A Julia package for bilevel optimization problems

Abstract

Mathematical optimization is the discipline dealing with the determination of the best (or almost best) decision with respect to a specific cost function and to a set of constraints on the decision. Bilevel optimization is a class of mathematical optimization problems with the optimality conditions of a lower-level problem embedded in the constraints. BilevelOptimization.jl is a toolbox built on top of the JuMP.jl ecosystem for mathematical optimization (Dunning, Huchette, & Lubin, 2017). Bilevel optimization is used to tackle various problems in areas such as power systems, security applications, network design or market equilibria. See Dempe (2018) for an overview of applications and recent formulations and theoretical progress. The computation of an optimal solution to a bilevel problem is in general hard. Even with all the constraints and the objectives at the two levels being linear, the resulting problem is non-convex and NP-hard, with a possibly disjoint feasible set. Optimization practitioners often rely on problem-specific properties and modeling techniques or heuristics. The goal of this package is to offer a both flexible model of a general class of bilevel problems and a solver which is compatible with the JuMP workflow.