Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse

With stochastic integer programming as the motivating application, we investigate techniques to use integrality constraints to obtain improved cuts within a Benders decomposition algorithm. We compare the effect of using cuts in two ways: (i) cut-and-project, where integrality constraints are used to derive cuts in the extended variable space, and Benders cuts are then used to project the resulting improved relaxation, and (ii) project-and-cut, where integrality constraints are used to derive cuts directly in the Benders reformulation. For the case of split cuts, we demonstrate that although these approaches yield equivalent relaxations when considering a single split disjunction, cut-and-project yields stronger relaxations in general when using multiple split disjunctions. Computational results illustrate that the difference can be very large, and demonstrate that using split cuts within the cut-and-project framework can significantly improve the performance of Benders decomposition.