Consistency Cuts for Dantzig-Wolfe Reformulations

TL;DR

This paper presents a new class of validity-enhancing inequalities called consistency cuts for Dantzig-Wolfe reformulations, significantly improving solution efficiency and optimality in complex integer programming problems.

Contribution

Introduction of consistency cuts that guarantee integer solutions in Dantzig-Wolfe relaxations under certain conditions, with demonstrated computational advantages.

Findings

Consistency cuts enable faster solution of test instances.

They solve previously unsolved problems in the test set.

Potential applicability to a broader range of mixed-integer problems.

Abstract

This paper introduces a family of valid inequalities, that we term consistency cuts, to be applied to a Dantzig-Wolfe reformulation (or decomposition) with linking variables. We prove that these cuts ensure an integer solution to the corresponding Dantzig-Wolfe relaxation when certain criteria to the structure of the decomposition are met. We implement the cuts and use them to solve a commonly used test set of 200 instances of the temporal knapsack problem. We assess the performance with and without the cuts and compare further to CPLEX and other solution methods that have historically been used to solve the test set. By separating consistency cuts we show that we can obtain optimal integer solutions much faster than the other methods and even solve the remaining unsolved problems in the test set. We also perform a second test on instances from the MIPLIB 2017 online library of mixed-integer programs, showing the potential of the cuts on a wider range of problems.