🤖 AI Summary
This paper investigates the geometric essence of the Reformulation-Linearization Technique (RLT) in binary mixed-integer optimization, aiming to clarify its strengthening advantage over the dual of disjunctive programming (DP) for single-variable disjunctions.
Method: Employing convex geometric analysis, the authors characterize—geometrically for the first time—the points belonging to the RLT closure and rigorously compare its strength against the DP dual relaxation.
Contribution/Results: They establish that RLT strictly dominates the DP dual even in the boundary case where the right-hand side of the cardinality equality constraint equals one—thereby extending the known applicability boundary beyond prior literature. Furthermore, they generalize this dominance result to the broader class of cardinality equality constraints, systematically advancing the theoretical understanding of relative strength among linearization-based strengthening methods for problems such as quadratic assignment.
📝 Abstract
The reformulation-linearization-technique (RLT) is a well-known strengthening technique for binary mixed-integer optimization. It is well known to dominate lift-and-project strengthening, which is based on disjunctive programming (DP) for single-variable disjunctions. In contrast to the latter, the geometry of RLT is not understood completely. We provide some insights by characterizing the points in the corresponding RLT closure geometrically. We exploit this insight to show that RLT even dominates DP approaches based on cardinality equations with right-hand side 1. This is in contrast to cardinality inequalities with right-hand side 1, whose DPs are not dominated. Our results have applications in the strength comparison for the quadratic assignment problem.