/multiobjective optimization. In a finite-dimensional setting, one speaks of multiobjective optimization. Here, several objective functions, which are usually conflicting, are minimized in parallel. Almost any real-world application in mathematical optimization has multiple conflictive criteria; see, for example, the problem of choosing a portfolio in financial mathematics, where the risk is to be minimized while the returns should be maximized simultaneously. Multiobjective optimization is an interesting area ideal to directly build upon the knowledge gained in the Optimization courses at NTNU. In a more abstract setting, when the space is infinite dimensional, one speaks of vector optimization. Here, one is concerned with optimization in functional spaces. This area is interesting for students who have already gained knowledge in functional analysis and would like to deepen their studies on a more theoretical level.

In relation with vector optimization problems, one is often concerned with vector variational inequalities, which are regarded as a powerful tool to study vector optimization problems. For example, one can show the equivalence between optimal solutions of vector optimization problems with differentiable convex objective function and solutions of vector variational inequalities (of a so-called Minty type). This area offers various problems that are interesting to study for students with an interest in applied functional analysis.

An interesting topic associated with set optimization is programming under uncertainty, where one follows a so-called robust approach, which is set-based and non-probabilistic. Here, one assumes that all possible objective function values for each feasible solution are comprised in a set, and one looks for the minimal solution.

For students looking for a project or Master theme, I could offer the following topics:

• Set optimization (solution concepts; optimality conditions; methods for computing solutions)
• Multi-objective optimization (theory and applications; optimality conditions; real-world problems; algorithmic procedures: for example, Jahn-Graef-Younes methods – implementation and comparisons to other sorting procedures and extension to different solution concepts)
• Vector optimization (again with a focus on theory and applications)
• (Vector-) Variational Inequalities (concepts; existence results; topological and algebraic properties of the solution set; stability and sensitivity analysis; scalarization methods, i.e., the reduction to a problem that is easier to solve)
• Programming under uncertainty; robustness

I would be happy to guide students in their projects and theses. If you are interested in any of the topics above, please send me an email.