Disjunctive Benders Decomposition

TL;DR

This paper introduces a novel enhancement to Benders decomposition that leverages disjunctive programming to generate convex hull inequalities, significantly improving computational efficiency on large-scale mixed-binary linear programs.

Contribution

It integrates disjunctive programming with Benders decomposition, enabling the construction of convex hull inequalities without solving the master problem as a mixed-integer program.

Findings

Substantial reductions in branch-and-bound nodes, often by orders of magnitude.

Consistent outperformance of commercial solvers on selected large-scale instances.

Effective integration of existing cut-generating oracles for convex hull inequalities.

Abstract

We propose an enhancement to Benders decomposition (BD) that generates valid inequalities for the convex hull of the Benders reformulation, addressing the limitation that classical BD cuts are typically tight only for the continuous relaxation. Our method integrates disjunctive programming with BD and introduces a routine that leverages existing cut-generating oracles as-is to construct convex hull inequalities. For mixed-binary linear programs, the approach removes the need to solve the master problem as a mixed-integer program, even with separable subproblems. It builds on a unified normalization framework for cut-generating programs, encompassing norm-based, reverse polar, and right-hand-side normalization, and enabling the design of new normalization schemes with streamlined analysis of supporting cuts. Computational results on large-scale instances show substantial reductions in branch-and-bound nodes often by orders of magnitude, while consistently outperforming commercial solvers on selected problem classes.