Why there is no need to use a big-M in linear bilevel optimization: a computational study of two ready-to-use approaches

Abstract Linear bilevel optimization problems have gained increasing attention both in theory as well practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to ability this class model hierarchical decision processes. However, makes also very hard solve. Since no general-purpose solvers are available, a “best-practice” has developed applied OR community, which not all people want develop tailored algorithms but “just use” modeling tool for practice. This best-practice big- M reformulation Karush–Kuhn–Tucker (KKT) conditions lower-level problem—an approach that been shown be highly problematic by Pineda Morales (2019). Choosing invalid values yields solutions may arbitrarily bad. Checking validity s however solving original problem Kleinert et al. Nevertheless, its appealing simplicity, especially w.r.t. required implementation effort, ready-to-use still most popular method. Until now, there lack approaches competitive terms effort computational cost. In note we demonstrate indeed another approach: If SOS-1 technique KKT complementarity conditions, adding simple additional root-node inequality (2020) leads performance—without having possible theoretical disadvantages approach.