TY - INPR A1 - Goerigk, Marc A1 - Kurtz, Jannis A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Connections between Robust and Bilevel Optimization N2 - Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (static robust problems with and without decision-dependence of their uncertainty sets, worst-case regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields - in particular to use both theory and algorithms from one field in the other and vice versa. KW - Bilevel optimization KW - Robust optimization KW - Reformulations Y1 - 2023 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 - 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 - JOUR A1 - Çelebi, Emre A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Γ-Robust Electricity Market Equilibrium Models with Transmission and Generation Investments JF - Energy Systems N2 - We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Γ-robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Γ-robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse. KW - Robust optimization KW - Robust market equilibria KW - Electricity market equilibrium models KW - Transmission and generation investment KW - Perfect competition Y1 - 2020 ER -