Geometry of the Reformulation-Linearization-Technique: Domination of Disjunctions

TL;DR

This paper explores the geometric structure of the reformulation-linearization-technique (RLT) in binary mixed-integer optimization, revealing its dominance over certain disjunctive programming approaches and providing insights into its strength and applications.

Contribution

It characterizes the geometry of RLT closure and demonstrates its dominance over disjunctive programming methods based on cardinality equations.

Findings

RLT dominates disjunctive programming based on cardinality equations with RHS 1.

The geometry of RLT closure is characterized geometrically.

Results apply to strength comparison in quadratic assignment problems.

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.