Title: Pessimistic Bilevel Optimization
Pessimistic bilevel optimization problems, as optimistic ones, possess a structure involving
three interrelated optimization problems. Moreover, their finite infima are only
attained under strong conditions. We address these difficulties within a framework of moderate
assumptions and a perturbation approach which allow us to approximate such finite
infima arbitrarily well by minimal values of a sequence of solvable single-level problems.
To this end, as already done for optimistic problems, we introduce the standard version of
the pessimistic bilevel problem. For its algorithmic treatment, we reformulate it as a standard
optimistic bilevel program with a two follower Nash game in the lower level. The latter lower level
game, in turn, is replaced by its Karush-Kuhn-Tucker conditions, resulting in a single-level
mathematical program with complementarity constraints.
We show that the perturbed pessimistic bilevel problem, its standard version, the
two follower game as well as the mathematical program with complementarity constraints
are equivalent with respect to their global minimal points. We also highlight the more intricate
connections between their local minimal points. As an illustration, we consider a regulator problem
Oliver Stein is full professor at the Institute of Operations Research (IOR) at the Karlsruhe Institute of Technology (KIT).
He received his doctoral degree from the University of Trier in 1997, and his venia legendi from RWTH Aachen University in 2002.
His research covers algorithms and their theoretical foundation for continuous and mixed-integer nonlinear optimization problems,
parametric optimization, multi-leader-multi-follower games, and multi-objective optimization. Oliver was fellow of the Friedrich-Ebert
Foundation, the Alexander-von-Humboldt Foundation, and the German Research Foundation (Heisenberg followship), and received various
teaching awards. Oliver is member of MOS, SIAM, GAMM, GOR, and DMV. Since 2015 he acts as Editor-in-Chief of MMOR.