Within the framework of the Mathematical Colloquium on
Thursday, 22.06.2023
16.15 h
Room HS 9
the following lecture by Dr. Johannes Thürauf will take place:

Theory and Algorithms for Uncertain and Nonlinear Bilevel Problems: Recent Results and Future Ideas

Abstract:
Bilevel optimization problems are a class of mathematical programming problems in which some of the variables need to solve another optimization problem. On the one hand, this allows to model hierarchical decision processes. On the other hand, bilevel optimization problems are generally hard to solve. From atheoretical and computational point of view, there has been significant progress in solving bilevel optimization problems. Especially, for the class of mixed-integer linear and convex bilevel optimization problems, recent results from the literature allow to solve bilevel problems of moderate size.

Contrarily, the case of bilevel optimization problem with nonlinear and nonconvex constraints is less researched and the existing solution algorithms are much less mature. This is mainly based on the fact that for nonlinear and nonconvex optimization problems no compact optimality certificates such as the Karush-Kuhn-Tucker conditions are sufficient. Thus, no standard reformulations of, e.g., linear, bilevel optimization can be applied. In addition to nonlinearities, considering uncertainties in bilevel optimization problem is a very young field of research. Uncertainties often affect specific coefficients of constraints or of the objective function and mainly arise due to prediction or measurement errors. Considering such uncertainties within the solution approaches is of great importance since even small perturbations of the coefficients can have a large impact on the solution.  My research focus is on developing theoretical and algorithmic results for bilevel optimization problems that contain nonlinearities or uncertainties. Among others, I consider uncertain or nonlinear bilevel optimization problems in which an underlying graph structure is present. This class of problems contains a variety of wellknown problems such as network interdiction problems and plays an important role in many applications like gas or water networks. The additional structure provided by the underlying network is a key component to obtain theoretical results and algorithms that allow to handle nonlinearities or uncertainties. Along this line of research, I will present recent results for nonlinear network interdiction problems and the effect of small measurement errors in nonlinear bilevel optimization. Moreover, I will outline different directions of future research for uncertain and nonlinear bilevel problems on which we are currently working on.