DEPENDENCE OF BILEVEL PROGRAMMING ON IRRELEVANT DATA STEPHAN DEMPE, SEBASTIAN LOHSE Abstract. In 1997, Macal and Hurter [7] have found that adding a constraint to the lower level problem, which is not active at the computed global optimal solution, can destroy global optimality. In this paper this property is recon- sidered and it is shown that this solution remains locally optimal under inner semicontinuity of the original solution set mapping. In the second part of the paper we prove that adding a variable in the linear lower level problem can also destroy global optimality. But here the solution remains locally optimal, provided the optimal solution in the lower level was dual non-degenerated. 1. Introduction Bilevel programming problems are hierarchical optimization problems where the feasible set of the so-called upper level or leader’s problem is restricted in part by the graph of the solution set mapping of a second optimization problem. This latter problem is the follower’s or lower level problem. To formulate the bilevel programming problem formally, consider the parametric follower’s problem first: (1.1) min y {f (x, y): g(x, y) ≤ 0}, where f : R n × R m → R,g : R n × R m → R p . Let Ψ(x) := Argmin y {f (x, y): g(x, y) ≤ 0} be its solution set mapping. Then, the leader’s problem is given as (1.2) min x,y {F (x, y): x ∈ X, (x, y) ∈ gph Ψ}, where gph Ψ := {(x, y): y ∈ Ψ(x)} denotes the graph of the solution set mapping of the problem (1.1), X ⊆ R n is a closed set and F : R n × R m → R. This problem has been investigated in the monographs [1, 4]. It has many applications, [5] is an annotated bibliography on bilevel programming problems, [3] gives an overview of the problem. Problem (1.2) is a NP -hard problem [6], which makes its solution difficult, es- pecially for large problems. Hence, reduction of the dimension of the problem is desirable. Two such reductions are used: either only some part of the constraints in the lower level problem are used and more constraints are added if necessary. This is done e.g. in cutting plane methods for solving discrete linear optimization problems. Or, variables can be dropped and added only if necessary. This is e.g. the column generation approach for solving large linear optimization problems. We will investigate the implications of such approaches to bilevel programming problems where the lower level problem is reduced. 1