TY - GEN A1 - Beck, Yasmine A1 - Schmidt, Martin T1 - A Gentle and Incomplete Introduction to Bilevel Optimization N2 - These are lecture notes on bilevel optimization. The class of bilevel optimization problems is formally introduced and motivated using examples from different fields. Afterward, the main focus is on how to solve linear and mixed-integer linear bilevel optimization problems. To this end, we first consider various single-level reformulations of bilevel optimization problems with linear or convex follower problems, discuss geometric properties of linear bilevel problems, and study different algorithms for solving linear bilevel problems. Finally, we consider mixed-integer linear bilevel problems, discuss the main obstacles for deriving exact as well as effective solution methods, and derive a branch-and-bound method for solving these problems. KW - Bilevel Optimization KW - Lecture Notes Y1 - 2021 ER - TY - JOUR A1 - Beck, Yasmine A1 - Schmidt, Martin T1 - A Robust Approach for Modeling Limited Observability in Bilevel Optimization JF - Operations Research Letters N2 - Many applications of bilevel optimization contain a leader facing a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. We consider bilinear bilevel problems with follower's response uncertainty due to limited observability regarding the leader's decision and exploit robust optimization to model the decision making of the follower. We show that the robust counterpart of the lower level allows to tackle the problem via the lower level's KKT conditions. KW - Bilevel optimization KW - Robust optimization KW - Bounded rationality KW - Limited observability KW - Reformulations Y1 - 2021 IS - 49(5) SP - 752 EP - 758 ER - TY - INPR A1 - Beck, Yasmine A1 - Ljubic, Ivana A1 - Schmidt, Martin T1 - Exact Methods for Discrete Γ-Robust Interdiction Problems with an Application to the Bilevel Knapsack Problem N2 - Developing solution methods for discrete bilevel problems is known to be a challenging task - even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a Γ-robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone Γ-robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the Γ-robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches. KW - Bilevel optimization KW - Robust optimization KW - Knapsack interdiction KW - Mixed-integer programming KW - Branch-and-Cut Y1 - 2021 ER -