Fields Institute Communications Volume 55, 2009 Global Optimisation of Pessimistic Bi-Level Problems Angelos Tsoukalas Department of Computing Imperial College of Science, Technology and Medicine London SW7 2RH, UK at102@doc.ic.ac.uk Wolfram Wiesemann Department of Computing Imperial College of Science, Technology and Medicine London SW7 2RH, UK wwiesema@doc.ic.ac.uk Ber¸ c Rustem Department of Computing Imperial College of Science, Technology and Medicine London SW7 2RH, UK br@doc.ic.ac.uk Abstract. We study the solution of non-convex, pessimistic bi-level problems. After providing several motivating examples, we relate the problem to existing research in optimisation. We analyse key properties of the optimisation problem, such as closedness of the feasible region and computational complexity. We then present and investigate a semi- infinite solution approach that solves ǫ-approximations of the problem. To the best of our knowledge, this represents the first solution technique proposed for this problem class. We close with numerical results and a discussion on fruitful directions for future research. 1 Introduction This paper is concerned with problems of the following type: min x∈X f (x) (1.1a) subject to g(x,y) ≤ 0 ∀ y ∈ arg max y ′ ∈Y h(x,y ′ ), (1.1b) where X ⊆ R n , Y ⊆ R m , f : X → R and g,h : X × Y → R. We assume that X and Y are non-empty and compact, while f , g and h are continuous in their arguments. In case multiple constraints g 1 ,...,g p : X × Y → R should be considered, they 2000 Mathematics Subject Classification. Primary ; Secondary. c 2009 American Mathematical Society 1