Disjunctive Benders Decomposition

Abstract: We propose a novel enhancement to Benders Decomposition (BD) that generates valid inequalities for the convex hull of the Benders reformulation, addressing a key limitation of conventional BD-its cuts are typically tight only for the continuous relaxation. Our method efficiently integrates disjunctive programming theory with BD, introducing a new routine that leverages existing cut-generating oracles for uncovering constraints required to construct valid inequalities for the convex hull. For mixed-binary linear programs, this approach eliminates the need to solve the master problem as a mixed-integer program. Additionally, we extend the a posteriori strengthening and lifting procedure for lift-and-project cuts into the BD framework, and present an approximate routine for generating lift-and-project cuts. Numerical results on large-scale instances show that our approach significantly reduces the number of branch-and-bound nodes required to reach the lower bound achieved by conventional BD, often by orders of magnitude.