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Bilevel programming has multiple applications in areas such as traffic systems, natural gas market regulation, waste management, and chemical engineering. However, mixed-integer bilevel programs are notoriously difficult to solve because the leader's feasible region is defined in part by optimality conditions governing an embedded follower subproblem. In this setting, optimality conditions are difficult to characterize because of the nonconvexity of the subproblem. We propose an exact finite algorithm for a class of these problems based on an adaptive sampling scheme. We demonstrate how this algorithm can be tailored to accommodate either optimistic or pessimistic assumptions on the follower's behavior. Computational experiments show that the proposed approach outperforms an existing state-of-the-art algorithm, and that the approach is capable of solving instances of a competitive scheduling problem involving nonlinear objectives.

The elementary shortest path problem with resource constraints (ESPPRC) is an NP-hard problem that often arises in the context of column generation for vehicle routing problems. We propose an exact solution method that relies on implicit enumeration with a novel bounding scheme that dramatically narrows the search space. We embedded our algorithm within a column generation to solve the linear relaxation (root node) of the vehicle routing problem with time windows (VRPTW) and found that the proposed algorithm performs well when compared against state-of-the-art algorithms for the ESPPRC on the well-known Solomon’s test bed for the VRPTW.